% This plain-TeX file is my paper on exotic holonomy.
%Topmatter
\magnification=\magstep1
% Here are the fonts we need
\font\eu=eufm10 \font\bigbf=cmbx12
\font\sc=cmcsc10
\font\tenjap=msbm10\font\sevenjap=msbm7
\font\greekbold=eurb10
\font\chin=msam10
\font\teneur=eurm10\font\seveneur=eurm7
% Here are some common abbreviations
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\def\Gr{{\rm Gr}}\def\GL{{\rm GL}}\def\SL{{\rm SL}}
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\def\euso{\hbox{\eu so}}\def\euco{\hbox{\eu co}}
\def\eup{\hbox{\eu p}}
% Here are the calligraphic, bold and barred letters.
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\def\C{{\cal C}}\def\Q{{\cal Q}}\def\cE{{\cal E}}
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\def\J{{\bf J}} \def\L{{\bf L}} \def\R{{\bf R}}
\def\ba{\bar a} \def\bb{\bar b} \def\br{\bar r} \def\bJ{\bar J}
\def\bl{\bar\lambda} \def\bo{\bar\omega} \def\bph{\bar\phi}
\def\bPh{\bar\Phi}\def\bTh{\bar\Theta}
\def\bPs{\bar\Psi}\def\bOm{\bar\Omega}
\def\er{{\mathchoice{\hbox{\teneur r}}{\hbox{\teneur r}}
{\hbox{\seveneur r}}{\hbox{\seveneur r}}}}
\def\el{{\mathchoice{\hbox{\teneur l}}{\hbox{\teneur l}}
{\hbox{\seveneur l}}{\hbox{\seveneur l}}}}
% Here are the bold greek letters we need.
\def\Th{\hbox{\greekbold\char2}}%bold Greek Theta
\def\Lam{\hbox{\greekbold\char3}}%bold Greek Lambda
\def\Ups{\hbox{\greekbold\char7}}%bold Greek Upsilon
\def\Ph{\hbox{\greekbold\char8}}%bold Greek Phi
\def\Ps{\hbox{\greekbold\char9}}%bold Greek Psi
\def\Om{\hbox{\greekbold\char10}}%bold Greek Omega
\def\be{\hbox{\greekbold\char12}}%bold Greek beta
\def\lam{\hbox{\greekbold\char21}}%bold Greek lambda
\def\bmu{\hbox{\greekbold\char22}}%bold Greek mu
\def\bfpi{\hbox{\greekbold\char25}}%bold Greek pi
\def\rh{\hbox{\greekbold\char26}}%bold Greek rho
\def\ph{\hbox{\greekbold\char30}}%bold Greek phi
\def\om{\hbox{\greekbold\char33}}%bold Greek omega
% End of Topmatter
% Here is the title information
\centerline{\bigbf Two Exotic Holonomies in Dimension Four,}
\centerline{\bigbf Path Geometries, and Twistor Theory}
\vskip 20 pt
\centerline{by}
\centerline{
\sc Robert L. Bryant\footnote*{\rm partially supported by NSF
grants
DMS-8352009 and DMS-8905207}
}
\vskip 35 pt
\centerline{\bf Table of Contents}
\vskip 10pt
\itemitem{\bf \S0.}{Introduction}
\itemitem{\bf \S1.}{The Holonomy of a Torsion-Free
Connection}
\itemitem{\bf \S2.}{The Structure Equations of $H_3$- and
$G_3$-structures}
\itemitem{\bf \S3.}{The Existence Theorems}
\itemitem{\bf \S4.}{Path Geometries and Exotic Holonomy}
\itemitem{\bf \S5.}{Twistor Theory and Exotic Holonomy}
\itemitem{\bf \S6.}{Epilogue}
\vskip 35 pt
\centerline{\bf \S0. Introduction}
\bigskip
Since its introduction by \'Elie Cartan, the {\it holonomy} of a
connection has played an important role in differential
geometry. One of the best known results concerning holonomy
is Berger's classification of the possible holonomies of
Levi-Civita connections of Riemannian metrics. Since the
appearance of Berger [{\bf 1955}], much work has been done to
refine his list of possible Riemannian holonomies. See the
recent works by Besse [{\bf 1987}] or Salamon [{\bf 1989}] for
useful historical surveys and applications of holonomy to
Riemannian and algebraic geometry.
\smallskip
Less well known is that, at the same time, Berger also
classified the possible pseudo-Riemannian holonomies which
act irreducibly on each tangent space. The intervening years
have not completely resolved the question of whether all of
the subgroups of the general linear group on his
pseudo-Riemannian list actually occur as holonomy groups of
pseudo-Riemannian metrics, but, as of this writing, only one
class of examples on his list remains in doubt, namely ${\rm
SO}^\ast(2p)\subset \GL(4p)$ for $p\ge3$.
\smallskip
Perhaps the least-known aspect of Berger's work on holonomy
concerns his list of the possible irreducibly-acting holonomy
groups of torsion-free affine connections. (Torsion-free
connections are the natural generalization of Levi-Civita
connections in the metric case.) Part of the reason for this
relative obscurity is that affine geometry is not as well known
or widely used as Riemannian geometry and global results are
generally lacking. However, another reason is perhaps that
Berger's list is incomplete in the affine case; it omits a finite
number of possibilities. In fact, just how many are missing
from this list is not known. We shall refer to the missing
entries as the list of {\it exotic holonomies}.
\bigskip
In this paper, we produce two examples which show that, in
fact, the list of exotic holonomies is non-empty. Moreover, we
show that the geometry of connections with these holonomies
is related to classical work on the geometry of {\sc ode} in
terms of ``path geometries'' and to the modern theory of
twistors as initiated by Penrose and his co-workers.
\medskip
In \S1, we give a brief discussion of the two criteria which
Berger used to compile his lists. We then apply these criteria
to a simple collection of subgroups: For each $d\ge1$ we can
regard $\SL(2,\bbR)$ as a subgroup $H_d\subset
\GL(d+1,\bbR)$ via the (unique) $(d+1)$-dimensional
irreducible representation of $\SL(2,\bbR)$. Moreover, if we
let $G_d\subset \GL(d+1,\bbR)$ denote the centralizer of
$H_d$, then $G_d$ may be regarded as a representation of
$\GL(2,\bbR)$. We show that $H_3$ and $G_3$ satisfy Berger's
criteria even though they do not preserve any non-trivial
symmetric quadratic form (even up to a conformal factor) and
do not appear on Berger's list of possible holonomies of
torsion-free connections on 4-manifolds. Thus, these are
candidates for exotic holonomy groups.
\smallskip
In \S2, we investigate the structure equations for torsion-free
connections on 4-manifolds whose holonomy is (conjugate to)
a subgroup of either $H_3$ or $G_3$. We demonstrate the
basic relationship between connections with these holonomies
and torsion-free $H_3$- and $G_3$-structures on 4-manifolds.
Of course, this is quite straightforward. However, some
interesting things turn up. For example, it is a consequence of
our calculations in \S2 that, for $k\ge3$, the $k^{th}$
covariant derivatives of the curvature tensor of a connection
with holonomy $H_3$ are expressible as universal polynomials
in the curvature tensor and its first two covariant derivatives.
Generally, however, this section serves as a repository of the
formulas (some, unfortunately, quite elaborate) needed in later
sections.
\smallskip
In \S3, we prove that connections with either of these
holonomies do, indeed, exist. We first show that there is a
differential system with independence condition whose space
of integral manifolds can be interpreted as the space of
diffeomorphism classes of 4-manifolds endowed with certain
``non-degenerate'' torsion-free connections with holonomy
$G_3$. (For the meaning of ``non-degenerate'', see \S3.) We
then show that this system is involutive and conclude that, in
the real analytic category, the space of non-degenerate
torsion-free $G_3$-structures modulo diffeomorphism
depends on four functions of three variables. Of course, our
main tool is the Cartan-K\"ahler Theorem. The reader who is
familiar with Bryant [{\bf 1987}] should be cautioned that the
differential system constructed in this paper is not of the
same sort constructed in that paper. In fact, our approach is
more closely related to that of Cartan [{\bf 1943}]. It is worth
remarking that Theorem 3.3 is purely an existence result and
does not seem to be of much help in constructing explicit
examples.
\smallskip
We then turn our attention to the geometry of torsion-free
$H_3$-structures. It is in this section that the theory
exhibits a remarkable serendipity and the most unexpected
behavior. The upshot of our work is that a torsion-free
$H_3$-structure always has at least a one-parameter local
symmetry group, but (except for the flat structure) is never
locally symmetric. Moreover, the (non-Hausdorff) ``moduli
space'' of such structures is essentially composed of a disjoint
union of a two-dimensional space, two one-dimensional
spaces, and four points.
\smallskip
To see how unusual this is, consider that, in all previously
known cases of an irreducibly acting subgroup $G\subset
\GL(n,\bbR)$ which can be holonomy, either there exist only
locally symmetric examples or else there exist connections
with holonomy $G$ which have no local symmetries and,
moreover, there exist at least an ``arbitrary function's worth''
of inequivalent connections with holonomy $G$. This
``rigid-but-not-symmetric'' behavior of $H_3$-holonomy
connections is perhaps the most remarkable aspect of this
entire paper.
\smallskip
In \S4, we begin to construct the bridge between the purely
differential geometric treatment in the previous sections and
the twistor-theoretic treatment given in \S5. In particular,
we show how a torsion-free $G_3$-structure gives rise to a
(contact) path geometry in the sense of Cartan. This geometry
interprets the original 4-manifold as the space of solutions of
a fourth order {\sc ode} for one function of one variable.
\smallskip
This aspect of the theory fits into a program of
``geometrizing'' classical {\sc ode} as envisioned in Cartan
[{\bf 1938}] and Chern [{\bf1940}]. Cartan had observed that
his method of producing differential invariants of geometric
structures, known as the method of equivalence, could be used
to integrate certain classes of {\sc ode}. From the examples
he knew, he then abstracted a general approach which led him
to define certain classes of equations which he called {\it
classes C}.
\smallskip
It turns out that the class consisting of the fourth order {\sc
ode} which arise from torsion-free $G_3$-structures
constitutes an instance of {\it classe C}. Although we do not
pursue this in this paper, it can be shown that, in an
appropriate sense, this class is the largest {\it classe C}
among the fourth order {\sc ode}. In fact, it was in connection
with the larger problem of understanding Cartan's {\it classes
C} that the author was led to consider torsion-free
$G_3$-structures in the first place.
\smallskip
In more modern language, we use a torsion-free
$G_3$-structure to construct a double fibration $N$, one base
of which is the original 4-manifold $M$ on which the
connection is defined while the other base is the
3-dimensional contact manifold $Y$ of ``totally geodesic null
surfaces'' in $M$ where the notion of ``nullity'' is constructed
out of a {\sl quartic} form associated to the $G_3$-holonomy
connection. We show that this double fibration is
non-degenerate and apply the method of equivalence to
determine the conditions for a given double fibration to arise
from a torsion-free $G_3$-structure. These conditions take
the form of requiring that certain invariants, herein called the
primary and secondary invariants, vanish.
\smallskip
In \S5, we describe the relationship of this path geometry
with twistor theory proper by moving from the real category
to the holomorphic category. In this transition, the space $Y$
is replaced by a complex contact manifold $\bcY$ and the
points of $\bcM$, a ``complexification'' of $M$, are interpreted
as rational contact curves in $\bcY$ with normal bundle
$\cO(2)\oplus\cO(2)$. We show, by methods analogous to
those in Hitchin [{\bf 1982}], how the holomorphic geometry
encodes the conditions of vanishing of the primary and
secondary invariants of the double fibration and hence gives
rise to torsion-free ``complexified'' $G_3$-structures.
\smallskip
We close this section by applying the twistor description to
show how one can, in principle, construct many explicit
examples of torsion-free $G_3$-structures by considering the
geometry of rational holomorphic curves in $\bbC\bbP^2$. The
simplest example, that of smooth conics passing through a
fixed point in $\bbC\bbP^2$, corresponds to the unique
non-flat homogeneous torsion-free $H_3$-structure. We also
give a (very brief) discussion of how the Kodaira-Spencer
deformation theory of pseudo-group structures on complex
manifolds might be used to ``construct'' deformations of
torsion-free $G_3$-structures.
\smallskip
Finally, in \S6, we point out some of the interesting questions
which this investigation has raised.
\medskip
It would be remiss of this author indeed if he did not comment
on the method of calculation employed in this paper. The fact
is that the vast majority of these calculations were carried
out on computers using the symbolic manipulation program
{\sc Maple}. Many of the formulas, if written out in full, would
contain literally hundreds of terms and would be utterly
incomprehensible. This is particularly true of the calculations
in \S3 and \S4. This presented a problem for the author: How
can one explain the essential points without being able to
demonstrate them by (sample) calculation or without the
reader having the ability to check the calculations personally?
\smallskip
For example, the essential point in the analysis of torsion-free
$H_3$-structures is the existence of three remarkable
identities for which the author has been able to find no
conceptual justification. These identities were found by brute
force and seem to just happen to be true. For anyone to find or
verify them by hand seems extremely unlikely.
\smallskip
Thus, the author settled on the following strategy: Give the
reader enough detail to understand the structure of the
argument and to form an opinion as to the reasonableness of
the claims and then make available thoroughly documented {\sc
Maple} files which will allow the reader to check the
calculations by computer if so inclined. This the author has
done. Anyone wishing to obtain copies of these files is
encouraged to contact the author either by corporeal or
electronic mail at the addresses provided.
\medskip
Since the calculations in this paper make extensive use of the
representation theory of $\SL(2,\bbR)$, particularly an
explicit version of the Clebsch-Gordan formula, we will give a
short account of it here to establish notation. A good
reference for proofs is Humphreys [{\bf 1972}], though our
notation is different.
\smallskip
Let $x$ and $y$ denote two indeterminates and let $\bbR[x,y]$
denote the polynomial ring with real coefficients generated by
$x$ and $y$. Let $ \SL(2,\bbR) $ act on $\bbR[x,y]$ in the usual
way via unimodular linear substitutions in $x$ and $y$. The
infinitesimal version of this action is generated by the Lie
algebra ${\cal D}$ spanned by the three derivations
$$ X = x{{\partial}\over{\partial y}},\quad
H = x{{\partial}\over{\partial x}} - y{{\partial}\over{\partial
y}}, \quad Y = y{{\partial}\over{\partial x}}.$$
\smallskip
Let $\V_d \subset \bbR[x,y]$ denote the subspace consisting of
homogeneous polynomials of degree $d$. Then it is well-known
that $\V_d$ is an irreducible $ \SL(2,\bbR) $-module and that
every finite-dimensional, irreducible $ \SL(2,\bbR) $-module
is isomorphic to $\V_d$ for some $d\ge 0$.
\smallskip
It will occasionally be necessary to refer to a basis of $\V_d$.
For this purpose, we use the standard basis given by the
monomials $x^{(d-j)/2}y^{(d+j)/2}$ where $j$ ranges over the
integers which lie between $-d$ and $d$ and are congruent to
$d$ modulo 2. The $x^{(d-j)/2}y^{(d+j)/2}$-coefficient of an
arbitrary element $v\in\V_d$ will be denoted by $v_{-j}$.
Thus, $v$ has the expansion $v=v_{-d}\,x^d+v_{-d+2}\,x^{d-
1}y+\cdots+v_{d}\,y^d$.
\smallskip
The {\it Clebsch-Gordan formula\/} describes the irreducible
decomposition of a tensor product of irreducible
$\SL(2,\bbR)$-modules:
$$\V_m\otimes \V_n =
\V_{|m-n|}\oplus\V_{|m-n|+2}
\oplus\cdots\oplus\V_{m+n-2}
\oplus\V_{m+n}.$$
We shall need an explicit formula for this decomposition which
we now describe. For each $p\ge0$, define the bilinear pairing
$\la\,,\ra_p\colon\,\bbR[x,y]\times\bbR[x,y]\to\bbR[x,y]$ by
$$\la u,v\ra_p =
{1\over{p!}}\sum_{k=0}^p (-1)^k {p\choose k}\,
{{\partial^p\,u}\over{\partial x^{p-k}\,\partial y^{k}}}\>
{{\partial^p\,v}\over{\partial x^{k}\,\partial y^{p-k}}}.
$$
For example, $\la u,v\ra_0 = uv$ and
$\la u,v\ra_1 = u_x v_y - u_y v_x$.
\smallskip
It is easy to prove that
$D\la u,v\ra_p =
\la Du,v\ra_p + \la u,Dv\ra_p$
for any $p\ge 0$ and any $D\in {\cal D}$.
It follows that the pairings $\la\,,\ra_p$ are
$\SL(2,\bbR)$-equivariant. For $u\in \V_m$ and
$v\in \V_n$, we have
$\la u,v\ra_p\in \V_{m+n-2p}$ and, moreover, the induced
linear mapping $\V_m\otimes\V_n\to\V_{m+n-2p}$ is clearly
non-trivial for $0\le p\le \min(m,n)$. Thus, these pairings
give the non-trivial projections implicit in the Clebsch-Gordan
formula.
\smallskip
The following identities are easily verified:
$$\la u,v\ra_p = (-1)^p \la v,u\ra_p$$
$$
\la u,\la v,w\ra_1\ra_1 -
\la v,\la u,w\ra_1\ra_1 =
\la \la u,v\ra_1,w\ra_1$$
In particular, note that
$\la\,,\ra_p\colon \V_p\times\V_p\to
\V_0 = \bbR$ is non-trivial and hence is an
$\SL(2,\bbR)$-invariant symmetric or skew-symmetric form
on
$\V_p$ depending on whether $p$ is even or odd.
\smallskip
These pairings satisfy an enormous number of further
identities which we shall not attempt to enumerate, though we
have used them implicitly in our calculations. For example, a
frequently used identity is
$$\la u,\la v,w\ra_2 \ra_1-\la \la u,v \ra_1,w \ra_2
-\la v,\la u,w \ra_1 \ra_2
=v\,\la u,w \ra_3+w\,\la u,v \ra_3-\la u, vw \ra_3\,.$$
However, a complete list of the identities that we have used in
the course of our calculations would be so long that it would
not be comprehensible or useful. Unfortunately, a systematic
method of deriving these identities (other than brute force) is
not known to the author.
\medskip
We shall often work with $\V_p$-valued differential forms on
a smooth manifold $M$. In this case, we simply extend the
pairings
$\la\,,\ra_p\colon\,\V_m\times\V_n
\to\V_{m+n-2p}$ as graded $\Omega^\ast(M)$-module pairings
$$\la ,\ra_p\colon
\left(\Omega^\ast(M)\otimes\V_m\right)\times
\left(\Omega^\ast(M)\otimes\V_n\right)
\to\left(\Omega^\ast(M)\otimes\V_{m+n-2p}\right).$$
In particular, note that if $\omega$ is a $\V_m$-valued
$r$-form on $M$ and $\eta$ is a $\V_n$-valued $s$-form on
$M$, then
$$\la \omega,\eta\ra_p =
(-1)^{rs+p} \la \eta,\omega\ra_p.$$
\bigskip
\filbreak
\centerline{\bf \S1. The Holonomy of a Torsion-Free
Connection}
\bigskip
Let $M^n$ be a smooth, connected, and simply connected
$n$-manifold. (The assumption of simple connectivity is made
for the sake of convenience. For the problems we wish to
address, the more general case does not differ in any
significant way.) Let ${\cal P}(M)$ denote the set of piecewise
smooth paths $\gamma\colon [0,1]\to M$. For $x\in M$, let
${\cal L}_x(M)\subset{\cal P}(M)$ denote the set of
$x$-based {\it loops} in $M$, namely those
$\gamma\in {\cal P}(M)$ for which $\gamma(0)=\gamma(1)=x$.
\smallskip
Let $\nabla$ be a torsion-free affine connection on the tangent
bundle of $M$. (In order to avoid confusion, we shall not
follow the common practice of using the term {\it symmetric}
as a synonym for {\it torsion-free\/}.) For each $\gamma\in
{\cal P}(M)$, the connection $\nabla$ defines a linear
isomorphism
$P_\gamma\colon T_{\gamma(0)}M\to T_{\gamma(1)}M$
called {\it parallel translation} along $\gamma$. For each
$x\in M$, we define the {\it holonomy of $\nabla$ at $x$} to be
the subset of $\GL(T_xM)$ given by
$$
H_x= \{\,P_\gamma\ |\
\gamma\in {\cal L}_x(M)\,\}\subset \GL(T_xM).
$$
It is well known (see Kobayashi and Nomizu [{\bf 1963}]), that
$H_x$ is a connected Lie subgroup of $\GL(T_xM)$ and that,
for any $\gamma\in {\cal P}(M)$,
\ $P_\gamma$ induces an isomorphism of $T_{\gamma(0)}M$
with $T_{\gamma(1)}M$ which identifies $H_{\gamma(0)}$
with
$H_{\gamma(1)}$.
\smallskip
Choose an $x_0\in M$ and an isomorphism $\iota\colon
T_{x_0}M\to V$, where $V$ is a fixed real vector space of
dimension $n$. Then, because $M$ is connected, the conjugacy
class of the subgroup $H\subset \GL(V)$ which corresponds
under $\iota$ to $H_{x_0}\subset \GL(T_{x_0}M)$ is
independent of the choice of $x_0$ or $\iota$. By abuse of
language, we speak of $H$ as the holonomy of $\nabla$.
\bigskip
A basic question in the theory is this: {\sl Which (conjugacy
classes of) subgroups $H\subset \GL(V)$ can occur as the
holonomy of some torsion-free connection $\nabla$ on some
$n$-manifold $M\/$?\/} Two necessary conditions on $H$
were derived by M. Berger [{\bf 1955}] in his thesis and we
will now describe them.
\medskip
Let us write $T$, $T^\ast$, etc.~to denote the bundles $TM$,
$T^\ast\!M$, etc. Let $R^\nabla$ denote the section of
$\eugl(T)\otimes\Lambda^2(T^\ast) =
T\otimes T^\ast\otimes\Lambda^2(T^\ast)$ which represents
the curvature of $\nabla$. Let $\euh\subset \eugl(T)$ denote
the sub-bundle whose fiber $\euh_x$ at $x\in M$ is the Lie
algebra of $H_x(\nabla)$. Then $R^\nabla$ is a section of
$\euh\otimes\Lambda^2(T^\ast)$. Moreover, because $\nabla$
is a torsion-free connection, the first Bianchi identity states
that $R^\nabla$ takes values in the kernel bundle $\K\subset
\eugl(T)\otimes\Lambda^2(T^\ast)$ of the short exact
sequence
$$0\to\K\to \eugl(T)\otimes\Lambda^2(T^\ast)\to
T\otimes\Lambda^3(T^\ast)\to 0,$$
where the mapping
$\eugl(T)\otimes\Lambda^2(T^\ast) =
T\otimes T^\ast\otimes\Lambda^2(T^\ast)
\to T\otimes\Lambda^3(T^\ast)$ is simply
skew-symmetrization on the last three indices. In particular,
$R^\nabla$ takes values in the bundle
$\K(\euh) = \K\cap (\euh\otimes\Lambda^2(T^\ast))$. (Note
that this intersection has constant rank because all of the
subalgebras
$\euh_x$ are conjugate under suitable identifications of the
tangent spaces.)
\smallskip
Similarly, if we let $\nabla R^\nabla$ denote the covariant
derivative of the curvature tensor, then $\nabla R^\nabla$ can
be regarded as a section of the bundle
$\eugl(T)\otimes\Lambda^2(T^\ast)\otimes T^\ast$. According
to the second Bianchi identity, the condition that $\nabla$ be
torsion-free implies that $\nabla R^\nabla$ has values in the
kernel bundle $\K^1\subset
\eugl(T)\otimes\Lambda^2(T^\ast)\otimes T^\ast$ of the short
exact sequence
$$0\to\K^1\to
\eugl(T)\otimes\Lambda^2(T^\ast)\otimes T^\ast
\to \eugl(T)\otimes\Lambda^3(T^\ast)\to 0,$$
where the mapping
$\eugl(T)\otimes\Lambda^2(T^\ast)\otimes T^\ast
\to \eugl(T)\otimes\Lambda^3(T^\ast)$ is again defined by
skew-symmetrization on the last three indices. In particular,
$\nabla R^\nabla$ takes values in the bundle
$\K^1(\euh) = \K^1\cap (\K(\euh)\otimes T^\ast)$.
\medskip
This motivates the following definitions. For any
$n$-dimensional vector space $V$, let us define $\K(V)$ and
$\K^1(V)$ to be the vector spaces described by the exact
sequences
$$0\to\K(V)\to
\eugl(V)\otimes\Lambda^2(V^\ast)
\to V\otimes\Lambda^3(V^\ast)\to 0$$
and
$$0\to\K^1(V)\to
\eugl(V)\otimes\Lambda^2(V^\ast)\otimes V^\ast
\to \eugl(V)\otimes\Lambda^3(V^\ast)\to 0.$$
For any Lie subalgebra $\eug\subset \eugl(V)$, we define two
vector spaces
$$\K(\eug) = \K(V)\cap
(\eug\otimes\Lambda^2(V^\ast))$$
and
$$\K^1(\eug) = \K^1(V)\cap (\K(\eug)\otimes V^\ast).$$
\smallskip
If $\eug^\prime\subset\eug\subset\eugl(V)$ is a pair of
subalgebras of $\eugl(V)$, then we have
$\K(\eug^\prime)\subset\K(\eug)$ and
$\K^1(\eug^\prime)\subset\K^1(\eug)$. Moreover, if
$\eug^\prime\subset\eugl(V)$ is any Lie subalgebra which is
$\GL(V)$-conjugate to $\eug$, then there are isomorphisms
$\K(\eug^\prime)\simeq\K(\eug)$ and
$\K^1(\eug^\prime)\simeq\K^1(\eug)$. As we shall see, the
existence of a mere abstract isomorphism
$\eug^\prime\simeq\eug$ does {\sl not\/} imply any
relationship between $\K(\eug^\prime)$ and $\K(\eug)$ or
between $\K^1(\eug^\prime)$ and $\K^1(\eug)$.
\smallskip
Recall that a connection $\nabla$ is said to be {\it locally
symmetric\/} if the (local) $\nabla$-geodesic symmetry about
each point of $M$ preserves $\nabla$. As is well known (see
Kobayashi and Nomizu [{\bf 1963}]), the necessary and
sufficient condition for $\nabla$ to be locally symmetric is
that $\nabla$ be torsion-free and that $\nabla R^\nabla=0$.
\smallskip
By combining our discussion so far with the Ambrose-Singer
Holonomy Theorem, the following result of Berger is now
easily derived. Hence, we omit the proof.
\bigskip
{\sl
\noindent{\sc Theorem 1.1 (Berger):} Let $V$ be a real vector
space of dimension $n$. Let $\eug \subset \eugl(V)$ be the Lie
algebra of a Lie subgroup $G\subseteq \GL(V)$.
\smallskip
\item{$(i)$} If $\K(\eug) = \K(\eug^\prime)$ for any proper
subalgebra $\eug^\prime \subset \eug $, then $G$ is not
(conjugate to) the holonomy of any torsion-free connection on
any manifold $M$ of dimension $n$.\par
\smallskip
\item{$(ii)$} If $\K^1(\eug) = 0$, then any torsion-free
connection whose holonomy is (conjugate to) a subgroup of $G$
is locally symmetric.\par
}
\bigskip
Since locally symmetric connections are also locally
homogeneous, the study of locally symmetric connections can
be reduced to the study of certain (non-trivial) problems in the
theory of Lie algebras. Hence, we will not discuss the locally
symmetric case any further.
\medskip
It follows from Theorem 1.1 that two necessary conditions for
a connected Lie subgroup $G\subset \GL(V)$ with Lie algebra
$\eug\subset\eugl(V)$ to be the holonomy of a torsion-free
connection which is not locally symmetric are, first, that
$\K(\eug^\prime)$ be a proper subspace of $\K(\eug)$ for every
proper subalgebra $\eug^\prime$ of $\eug$, and, second, that
$\K^1(\eug)\not=0$. We shall refer to these two conditions as
{\it Berger's first and second criteria}.
\medskip
\noindent{\sc Examples:}
To get some feel for these criteria (and because we shall need
the results in subsequent sections), let us compute some
examples.
\smallskip
For $n\ge1$, let $\V_n$ denote the irreducible
$\SL(2,\bbR)$-module of dimension $n+1$ described in \S0.
Let $\euh_n \subset \eugl(\V_n)$ denote the Lie algebra
generated by the action of $\SL(2,\bbR)$ on $\V_n$ and let
$H_n\subset \GL(\V_n)$ denote the connected Lie subgroup
whose Lie algebra is $\euh_n$. It is easy to see that $H_n$ is
abstractly isomorphic to either $\SL(2,\bbR)$ or ${\rm
PSL}(2,\bbR)$ accordingly as $n$ is odd or even. Using the
Clebsch-Gordan formula and {\sc Maple}, it is easy to compute
the entries in Table I.
\bigskip
\centerline{\bf Table I}
\nobreak
\vskip 2 pt
\nobreak
\centerline{
\vbox{\offinterlineskip
\hrule
\halign{
&\vrule#&\strut\quad\hfil#\hfil\quad\cr %The preamble
height 2 pt&\omit&&\omit&&\omit&\cr
&$n$&&$\K(\euh_n)$&&$\K^1(\euh_n)$&\cr
height 2 pt&\omit&&\omit&&\omit&\cr
\noalign{\hrule}
height 1 pt&\omit&&\omit&&\omit&\cr
\noalign{\hrule}
height 2 pt&\omit&&\omit&&\omit&\cr
&$1$&&$\V_2$&&$\V_1\oplus\V_3$&\cr
&$2$&&$\V_0\oplus\V_4$&&$\V_2\oplus\V_4\oplus\V_6$&\cr
&$3$&&$\V_2$&&$\V_3$&\cr
&$4$&&$\V_0$&&0&\cr
&$\ge 5 $&&0&&0&\cr
height 2 pt&\omit&&\omit&&\omit&\cr}
\hrule}
}
\bigskip
Similarly, let $\eug_n\subset\eugl(\V_n)$ denote the Lie algebra of
dimension 4 which is generated by $\euh_n$ and the multiples of the
identity mapping in $\eugl(\V_n)$, and let $G_n\subset \GL(\V_n)$
denote the connected subgroup whose Lie algebra is $\eug_n$.
Applying the same methods as in the case of $\euh_n$, we can derive
the entries of Table II.
\bigskip
\centerline{\bf Table II}
\vskip 2 pt
\centerline{
\vbox{\offinterlineskip
\hrule
\halign{
&\vrule#&\strut\quad\hfil#\hfil\quad\cr %The preamble
height 2 pt&\omit&&\omit&&\omit&\cr
&$n$&&$\K(\eug_n)$&&$\K^1(\eug_n)$&\cr
height 2 pt&\omit&&\omit&&\omit&\cr
\noalign{\hrule}
height 1 pt&\omit&&\omit&&\omit&\cr
\noalign{\hrule}
height 2 pt&\omit&&\omit&&\omit&\cr
&$1$&&$\V_0\oplus\V_2$&&$2\V_1\oplus\V_3$&\cr
&$2$&&$\V_0\oplus\V_2\oplus\V_4$&
&$2\V_2\oplus 2\V_4\oplus\V_6$&\cr
&$3$&&$\V_2\oplus\V_4$&
&$\V_1\oplus\V_3\oplus\V_5\oplus\V_7$&\cr
&$4$&&$\V_0$&&0&\cr
&$\ge 5 $&&0&&0&\cr
height 2 pt&\omit&&\omit&&\omit&\cr}
\hrule}
}
\bigskip
When $n\ge 5$, neither $\euh_n$ nor $\eug_n$ satisfy Berger's first
criterion. In fact, since $\K(\eug_n)=\K(\euh_n)=0$ for all $n\ge 5$,
any torsion-free connection on an $(n+1)$-manifold whose holonomy
is conjugate to any subgroup of $G_n\subset \GL(\V_n)$ is actually
flat.
\medskip
When $n=4$, since $\K(\eug_4)=\K(\euh_4)$ and $\euh_4\subset
\eug_4$, it follows that $\eug_4$ does not satisfy Berger's first
criterion.
\smallskip
On the other hand, $\euh_4$ does satisfy Berger's first criterion
since, as is easily seen, $\K(\eus)=0$ for any proper subalgebra
$\eus\subset\euh_4$.
\smallskip
However, $\euh_4$ does not satisfy Berger's second criterion since
$\K^1(\euh_4)=0$. Thus, any connection on a 5-manifold with
holonomy conjugate to $H_4\subset \GL(\V_4)$ must be locally
symmetric. In fact, it is easily shown that any such connection on a
5-manifold is locally equivalent to the canonical connection on one
of the irreducible affine symmetric spaces $\SL(3,\bbR)/\SO(2,1)$
or ${\rm SU}(2,1)/\SO(2,1)$.
\medskip
In the remaining dimensions, things are more interesting:
\smallskip
When $n=1$, we have $\eug_1=\eugl(\V_1)$ and
$\euh_1=\eusl(\V_1)$. A glance at the tables shows that these two
algebras satisfy Berger's criteria. The corresponding torsion-free
connections on 2-manifolds are, respectively, the ``generic'' affine
connections and the ``generic'' affine connections preserving a
non-zero parallel area form. It is easy to see that
non-locally-symmetric connections exist with holonomy either
$H_1=\SL(\V_1)$ or $G_1=GL^+(\V_1)$.
\smallskip
When $n=2$, we have $\euh_2=\euso(\V_2)$, the Lie algebra of
linear transformations which preserve the $\SL(2,\bbR)$-invariant
(indefinite) quadratic form $\la\,,\ra_2$ on $\V_2$, and
$\eug_2=\euco(\V_2)$, the Lie algebra of linear transformations
which preserve $\la\,,\ra_2$ up to a scalar multiple. Again, a
glance at the tables shows that these two algebras satisfy Berger's
criteria. In the case of $\euh_2$, the corresponding connections on
3-manifolds are the Levi-Civita connections of Lorentzian metrics.
Of course, these do not have to be locally symmetric. A 3-manifold
$M$ endowed with a torsion-free connection $\nabla$ whose
holonomy is conjugate to a subgroup of $G_2=CO(\V_2)$ is known in
the classical literature as a {\it Weyl space}, after Weyl's work on
conformal geometry. It is easy to construct examples with
holonomy $G_2$ which are not locally symmetric.
\smallskip
When $n=3$, the situation is the least understood. Note that both
$\eug_3$ and $\euh_3$ satisfy Berger's second criterion. It is not
difficult to show that they both satisfy Berger's first criterion. In
the remaining sections of this paper, we will show that, for each of
$H_3$ and $G_3$, a torsion-free connection with this holonomy
does, in fact, exist. Moreover, as we shall show, these connections
have interesting relationships with path geometry, twistor theory,
and algebraic geometry.
\bigskip
We close this section with a short discussion of the general
case. The original question can now be refined to the
following one:
\medskip
{\sl
For which connected Lie subgroups $G\subset \GL(V)$
satisfying Berger's criteria do there exist torsion-free
connections which are not locally symmetric and whose
holonomy is (conjugate to) $G$?
}
\medskip
To the author's knowledge, there is no $G\subset \GL(V)$
satisfying Berger's criteria which is known not to occur as the
holonomy of any torsion-free connection which is not locally
symmetric. Nevertheless, it seems too much to hope that
Berger's necessary criteria are sufficient.
\smallskip
An obvious strategy for solving this problem is to first make a
list of the subgroups of $\GL(V)$ which satisfy Berger's
criteria and then examine each case separately. A natural
place to start is to classify those subgroups which, in
addition, act {\sl irreducibly} on $V$. Berger [{\bf 1955}]
employed an extensive series of representation-theoretic
calculations to make a list of (nearly) all of the connected
groups $G\subset \GL(V)$ which act irreducibly on $V$ and
which satisfy his two criteria. His list falls naturally into
two parts (see Theorems 3 and 4 of Chapter III of Berger [{\bf
1955}]).
\smallskip
The first part, the metric list, consists of the irreducibly
acting $G$ which satisfy Berger's criteria and which fix some
non-trivial symmetric quadratic form on $V$. Actually,
Berger's metric list contains three spurious real forms of
${\rm Spin}(9,\bbC)$ occurring as subgroups of $\GL(16,\bbR)$
which do not satisfy his second criterion. Due to the efforts
of several persons (see Salamon [{\bf 1989}] for an overview),
every $G$ on the corrected list save one is now known to be
the holonomy of some torsion-free connection which is not
locally symmetric. (The remaining open case is that of ${\rm
SO}^\ast(2p)\subset \GL(4p,\bbR)$ where $p\ge3$.)
\smallskip
The second part, the ``non-metric'' list, consists of {\it all but
a finite number\/} of the irreducibly acting $G$ which satisfy
Berger's criteria but which do not fix any non-trivial
symmetric quadratic form on $V$. Unfortunately, because
Berger gives no indication of the proof of his Theorem 4, we do
not have an estimate of how many cases are missing from the
non-metric list.
\smallskip
We shall refer to these missing subgroups as {\it exotic}.
Since the subgroups $H_3$ and $G_3$ of $\GL(\V_3)$ do not
appear on Berger's non-metric list, the set of exotic
holonomies is non-empty.
\smallskip
Finally, let us note that, even among the groups which do
appear on Berger's non-metric list, there are many which are
not known to be the holonomy group of any torsion-free
connection which is not locally symmetric. Thus, the
non-metric holonomy problem is far from being solved.
\bigskip
\bigskip
\centerline{\bf \S2. The Structure Equations of $H_3$- and
$G_3$-structures}
\bigskip
Let us begin by discussing the geometry of $H_3$-structures
on 4-manifolds. Let $M$ be a 4-manifold and let
$\pi\colon\F\to M$ be the $\V_3$-coframe bundle, i.e., each
$u\in\F$ is a linear isomorphism $u\colon
T_{\pi(u)}M\ismto\V_3.$ Then $\F$ is naturally a principal
right $\GL(\V_3)$-bundle over $M$ where the right action
$R_g\colon\F\to\F$ is defined by $R_g(u) = g^{-1}\circ u$. A
$\V_3$-valued 1-form $\omega$ on $\F$, called the
tautological 1-form, is defined by letting
$\omega(v) = u(\pi_\ast(v))$ for $v\in T_u\F$. The 1-form
$\omega$ has the $\GL(\V_3)$-equivariance
${R_g}^\ast(\omega) = g^{-1} \omega$.
\smallskip
An $H_3$-structure on $M$ is, by definition, an
$H_3$-subbundle $F\subset\F$. Note that the set of such
$H_3$-structures on $M$ is in one-to-one correspondence with
the set of sections of the quotient bundle
$\bar\pi\colon\F/H_3\to M$ whose general fiber is isomorphic
to $\GL(\V_3)/H_3$, a homogeneous space of dimension 13.
For any $H_3$-structure $F$, we will denote the restriction of
$\pi$ and $\omega$ to $F$ by the same letters.
\medskip
We shall first show that an $H_3$-structure $F$ has a
canonical connection. Since $H_3$ is canonically isomorphic
to $\SL(2,\bbR)$, we may regard the
$\SL(2,\bbR)$-representations $\V_d$ equally well as
$H_3$-representations. Moreover, since
$\eusl(2,\bbR)\simeq\V_2$, it is easily seen that the map
$\rho'_3\colon\V_2\to {\rm End}(\V_3)$ defined, for
$a\in\V_2$, by $\rho_3(a)(b)=\la a,b\ra_1$ establishes an
isomorphism $\euh_3\simeq\V_2$. We will use this to regard
a connection on $F$ as an $H_3$-equivariant,
$\V_2$-valued 1-form $\phi$ on $F$. The {\it torsion} of
$\phi$ is then represented by the $\V_3$-valued 2-form
$T(\phi)=d\omega + \la \phi,\omega\ra_1$ and the {\it
curvature} of $\phi$ is represented by the $\V_2$-valued
2-form $R(\phi)=d\phi + {1\over2}\la \phi,\phi\ra_1$.
\smallskip
By Clebsch-Gordan, ${\rm hom}(\V_3,\V_2)\simeq
\V_1\oplus\V_3\oplus\V_5$, so once one connection $\phi_0$
has been chosen, any other connection $\phi$ can be written
uniquely in the form
$$\phi = \phi_0 + \la p^1,\omega\ra_1
+ \la p^3,\omega\ra_2 + \la p^5,\omega\ra_3,$$ where each
$p^i$ is an $H_3$-equivariant, $\V_i$-valued function on $F$.
Using appropriate identities on the pairings $\la\,,\ra_i$ we
then have
$$\eqalign{T(\phi)&=T(\phi_0)+\la \la p^1,\omega\ra_1
+ \la p^3,\omega\ra_2 + \la p^5,\omega\ra_3,\omega\ra_1\cr
&=T(\phi_0)
+{\textstyle{1\over2}}\la p^1,\la \omega,\omega\ra_1\ra_1
+{\textstyle{15\over4}}p^3\la \omega,\omega\ra_3
+{\textstyle{1\over12}}\la p^3,\la \omega,\omega\ra_1\ra_2
-{\textstyle{1\over6}}\la p^5,\la \omega,\omega\ra_1\ra_3.
\cr}$$
Now, $T(\phi_0)$ can be written uniquely in the form
$$T(\phi_0)=
\la t^1,\la \omega,\omega\ra_1\ra_1
+s^3\la \omega,\omega\ra_3
+\la t^3,\la \omega,\omega\ra_1\ra_2
+\la t^5,\la \omega,\omega\ra_1\ra_3
+\la t^7,\la \omega,\omega\ra_1\ra_4,$$
where each $t^i$ is a $\V_i$-valued function on $F$ and $s^3$
is a $\V_3$-valued function on $F$. It follows that there is a
unique connection $\phi$ on $F$ for which
$$T(\phi) = \la \tau^3,\la \omega,\omega\ra_1\ra_2 +
\la \tau^7,\la \omega,\omega\ra_1\ra_4$$
for some $\V_3$-valued function $\tau^3$ and some
$\V_7$-valued function $\tau^7$ on $F$.
\smallskip
We call this $\phi$ the {\it intrinsic connection} of $F$ and we
call the resulting torsion the {\it intrinsic torsion\/} of $F$.
We shall say that $F$ is {\it 1-flat} or {\it torsion-free} if its
intrinsic torsion vanishes. (For an explanation of the term
``1-flat,'' see Bryant [{\bf 1987}]).
\smallskip
The equations $\tau^3 = \tau^7 = 0$ may be regarded as a set of
12 first order {\sc pde} for the section of the bundle $\F/H_3$
which determines $F$. Thus, locally, these represent 12 first
order equations for 13 unknowns. However, because this
``underdetermined'' system is invariant under the
diffeomorphism group of $M$, its behavior is hard to
understand directly via the classical approaches to {\sc pde}
systems. In particular, proving local existence of any local
solutions other than the flat solution is non-trivial and will be
taken up in \S3.
\medskip
There is a natural equivalence relation, {\it homothety,\/} on
$H_3$-structures which is defined as follows. Regard
$\bbR^+$, the positive real numbers, as a subgroup of
$\GL(\V_3)$ by regarding $t\in\bbR^+$ as $t$ times the
identity mapping. Then, on $\F$, ${R_t}^\ast(\omega) = t^{-
1}\omega$. For any $H_3$-structure $F$, we can define $F^t =
R_t(F)$. We say that $F^t$ and $F$ are {\it homothetic} and the
set of $H_3$-structures homothetic to $F$ will be called its
{\it homothety class}. The mapping $R_t\colon F\to F^t$ is a
bundle mapping and satisfies
${R_t}^\ast(\omega^t) = t^{-1}\omega$ and
${R_t}^\ast(\phi^t) = \phi$ where $\omega^t$ and $\phi^t$ are
the tautological form and intrinsic connection, respectively, of
$F^t$. It follows that ${R_t}^\ast((\tau^3)^t) = t\,\tau^3$ and
${R_t}^\ast((\tau^7)^t) = t\,\tau^7$. In particular, the
condition of being torsion-free is a homothety invariant.
\medskip
Henceforth, $F$ will denote a torsion-free $H_3$-structure on
$M$ and $\phi$ will denote its intrinsic connection. We thus
have the {\it first structure equation:}
$$d\,\omega = -\la \phi,\omega\ra_1.\leqno(1)$$
Differentiating this equation then yields the relation
$\la R(\phi),\omega\ra_1=0$, which is the first Bianchi
identity. According to Table I of \S1, the curvature of $\phi$
must be represented by a $\V_2$-valued function on $F$. In
fact, solving the first Bianchi identity shows that there is a
unique $\V_2$-valued function $a$ on $F$ for which the
following relation, known as the {\it second structure
equation,} holds:
$$\eqalign{d\,\phi&=-{\textstyle{1\over2}}\la
\phi,\phi\ra_1+R(\phi)\cr
&=-{\textstyle{1\over2}}\la \phi,\phi\ra_1+
a\la \omega,\omega\ra_3
-{\textstyle{1\over12}}\la a,\la
\omega,\omega\ra_1\ra_2.\cr}
\leqno(2)$$
The following proposition explains the relationship of
torsion-free $H_3$-structures on $M$ with torsion-free
connections on $M$ whose holonomy is conjugate to $H_3$.
\medskip
{\sl
\noindent{\sc Proposition 2.1:} Let $M$ be a smooth, simply
connected 4-manifold. There is a one-to-one correspondence
between the set of torsion-free affine connections on $M$
whose holonomy is conjugate to a non-trivial subgroup of
$H_3$ and the homothety classes of torsion-free
$H_3$-structures on $M$ whose canonical connections are not
flat. \par
}
\medskip
\noindent{\sc Proof:} Let $F$ be a torsion-free
$H_3$-structure on $M$ whose canonical connection $\phi$ is
not flat. Let $\psi$ denote the $\eugl(\V_3)$-valued
connection 1-form on $\F$ which restricts to $F$ to become
$\phi$. Then $\psi$ is a torsion-free connection on $\F$ and
hence corresponds to a unique torsion-free affine connection
$\nabla_F$ on $M$. By construction, $F$ is invariant under
$\nabla_F$-parallel transport and it follows that the holonomy
of $\nabla_F$ is conjugate to a subgroup of $H_3$. Note that
$\nabla_{F^t} = \nabla_F$, so $\nabla_F$ depends only on the
homothety class of $F$.
\smallskip
Conversely, let $\nabla$ be a torsion-free affine connection on
$M$ whose holonomy is conjugate to a non-trivial subgroup of
$H_3$. Since the holonomy of $\nabla$ is non-trivial, by the
Ambrose-Singer Holonomy Theorem, there must be a point
$x\in M$ at which the curvature $R^\nabla$ is not zero. Let
$u\in\F_x$ be a linear isomorphism $u\colon
T_{\pi(u)}M\ismto\V_3$ which induces an isomorphism
$H_x\ismto H\subset H_3$, where $H$ is a non-trivial
connected Lie subgroup of $H_3$. Let $P(u)\subset\F$ denote
the $\nabla$-holonomy bundle through $u$, and let $P_+(u) =
P(u)\cdot H_3$ denote its extension to an $H_3$-subbundle of
$\F$. Let $\psi$ denote the $\eugl(\V_3)$-valued 1-form on
$\F$ which corresponds to $\nabla$.
\smallskip
Because the $H_3$-structure $P_+(u)$ contains $P(u)$, it
follows that $P_+(u)$ is preserved by $\nabla$-parallel
transport and hence that $\psi$ restricts to $P_+(u)$ to
become a 1-form $\phi$ with values in $\euh_3\simeq\V_2$
and which is a connection on $P_+(u)$. Since $\psi$ is
torsion-free, it follows that $\phi$ is also. It follows that
$\phi$ is the intrinsic connection of $F=P_+(u)$ and that $F$ is
torsion-free. In particular, (1) and (2) hold.
\smallskip
If, on $F$, we write
$$\eqalignno{\omega&=\omega_{-3}\,x^3 + \omega_{-1}\,x^2y
+ \omega_{1}\,xy^2 +\omega_{3}\,y^3\cr
\noalign{\hbox{and}}
a&=a_{-2}\,x^2 + a_{0}\,xy + a_{2}\,y^2,\cr}$$
then the formula for $R(\phi)$ can be expressed as
$$\eqalign{R(\phi) &=\phantom{+}
( a_{-2} (9\,\omega_{-3}\wedge \omega_{3}
- 5\,\omega_{-1}\wedge \omega_{1})
+ 3 a_{0}\,\omega_{-3}\wedge \omega_{1}
- 6 a_{2}\,\omega_{-3}\wedge\omega_{-1} )\,x^{2}\cr
&\quad+ (- 6 a_{-2}\,\omega_{-1}\wedge \omega_{3}
+ a_{0} (18\,\omega_{-3}\wedge \omega_{3}
- 2\,\omega_{-1}\wedge \omega_{1})
- 6 a_{2}\,\omega_{-3}\wedge \omega_{1})\,xy\cr
&\quad+ ( - 6 a_{-2}\,\omega_{1}\wedge \omega_{3}
+ 3 a_{0}\,\omega_{-1}\wedge \omega_{3}
+ a_{2}(9\,\omega_{-3}\wedge \omega_{3}
- 5\,\omega_{-1}\wedge \omega_{1})\,)\,y^{2}.\cr}$$
Since $R^\nabla$ is non-zero at $u\in F$, it follows that $a(u)$
is non-zero. From this explicit formula for $R(\phi)$ it
follows that, at $u$, this 2-form does not have values in any
proper subspace of $\V_2$ and hence that the holonomy
algebra $\euh$ cannot be a proper subspace of $\euh_3$. In
particular, again by an application of the Ambrose-Singer
Holonomy Theorem, we must have $H = H_3$ and $F = P_+(u) =
P(u)$.
\smallskip
Now let $u'\in\F$ be any other coframe
$u'\colon T_{\pi(u')}M\ismto\V_3$ which induces an
isomorphism ${\cal H}_{\pi(u')}(\nabla)\ismto H_3$. By
parallel transport, we may assume that $\pi(u')=\pi(u)=x$. It
follows that $u' = A\circ u$ where $A$ is an element of the
normalizer of $H_3$ in $\GL(\V_3)$. However, it is easily
seen that this normalizer is $G_3$. In particular, $A=t\,g$
where $g\in H_3$ and $t\in\bbR^+$. It follows that
$P(u')=R_t(P(u)) = F^t$. Thus, $\nabla$ canonically determines
the homothety class of $F$.
\smallskip
The monicity of the correspondence just constructed is now
immediate. \hfill\square
\bigskip
It is worth remarking that, in the course of the above proof,
we showed that no non-trivial proper subgroup of $H_3$ can be
the holonomy of an affine torsion-free connection on $M$.
\medskip
Let us now explore the consequences of the structure
equations (1) and (2). If we write $Da =da +\la \phi,a\ra_1$,
then the derivative of (2) becomes the relation $Da\wedge\la
\omega,\omega\ra_3
-{\textstyle{1\over12}}\la Da,\la
\omega,\omega\ra_1\ra_2=0$. This relation can be solved,
showing that $Da = \la b,\omega\ra_2$ for some unique
$\V_3$-valued function $b$ on $F$. This gives the {\it third
structure equation,}
$$d\,a = - \la \phi,a\ra_1 + \la b,\omega\ra_2.\leqno(3)$$
Setting $Db = d\,b +\la \phi,b\ra_1$ and differentiating (3),
gives the relation
$-\la R(\phi),a\ra_1+\la Db,\omega\ra_2=0$, which can be
solved for $Db$ to show that there exists a
$\V_0$-valued function $c$ on $F$ so that the {\it fourth
structure equation\/} holds:
$$d\,b = - \la \phi,b\ra_1 + (c-\la a,a\ra_2)\,\omega
+{\textstyle{1\over12}}\la \la a,a\ra_0,\omega\ra_2.
\leqno(4)$$
Finally, differentiation of (4) gives the {\it fifth structure
equation\/}
$$d\,c=0.\leqno(5)$$
No further relations can be deduced by differentiation of the
equations (1)--(5). In the next section, we will use these
equations to establish the existence of non-flat, torsion-free
$H_3$-structures and determine their ``generality.''
\bigskip
\smallskip
We now turn to the case of $G_3\subset \GL(\V_3)$ which, at
first, is similar to that of $H_3$. Let $F\subset\F$ be an
$G_3$ subbundle, i.e., a $G_3$-structure on $M$. We denote the
restriction of $\pi$ and $\omega$ to $F$ by the same letters.
Note that the set of $G_3$-structures on $M$ is in one-to-one
correspondence with the set of sections of the quotient bundle
$\bar\pi\colon\F/G_3\to M$ whose general fiber is isomorphic
to $\GL(\V_3)/G_3$, a homogeneous space of dimension 12.
\smallskip
Again, we wish to introduce a canonical connection on $F$.
Now $\eug_3=\bbR\cdot{\rm id_{\V_3}}+\euh_3 \simeq
\V_0\oplus\V_2$.
We will use these identifications to regard a connection on $F$
as an equivariant $(\V_0\oplus\V_2)$-valued
1-form $\varphi = \lambda + \phi$ on $F$, where $\lambda$ is
a $\V_0$-valued 1-form on $F$ and $\phi$ is, as before, a
$\V_2$-valued 1-form on $F$. The {\it torsion} of $\varphi$
is represented by the $\V_3$-valued 2-form $T(\varphi)=
d\omega+\lambda\wedge\omega+\la \phi,\omega\ra_1$ and
the {\it curvature} of $\varphi$ is represented by the
$(\V_0\oplus\V_2)$-valued 2-form
$R(\varphi)=d\varphi + {1\over2}\la \varphi,\varphi\ra_1
= d\lambda + d\phi + {1\over2}\la \phi,\phi\ra_1$.
\smallskip
By an argument entirely analogous to the one for $H_3$ and
which need not be recounted here, it can be shown that there is
a unique connection $\varphi$ on $F$ for which the torsion
takes the form
$$T(\varphi) = \la \tau,\la \omega,\omega\ra_1\ra_4$$
for some some $\V_7$-valued function $\tau$ on $F$. We call
the resulting connection $\varphi$ the {\it intrinsic
connection} of $F$ and the resulting torsion the {\it intrinsic
torsion\/} of $F$. We say that $F$ is {\it 1-flat} or {\it
torsion-free} if its intrinsic torsion vanishes.
\smallskip
The equations $\tau = 0$ represent a set of 8 first order {\sc
pde} for the section of the bundle $\F/G_3$ which determines
$F$. Thus, locally, these represent 8 first order equations for
12 unknowns. As in the $H_3$ case, because these equations
are invariant under the diffeomorphism group of $M$, they
cannot be written locally as an ``underdetermined
Cauchy-Kowalewski system,'' in the usual sense. Thus, a
direct application of {\sc pde} techniques to the study of the
``generality'' of the solutions of these equations modulo
diffeomorphism equivalence is not easy. It is not at all
obvious that this ``solution space'' is even non-empty.
\smallskip
For the remainder of this section, $F$ will denote a
torsion-free $G_3$-structure on $M$ and $\varphi$ will denote
its intrinsic connection. The {\it first structure equation} is
then
$$d\omega
= -\lambda\wedge\omega-\la \phi,\omega\ra_1
.\leqno(6)$$
Differentiating (6) then yields the relation
$$d\lambda\wedge\omega
+\la d\phi + {\ts{1\over2}}\la
\phi,\phi\ra_1,\omega\ra_1=0,$$
which is the first Bianchi identity. According to Table II of
\S1, the curvature of $\varphi$ must be represented by a
$(\V_2\oplus\V_4)$-valued function on $F$. In fact, solving
the first Bianchi identity shows that there exist on $F$ a
unique $\V_2$-valued function $a^2$ and a unique
$\V_4$-valued function $a^4$ for which the following
relations, the {\it second structure equations,} hold:
$$\eqalign{d\lambda&=\la a^4,\la
\omega,\omega\ra_1\ra_4\cr
d\phi&=-{\textstyle{1\over2}}\la \phi,\phi\ra_1+
a^2\,\la \omega,\omega\ra_3
-{\textstyle{1\over12}}\la a^2,\la \omega,\omega\ra_1\ra_2
+{\textstyle{1\over12}}\la a^4,\la
\omega,\omega\ra_1\ra_3.\cr}
\leqno(7)$$
Unfortunately, there does not seem to be a simple analogue of
Proposition 2.1 for the case of $G_3$. This is due, in part, to
the fact that $\eug_3$ contains non-trivial algebras $\euh$
(besides $\euh_3$) for which ${\cal K}(\euh)\ne0$. However,
we do have the following statement.
\medskip
{\sl
\noindent{\sc Proposition 2.2:} Let $M$ be a smooth, simply
connected 4-manifold. Any torsion-free
$G_3$-structure $F$ on $M$ determines a unique torsion-free
affine connection on $M$ whose holonomy is (conjugate to) a
subgroup of $G_3$. Conversely, any torsion-free affine
connection on $M$ whose holonomy is (conjugate to) $G_3$
corresponds to a unique torsion-free $G_3$-structure on $M$.
}
\medskip
\noindent{\sc Proof:} The construction of the correspondence
and proof of this proposition are completely analogous to those
of Proposition 2.1. The essential point in the proof of the
second statement of the proposition is that $G_3$ is its own
normalizer in $\GL(\V_3)$. \hfill\square
\medskip
Finally, for use in the next section, we derive the
$G_3$-analogues of the third and fourth structure equations. A
glance at Table II of \S1 shows that we should expect the first
covariant derivative of the curvature tensor of $\varphi$ to be
represented by a function on $F$ which takes values in the
vector space $\V_1\oplus\V_3\oplus\V_5\oplus\V_7$. In
fact, computing the exterior derivative of (7) and solving the
resulting relations allows us to show that, for each $i=1$, 3,
5, or 7, there exists a unique $\V_i$-function $b^i$ on $F$ so
that the following equations hold:
$$\eqalign{
d\,a^2&=2\lambda\wedge a^2 -\la \phi,a^2\ra_1 +
10\la b^1,\omega\ra_1+\la b^3,\omega\ra_2+
14\la b^5,\omega\ra_3\cr
d\,a^4&=2\lambda\wedge a^4 -\la \phi,a^4\ra_1 +
9\la b^1,\omega\ra_0-5\la b^5,\omega\ra_2+
\la b^7,\omega\ra_3\cr}
\leqno(8)$$
We shall need formulas for the derivatives of the functions
$b^i$ as well. This leads to a rather formidable linear algebra
problem. Fortunately, however, this can be solved easily by
simple {\sc Maple} procedures. We merely record the result
here: Set
$$\eqalign{
\beta^1&=d\,b^1-3\lambda\wedge b^1 +\la \phi,b^1\ra_1\cr
&\quad
+{\ts{1\over30}}\la\la a^2,a^4\ra_2,\omega\ra_2
-\la{\ts{1\over108}}\la a^4,a^4\ra_2
-{\ts{1\over16 }}\la a^2,a^4\ra_1,\omega\ra_3\cr
\beta^3&=d\,b^3-3\lambda\wedge b^3 +\la \phi,b^3\ra_1
+{\ts{1\over15}}\la\la a^2,a^4\ra_2,\omega\ra_1\cr
&\quad
-\la{\ts{1\over405}}\la a^4,a^4\ra_2
-{\ts{1\over30}}\la a^2,a^4\ra_1
+{\ts{1\over12}}\la a^2,a^2\ra_0,\omega\ra_2
-{\ts{4\over25}}\la\la a^2,a^4\ra_0,\omega\ra_3\cr
\beta^5&=d\,b^5-3\lambda\wedge b^5 +\la \phi,b^5\ra_1
-\la {\ts{1\over1260}}\la a^4,a^4\ra_2
+ {\ts{1\over240}}\la a^2,a^4\ra_1,\omega\ra_1\cr
&\quad
+{\ts{1\over150}}\la\la a^2,a^4\ra_0,\omega\ra_2
+{\ts{1\over70}}\la\la a^4,a^4\ra_0,\omega\ra_3
\cr
\beta^7&=d\,b^7-3\lambda\wedge b^7 +\la \phi,b^7\ra_1.
\cr}
\leqno(9)$$
A calculation then shows that the exterior derivatives of the
equations (8) can be written in the form
\vskip-1\jot
$$\vcenter{\openup 1\jot
\halign{$#$&&\ $#$\ &\hfil$#$\cr
0
&=&10\la \beta^1,\omega\ra_1
&+&\la \beta^3,\omega\ra_2
&+&14\la \beta^5,\omega\ra_3\cr
0
&=&9\la \beta^1,\omega\ra_0
&&
&-&5\la \beta^5,\omega\ra_2
&+&\la \beta^7,\omega\ra_3.\cr
}
}
\leqno(10)
$$
Another calculation then shows that, for each $i=0$, 2, 4, 6, 8,
or, 10 there exists a unique $\V_i$-valued function $c^i$ on
$F$ so that
\vskip-1\jot
$$\vcenter{\openup 1\jot
\halign{$#$&&\ $#$\ &\hfil$#$\cr
\beta^1
&=&
&&
&-& 35\la c^2,\omega\ra_2
&-&147\la c^4,\omega\ra_3\phantom{.}\cr
\beta^3
&=&\la c^0,\omega\ra_0
&+&56\la c^2,\omega\ra_1
&-&392\la c^4,\omega\ra_2
&+&168\la c^6,\omega\ra_3\phantom{.}\cr
\beta^5
&=&9\la c^2,\omega\ra_0
&+&9\la c^4,\omega\ra_1
&+&3\la c^6,\omega\ra_2
&+&\la c^8,\omega\ra_3\phantom{.}\cr
\beta^7
&=&-162\la c^4,\omega\ra_0
&+&10\la c^6,\omega\ra_1
&-&2\la c^8,\omega\ra_2
&+&\la c^{10},\omega\ra_3.\cr}
}\leqno(11)$$
\smallskip
The main importance of this formula (as we shall see in the
next section) is that the space of solutions of the equations
(10) at each point of $F$ has dimension
$$\dim(\V_0\oplus\V_2\oplus\V_4
\oplus\V_6\oplus\V_8\oplus\V_{10}) = 36.$$
\bigskip
\bigskip
\centerline{\bf \S3. The Existence Theorems}
\bigskip
In this section, we turn to the question of the existence and
``generality'' of the set of torsion-free $H_3$- or
$G_3$-structures on 4-manifolds. These problems, though
expressible locally in terms of {\sc pde}, are rather difficult
to treat directly because of their diffeomorphism invariance.
Our approach will be to cast these problems as problems in
exterior differential systems via a technique due originally to
\'Elie Cartan [{\bf 1943}].
\smallskip
We will begin by treating the case of $G_3$. First, we note
that we can just as well regard $\V_p$ for each $p\ge0$ as a
$GL^+(2,\bbR)$ representation where $GL^+(2,\bbR)$ is the
group of linear transformations in the two variables $x$ and
$y$ with positive determinant. However, note that the
pairings $\la\,,\ra_p$ are {\it not\/}
$GL^+(2,\bbR)$-equivariant for $p>0$. For the sake of
simplicity, we shall denote the action of $g\in GL^+(2,\bbR)$
on $v\in\V_p$ by $g\cdot v$. Note that in the case $p=3$ this
establishes a canonical isomorphism $G_3 = GL^+(2,\bbR)$. We
shall identify these two groups via this isomorphism from now
on when there is no possibility of confusion.
\smallskip
Suppose that $F$ is a torsion-free $G_3$-structure on a
4-manifold $M$. The structure equations derived in the last
section then show that we have $G_3$-equivariant mappings
$a^i\colon F\to\V_i$ for $i=2$ or 4, and $b^j\colon F\to\V_j$
for $j=1$, 3, 5 or 7. To simplify our notation, let us set $\V =
\V_2\oplus\V_4$ and
$\W = \V_1\oplus\V_3\oplus\V_5\oplus\V_7$ and let
$a=a^2 + a^4$ and $b=b^1+b^3+b^5+b^7$ be regarded as mappings
of $F$ into $\V$ and $\W$ respectively. We define the {\it
total curvature mapping\/} of $F$ to be the map $K\colon F\to
\V\oplus\W$ where $K = a+b$.
\smallskip
We shall say that $F$ is {\it non-degenerate} if the mapping
$a\colon F\to \V$ is a local diffeomorphism. Since $F$ and
$\V$ both have dimension 8, this is not an unreasonable notion.
Indeed, if non-degenerate, torsion-free $G_3$-structures
exist, one expects them, in some sense, to be ``generic'' among
torsion-free $G_3$-structures. If $F$ is non-degenerate, the
mapping $K\colon F\to\V\oplus\W$ determines an
8-dimensional, $G_3$-invariant, immersed submanifold of
$\V\oplus\W$.
\medskip
We shall now show how this process can be reversed, at least
locally. That is, we are going to show that the image $K(F)$
determines $M$ and $F$ locally up to diffeomorphism.
Moreover, we shall show that $K(F)$ is characterized as an
integral manifold of a certain exterior differential system
with independence condition.
\smallskip
We now turn to the construction of this differential system.
Let $\a^i\colon\V\oplus\W\to\V_i$ for $i=2$ or 4, and
$\b^j\colon\V\oplus\W\to\V_j$ for $j=1$, 3, 5 or 7 denote the
projections thought of as vector-valued functions on
$\V\oplus\W$. We would like to define 1-forms $\lam$,
$\ph$, and $\om$ on $\V\oplus\W$ with values in $\V_0$,
$\V_2$, and $\V_3$ respectively, so that the following
equations hold:
$$\eqalign{
d\,\a^2&=2\lam\wedge \a^2 -\la \ph,\a^2\ra_1 +
10\la \b^1,\om\ra_1+\la \b^3,\om\ra_2+
14\la \b^5,\om\ra_3\cr
d\,\a^4&=2\lam\wedge \a^4 -\la \ph,\a^4\ra_1 +
9\la \b^1,\om\ra_0-5\la \b^5,\om\ra_2+
\la \b^7,\om\ra_3.\cr}
\leqno(1)$$
Setting $\a=\a^2 + \a^4$ and $\b=\b^1+\b^3+\b^5+\b^7$, this
set of equations can be written in the form
$$d\,\a =\A(\a)(\lam+\ph)+\B(\b)(\om)\leqno(1^\ast)$$
where $\A$ is linear mapping from $\V$ to
$\hom(\V_0\oplus\V_2,\V_2\oplus\V_4)$ and $\B$ is linear
mapping from $\W$ to $\hom(\V_3,\V_2\oplus\V_4)$.
Regarding $\A(\a)$ and $\B(\b)$ as vector-valued functions on
$\V\oplus\W$, we can define a function $\J$ on $\V\oplus\W$
with values in
$\hom(\V_0\oplus\V_2\oplus\V_3,\V_2\oplus\V_4)$ by the
property that $\J(u+v) = \A(\a)(u) + \B(\b)(v)$ for all
$u\in\V_0\oplus\V_2$ and $v\in\V_3$. Thus, the above
equation can also be written as $d\,\a =\J(\lam+\ph+\om)$.
Since the vector spaces $\V_0\oplus\V_2\oplus\V_3$ and
$\V_2\oplus\V_4$ both have dimension 8, it follows that,
relative to the standard bases of these two spaces, $\J$ has a
representation as a square matrix (whose entries are linear
functions on $\V\oplus\W$). Let $\D$ be the determinant of
this matrix representation of $\J$. Then $\D$ is a polynomial
function of degree 8 on $\V\oplus\W$ and it is not hard to
check (using {\sc Maple}) that $\D$ does not vanish identically.
\smallskip
Let ${\cal O}\subset\V\oplus\W$ denote the open set where
$\D$ is non-zero. Then, on ${\cal O}$, we {\sl define} $\lam$,
$\ph$, and $\om$ by the formula
$$\lam+\ph+\om= \J^{-1}d\,\a.\leqno(2)$$
Note that, with this definition, the equations (1) become {\it
identities\/}. Moreover, if we set
$$\Omega = \lam_0\wedge\ph_{-
2}\wedge\ph_{0}\wedge\ph_{2}\wedge
\om_{-3}\wedge\om_{-1}\wedge\om_{1}\wedge\om_{3},$$
then we have
$$ \Omega =
\D^{-1}d\a^2_{-2}\wedge d\a^2_{0}\wedge d\a^2_{2}\wedge
d\a^4_{-4}\wedge d\a^4_{-2}\wedge d\a^4_{0}\wedge
d\a^4_{2}\wedge d\a^4_{4}.$$
\smallskip
We are now ready to define an exterior differential system on
${\cal O}$. First, we define the following 2-forms on ${\cal
O}$:
$$\eqalign{
\Th&=d\om + \lam\wedge\om +\la\ph,\om\ra_1,\cr
\Lam&=d\lam-\la \a^4,\la \om,\om\ra_1\ra_4,\cr
\Ph&=d\ph+{\textstyle{1\over2}}\la \ph,\ph\ra_1
-\a^2\,\la \om,\om\ra_3
+{\textstyle{1\over12}}\la \a^2,\la \om,\om\ra_1\ra_2
-{\textstyle{1\over12}}\la \a^4,\la \om,\om\ra_1\ra_3.\cr}
\leqno(3)$$
Using the identities (1), it is straightforward to compute that
the exterior derivatives of these 2-forms are given by the
formulas
$$\eqalign{
d\Th&=\Lam\wedge\om - \lam\wedge\Th
+\la\Ph,\om\ra_1-\la\ph,\Th\ra_1,\cr
d\Lam&=2\la \a^4,\la \om,\Th\ra_1\ra_4,\cr
d\Ph&=-\la \ph,\Ph\ra_1
+2\a^2\,\la \om,\Th\ra_3
-{\textstyle{1\over6}}\la \a^2,\la \om,\Th\ra_1\ra_2
+{\textstyle{1\over6}}\la \a^4,\la \om,\Th\ra_1\ra_3.\cr}
\leqno(4)$$
It follows that the exterior ideal ${\cal I}\subset
\Omega^\ast({\cal O})$ generated by the eight 2-forms
$$\Lam_0,\Ph_{-2},\Ph_{0},\Ph_{2},
\Th_{-3},\Th_{-1},\Th_{1},\Th_{3}$$
is differentially closed.
\medbreak
{\sl
\noindent{\sc Proposition 3.1:} Let $F$ be a non-degenerate,
torsion-free $G_3$-structure on a 4-manifold $M$. Then the
curvature mapping $a+b\colon F\to \V\oplus\W$ has its image
in ${\cal O}\subset \V\oplus\W$ and is an integral manifold of
the differential system with independence condition $({\cal
I},\Omega)$. Conversely, every integral manifold of $({\cal
I},\Omega)$ is locally the image of a non-degenerate,
torsion-free $G_3$-structure on some 4-manifold $M$ and
this $G_3$-structure is unique up to diffeomorphism.
}
\medskip
\noindent{\sc Proof:} First, suppose that $F$ is a
non-degenerate, torsion-free $G_3$-structure on $M$. Then
the structure equations (2.6--8) hold for the canonical
1-forms $\omega$, $\lambda$, and $\phi$ and functions $a^i$
and $b^j$ on $F$. The curvature mapping $K=a+b$ satisfies
$K\colon F\to \V\oplus\W$ and clearly satisfies
$K^\ast(\a^i)=a^i$ and $K^\ast(\b^j)=b^j$. Now, (2.8) takes the
form $da=K^\ast(\J)(\lambda+\phi+\omega)$. By the
hypothesis that $F$ is non-degenerate, we know that the rank
of the differential of the mapping $a\colon F\to\V$ is 8, so it
follows that $K^\ast(\J)$ must be invertible. Thus,
$K^\ast(\D)\ne0$, so $K(F)\subset{\cal O}$. Now, pulling back
the identity (2), we get
$$\eqalign{K^\ast(\lam+\ph+\om)
&=K^\ast(\J)^{-1}d\,K^\ast(\a)\cr
&=K^\ast(\J)^{-1} da=(\lambda+\phi+\omega).\cr
}\leqno(5)$$
Of course, it then follows that $K^\ast(\lam)=\lambda$,
$K^\ast(\ph)=\phi$, and $K^\ast(\om)=\omega$. Then the
structure equations (2.6,7) imply that $K^\ast(\Lam)=0$,
$K^\ast(\Ph)=0$, and $K^\ast(\Th)=0$. Moreover, we clearly
have $K^\ast(\Omega)\ne0$. Thus, $K\colon F\to\V\oplus\W$
is an integral manifold of $({\cal I},\Omega)$, as desired.
\smallskip
Now, for the converse. Let $P^8\subset{\cal O}$ be a connected
integral manifold of $({\cal I},\Omega)$ and let $\iota\colon
P\hookrightarrow{\cal O}$ denote the inclusion mapping. Let
us define $\iota^\ast(\lam)=\bl$, $\iota^\ast(\ph)=\bph$,
$\iota^\ast(\om)=\bo$, $\iota^\ast(\a^i)=\ba^i$, and
$\iota^\ast(\b^j)=\bb^j$. Since $P$ is an integral manifold of
$({\cal I},\Omega)$, it follows that these quantities satisfy
the barred versions of the equations (2.6--8), which we will
denote by $(2.\bar6$--$\bar8)$. Moreover, the eight
1-form components of $\bl$, $\bph$, and $\bo$ are independent
on $P$ since $\iota^\ast(\Omega)\ne0$ and hence they form a
coframing on $P$.
For any element $v=v^0+v^2\in\V_0\oplus\V_2$, let $X_v$
denote the unique vector field on $P$ which satisfies the
equations $(\bl+\bph)(X_v)=v$ and $\bo(X_v)=0$. The
equations $(2.\bar6$--$\bar8)$ then yield the formulas
$$\def\temp{\phantom{^i}}
\eqalign{
\Lie_{X_v}\,\bo\temp&= -v^0\bo - \la v^2,\bo\ra_1\cr
\Lie_{X_v}\,\bl\temp&= \phantom{2 v^0\ba^i}0\cr
\Lie_{X_v}\,\bph\temp&=\phantom{2 v^0\ba^i}-\la v^2,\bph\ra_1\cr
\Lie_{X_v}\,\ba^i &= 2 v^0\ba^i - \la v^2,\ba^i\ra_1\cr
\Lie_{X_v}\,\bb^i &= 3 v^0\bb^j - \la v^2,\bb^j\ra_1\cr
}\leqno(6)$$
where $\Lie_{X_v}$ denotes the Lie derivative with respect to
$X_v$. From these formulas, it follows easily that the vector
fields $\{\,X_v\ |\ v\in\V_0\oplus\V_2\,\}$ generate a
locally free right action of the group $GL^+(2,\bbR)$ on $P$ in
such a way that $\iota = \ba+\bb$ is equivariant with respect
to the natural right $GL^+(2,\bbR)$ action on $\V\oplus\W$
generated by the right action on each summand given by
$R_g(u)=g^{-1}\cdot u$ for $u\in\V_p$. In particular, note
that, for each $z\in P$, the intersection $P\cap \left(z\cdot
GL^+(2,\bbR)\right)$ is of dimension 4 and is an open subset of
the orbit $z\cdot GL^+(2,\bbR)$.
\smallskip
Now let $z_0\in P$ be fixed. It follows from the above
discussion that there is a neighborhood $U$ of $z_0$ in $P$
which can be written in the form $U = M\cdot V$ where
$M\subset P$ is a connected submanifold of dimension 4 which
passes through $z_0$ and which is transverse to the local
$GL^+(2,\bbR)$ orbits and $V$ is a connected open neighborhood
of the identity in $GL^+(2,\bbR)$. Let $\bar\pi\colon U\to M$
be the projection onto the first factor. Let $\pi\colon\F\to M$
denote, as usual, the $\GL(\V_3)$ coframe bundle of $M$.
There is a natural mapping $\tau\colon U\to\F$ which is
defined as follows: For $u\in U$, we let $\tau(u)$ denote the
linear isomorphism $\tau(u)\colon T_{\bar\pi(u)}\ismto\V_3$
which satisfies $\bo(v)=\tau(u)(\bar\pi_\ast(v))$ for all
$v\in T_uU$.
\smallskip
It is now easy to check that $\tau$ is a
$GL^+(2,\bbR)$-equivariant embedding and that there exists a
unique $G_3$-structure $F$ on $M$ which contains $\tau(U)$
(as an open subset). Moreover, if $\omega$ represents the
tautological form on $F$, then $\tau^\ast(\omega)=\bo$. It
follows easily that if $\phi$ and $\lambda$ represent the
intrinsic connection and torsion on $F$, then they satisfy
$\tau^\ast(\phi)=\bph$ and $\tau^\ast(\lambda)=\bl$. We then
conclude from the equations $(2.\bar6$--$\bar8)$, that $F$ is
a torsion-free $G_3$-structure. Moreover, we must have
$\tau^\ast(a^i)=\ba^i$ and $\tau^\ast(b^j)=\bb^j$. It then
follows that $\iota|_U = \ba+\bb = K\circ\tau$, where $K=a+b$
is the curvature mapping of $F$. Moreover, since
$\ba=a\circ\tau$, we see that $a\colon F\to\V$ is an
immersion. Thus, $F$ is non-degenerate.
\smallskip
We have shown that $P$ is locally of the form $K(F)$ for a
non-degenerate $G_3$-structure $F$. Local uniqueness of $F$
up to diffeomorphism can safely be left to the reader.
\hfill\square
\medskip
We now turn to the problem of proving the existence of
non-degenerate torsion-free $G_3$-structures. By Proposition
3.1, it suffices to prove the existence of integral manifolds of
$({\cal I},\Omega)$.
\smallskip
As a first step in this analysis, let us define, for each $i=1$,
3, 5, or 7, the $\V_i$-valued 1-form $\be^i$ on ${\cal O}$ by
``emboldening'' the definition of $\beta^i$ in (2.9). For
example,
$\be^i=d\,\b^7-3\,\lam\wedge\b^7+\la\ph,\b^7\ra_1$. We will
also use the notation $\be=\be^1+\be^3+\be^5+\be^7$.
\smallskip
If we differentiate $(1^\ast)$, then, after a considerable
amount of linear algebra, we can express the result in the form
$$\A(\a)(\Lam+\Ph)+\B(\b)(\Th) = -\B(\be)(\om).\leqno(7)$$
Since $\J(\Lam+\Ph+\Th)=\A(\a)(\Lam+\Ph)+\B(\b)(\Th)$ and
$\J$ is invertible on ${\cal O}$, it follows that the
differential system ${\cal I}$ can equally well be generated by
the eight components of the $(\V_2\oplus\V_4)$-valued
2-form $\B(\be)(\om)$.
\medskip
{\sl
\noindent{\sc Proposition 3.2:} The differential system with
independence condition $({\cal I},\Omega)$ is in linear form
and involutive on ${\cal O}$ with Cartan characters $s_0=0$,
$s_1=s_2=8$, $s_3=4$ and $s_i=0$ for $i>3$.
}
\medskip
\noindent{\sc Proof:} According to the discussion just above,
${\cal I}$ is generated by the components of the 2-forms
$\Ups^2$ and $\Ups^4$ where
\vskip-2\jot
$$\vcenter{\openup 1\jot
\halign{$#$&&\ $#$\ &\hfil$#$\cr
\Ups^2
&=&10\la \be^1,\om\ra_1
&+&\la \be^3,\om\ra_2
&+&14\la \be^5,\om\ra_3\cr
\Ups^4
&=&9\la \be^1,\om\ra_0
&&
&-&5\la \be^5,\om\ra_2
&+&\la \be^7,\om\ra_3.\cr
}
}
\leqno(8)
$$
This clearly shows that ${\cal I}$ is in linear form.
\smallskip
Now, according the calculation done at the end of \S2 which
resulted in (2.11), the space of integral elements of $({\cal
I},\Omega)$ at each point of ${\cal O}$ is an affine space of
dimension $S=36$.
\smallskip
It remains to calculate the Cartan characters. Relative to the
sequence
$$\sigma=(\om_{-3}, \om_{3}, \om_{-1},\om_{1},
\lam_0, \ph_{-2}, \ph_{0}, \ph_{2})$$
({\it note the ordering\/}), the reduced character sequence is
easily shown to be $s'_0=0$, $s'_1=s'_2=8$, $s'_3=4$ and
$s'_i=0$ for $i>3$. Since
$S=36=s'_1+2s'_2+3s'_3+4s'_4
+5s'_5+6s'_6+7s'_7+8s'_8$,
it follows that Cartan's Test is verified, so the sequence
$\sigma$ is regular
\smallskip
Thus, $({\cal I},\Omega)$ is involutive. Moreover, $s_i=s'_i$
for $0\le i\le 8$. \hfill\square
\medskip
{\sl
\noindent{\sc Theorem 3.3:} There exist torsion-free,
non-degenerate $G_3$-structures whose holonomy is equal to
$G_3$. In fact, modulo diffeomorphism, the general such
structure depends on four functions of three variables.
}
\medskip
\noindent{\sc Proof:} It is clear that, for an open set $U$ of
values $a=a^2+a^4\in\V_2\oplus\V_4$, the associated
curvature tensor in $\K(\eug_3)$ does not lie in $\K(\eug)$
for any proper subalgebra $\eug\subset\eug_3$. Thus, if $F$
is a torsion-free, non-degenerate $G_3$-structure on a
4-manifold $M$ whose curvature function $a$ takes values in
$U$, then the holonomy of $F$ will be equal to $G_3$.
\smallskip
By the Cartan-K\"ahler theorem, since the differential system
$({\cal I},\Omega)$ is involutive and real-analytic on ${\cal
O}$, there exist integral manifolds of $({\cal I},\Omega)$
passing through every point of ${\cal O}$. It follows easily
that there exists a torsion-free, non-degenerate
$G_3$-structure on a 4-manifold $M$ whose curvature
function $a$ takes values in $U$. The statement about the
generality of such structures up to diffeomorphism follows
immediately from Proposition 3.2 in which it was shown that
the last non-zero character of $({\cal I},\Omega)$ is
$s_3=4$.\hfill\square
\medskip
Note that, in view of Proposition 2.2, one effect of Theorem
3.3 is to prove that there do indeed exist
torsion-free affine connections whose holonomy is $G_3$.
\medskip
Before leaving the differential system ${\cal I}$, let us note
that it has four-dimensional Cauchy characteristics, namely
the orbits of the action of $GL^+(2,\bbR)$ on ${\cal O}$.
Unfortunately, this action is not free, so passing to the orbit
space will not generally yield a smooth manifold, only an
orbifold. However, it is not difficult to show that the {\sl
generic} point of ${\cal O}$ has trivial
$GL^+(2,\bbR)$-stabilizer.
\smallskip
Let ${\cal O}^\ast\subset{\cal O}$ denote the open set on
which $GL^+(2,\bbR)$ acts freely. The quotient $X^\ast = {\cal
O}^\ast/GL^+(2,\bbR)$ is then a smooth manifold. If we let
$\bfpi\colon{\cal O}^\ast\to X^\ast$ denote the canonical
projection, then the fibers of $\bfpi$ are the Cauchy
characteristics of the system ${\cal I}$. Thus, there is an
exterior differential system with independence condition
$(\bar{\cal I},\bar\Omega)$ on $X^\ast$ whose local integral
manifolds are of dimension 4 and are the
$GL^+(2,\bbR)$-quotients of the local integral manifolds of
$({\cal I},\Omega)$. This ``reduced'' system is the natural one
for discussing the geometry of the integral manifolds.
However, working directly on this space is awkward because
there is no natural basis for the generators of $(\bar{\cal
I},\bar\Omega)$.
\medskip
We will close our discussion of the $G_3$ case in this section
by briefly describing the characteristic variety of the integral
manifolds.
\smallskip
Although the action of $\SL(2,\bbR)$ on $\V_3$ does not
preserve any symmetric quadratic or cubic forms, it does
preserve a quartic form $Q$ given by
$Q(v)=\la \la v,v\ra_2,\la v,v\ra_2\ra_2$ for all $v\in\V_3$.
It is easy to see that $Q(v)=0$ if and only if $v$ has a double or
triple linear factor when regarded as a homogeneous cubic
polynomial in $x$ and $y$. Thus, $Q$ is essentially the {\it
discriminant} of $v$.
\smallskip
If a 4-manifold $M$ is endowed with a $G_3$-structure $F$,
then each tangent space $T_xM$ can be identified with $\V_3$
up to an action of $G_3$, and thus, the polynomial $Q$ is
well-defined up to a scalar multiple on $T_xM$. In particular,
$\Xi_x\subset \bbP(T_xM)$, the zero locus of $Q$, is a well
defined algebraic hypersurface of degree 4. The union of all of
these varieties as $x\in M$ varies is a subset $\Xi\subset
\bbP(TM)$. It is not difficult to show that if $F$ is a
torsion-free, non-degenerate $G_3$-structure, then $\Xi$ is
the characteristic variety of the corresponding integral of the
Cauchy-reduced exterior differential system $\bar{\cal I}$.
\bigskip
We now turn to the case of $H_3$. The remainder of this
section will be devoted to a proof that torsion-free
$H_3$-structures do, indeed, exist. Moreover, in a certain
sense, the moduli space of torsion-free $H_3$-structures
modulo diffeomorphism will be shown to consist of the
disjoint union of a two-dimensional space, two
one-dimensional spaces, and four points.
\smallskip
It is natural to attempt an analysis similar to the one which
worked for $G_3$. However, due to a remarkable identity to be
described below, the $H_3$ situation is quite different.
\smallskip
Let $F$ be a torsion-free $H_3$-structure on a 4-manifold
$M$. Using the notation from \S2, there are well-defined
functions $a$ and $b$ on $F$ with values in $\V_2$ and $\V_3$
respectively. Note that, under the homothety action, we have
${R_t}^\ast(a^t) = t^2\,a$, ${R_t}^\ast(b^t) = t^3\,b$, and
${R_t}^\ast(c^t) = t^4\,c$. Thus, we will assign the functions
$a$, $b$, and $c$ the weights 2, 3, and 4, respectively.
Similarly, we assign $\omega$ and $\phi$ the respective
weights $-1$ and 0. Of course, by homothety, we could now
reduce to the cases where $c = -1$, 0, or 1, but this does not
significantly simplify the calculations, so we will carry $c$
along as a parameter.
\smallskip
For simplicity of notation, let us set $\V = \V_2\oplus\V_3$
and let $\a\colon \V\to\V_2$ and $\b\colon \V\to\V_3$
denote the projections thought of as vector-valued functions
on $\V$. If we set $K = a + b$, then $K\colon F\to \V$ is a
$\SL(2,\bbR)$-equivariant mapping satisfying $K^\ast(\a)=a$
and $K^\ast(\b)=b$. Since both $F$ and $\V$ have dimension 7,
one might hope to use this mapping and the structure equations
(2.3--4) to embed (or at least immerse) $F$ into $\V$ and then
mimic the procedure which worked for $G_3$.
\smallskip
However, this fails for the following reason: Let us write
(2.3--4) in the form
$$da + db = J\,(\phi + \omega).\leqno(9)$$
where $J$ is a function on $F$ with values in $\hom(\V,\V)$.
Now $J = K^\ast(\J_c)$ where $\J_c\colon\V\to\hom(\V,\V)$
is a polynomial mapping which depends upon a real parameter
$c$. Relative to the standard basis
$(x^2,xy,y^2,x^3,x^2y,xy^2,y^3)$ of $\V$, the linear
transformation $\J_c$ has the matrix representation
$$\J_c=\left(\matrix{
- 2 \a_{0} & 2 \a_{-2} & 0
& 6 \b_{1} & - 4 \b_{-1} & 6 \b_{-3} & 0 \cr
- 4 \a_{2} & 0 & 4 \a_{-2}
& 18 \b_{3} & -2 \b_{1} & -2 \b_{-1} & 18 \b_{-3} \cr
0 & -2 \a_{2} & 2 \a_{0} & 0
& 6 \b_{3} & - 4 \b_{1} & 6 \b_{-1} \cr
- 2 \b_{-1} & 3 \b_{-3} & 0
& c - {\bf q}_1 & - \a_{-2} \a_{0} & \a_{-2}^2 & 0 \cr
- 4 \b_{1} & \b_{-1} & 6 \b_{-3}
& 3 \a_{0} \a_{2} & c - {\bf q}_2 & 0 & 3 \a_{-2}^2 \cr
- 6 \b_{3} & - \b_{1} & 4 \b_{-1}
& 3 \a_2^2 & 0 & c - {\bf q}_2 & 3 \a_{-2} \a_{0} \cr
0 & - 3 \b_{3} & 2 \b_{1}
& 0 & \a_2^2 & -\a_{0} \a_{2} & c - {\bf q}_1 \cr
} \right)
\leqno(10)$$
where ${\bf q}_1 = 3 \a_{-2} \a_{2} - {3\over2} \a_{0}^2$ and
${\bf q}_2 = 5 \a_{-2} \a_{2} - { 1\over2} \a_{0}^2$.
\smallskip
If $J$ were invertible, then we could regard the components of
the functions $a$ and $b$ as a natural coordinate system on
$F$ and define the forms $\phi$ and $\omega$ by the formula
$(\phi + \omega)=J^{-1}\,(da + db)$. However, by direct {\sc
Maple} calculation, one reaches the remarkable conclusion that
$\det(\J_c)\equiv0$.
\medskip
Thus, we must seek other methods for determining whether
there exist local 1-forms $\phi$ and $\omega$ with linearly
independent components satisfying (2.1--4) on open domains in
$\bbR^7$ for any value of the constant $c$. This class of
problem was addressed by \'E. Cartan via his generalization of
the third fundamental theorem of Lie, (Cartan [{\bf 1904}]).
However, appealing to Cartan's general results at this point is
somewhat unsatisfactory. First, his treatment is rather
sketchy in the ``intransitive'' case (he does not even state a
theorem explicitly) and this is the case into which our problem
falls. Second, in our case, much more information can be got
by direct methods.
\medskip
Since $\det(\J_c)\equiv0$, it follows that, if we let $\L_c$
denote the classical adjoint matrix of $\J_c$ (i.e., the matrix
of 6-by-6 minors), then $\L_c \J_c = \J_c \L_c = 0$.
Computation of $\L_c$ as a matrix with entries which are
polynomial in the quantities $\a$, $\b$, and $c$ shows that
$\L_c$ does not vanish identically. (The reader should be
careful to distinguish $L = K^\ast(\L_c)$ from $\L_c$, for it
{\sl can} happen that $L$ vanishes identically.) Let
$\Sigma_c\subset\V$ denote the subset on which $\L_c$
vanishes. For ${\bf x}\in \V\setminus\Sigma_c$, the linear
mapping $\J_c({\bf x})\colon\V\to\V$ has rank 6 and the
mapping $\L_c$ has rank 1.
\smallskip
It is a standard (and easy) result in commutative algebra that
a rectangular matrix with polynomial entries which has rank
at most 1 for all values of the indeterminates in the
polynomials can be factored into the product of a column and a
row, each of whose entries are also polynomials. This applies
to $\L_c$ to show that there must exist polynomial mappings
$\r_c\colon \V\to\V$ and $\rh_c\colon \V\to\V^\ast$ so that
$\L_c(v) = \rh_c(v)\,\r_c$ for all $v\in\V$. Of course,
actually {\sl finding\/} $\r_c$ and $\rh_c$ is an enormous
algebra problem which requires factoring the entries of
$\L_c$.
\smallskip
Just this once, we will comment on the mechanical
calculations: The typical entry of $\L_c$ has between 150 and
200 terms in the $\a$ and $\b$ variables. For some
inexplicable reason having to do with the algorithm used to
find factors, {\sc Maple} was unable to factor about half of the
entries. Fortunately, as the reader will have realized, it
suffices to factor at least one entry in each row and column.
This {\sc Maple} was able to do.
The typical entry is the product of two irreducible factors of
degree 5. In the cases where {\sc Maple} was able to factor
the entry, the process took about 10 to 15 minutes on a
Macintosh SE/30. When {\sc Maple} failed to find a
factorization, it typically reported failure or returned errors
in about 5 minutes. For a {\sc SUN} 3/60 implementation of
the same problem, corresponding processes were about four
times as fast. Reassuringly(?), failures and errors were
reported for the same entries on both machines.
Once enough entries of $\L_c$ had been factored to determine
candidates for $\r_c$ and $\rh_c$, factorizations of the
remaining entries were easily checked by multiplication.
\smallskip
The result of this calculation can be described as follows: Let
$\p_c\colon\V\to \V_2$, $\q\colon\V\to \V_1$,
$\r^2_c\colon\V\to \V_2$, and $\r^3_c\colon\V\to \V_3$ be
the following polynomial functions:
$$\eqalign{
\p_c &=(2c-\la\a,\a\ra_2) \a - \la\b,\b\ra_2\,,\cr
\q\phantom{_c}&=\la\a,\b\ra_2\,,\cr
\r^2_c&={\ts{7\over6}}\la\q,\q\ra_0
+{\ts{1\over3}}\la\la\a,\b\ra_1,\q\ra_1
+{\ts{1\over2}}(2c-\la\a,\a\ra_2)\p_c-\la\a,\p_c\ra_2\a\,,\cr
\r^3_c&= -\la\b,\p_c\ra_1 - \a\la\a,\q\ra_1\,.\cr
}\leqno(11)$$
Then for all $v=v^2+v^3\in\V$ with $v^i\in\V^i$, we have
$$\L_c(v)=
\left(\la\r^2_c,v^2\ra_2-
\la\r^3_c,v^3\ra_3\right)\,(\r^2_c+\r^3_c).
\leqno(12)$$
In particular, note that
$$\Sigma_c = \left\{ {\bf x}\in\V\,|\,
\r^2_c({\bf x})= \r^3_c({\bf x})=0\,\right\}.
\leqno(13)$$
Now a second remarkable identity occurs: If we set
$$\rh_c = \la\r^2_c,d\,\a\ra_2
-\la\r^3_c,d\,\b\ra_3\,,\leqno(14)$$
{\it then the 1-form $\rh_c$ is closed.\/} In particular, by
the polynomial Poincar\'e lemma, there must exist a
polynomial $\R_c$ on $\V$ so that $d\,\R_c = \rh_c$. If we
normalize this polynomial by requiring that it vanish at
$0\in\V$, then
$$\R_c={\ts{1\over4}}\la\p_c,\p_c\ra_2
+{\ts{1\over2}}\la\a,\la\q,\q\ra_0\ra_2
\leqno(15)
$$
This function $\R_c$ has the following significance: By
construction, its differential $\rh_c$ at each point ${\bf
x}\in\V$ is a linear form on $\V$ whose kernel contains the
image of $\J_c({\bf x})$. Since $K = a + b = J(\phi + \omega) =
K^\ast(\J_c)(\phi + \omega)$, it follows that for each $f\in F$,
the image of the differential mapping $dK_f\colon T_fF\to
T_{K(f)}\V$ lies in the kernel of $\rh_c$ at $K(f)$. In other
words, $K^\ast(\rh_c) = 0$. Thus, {\sl $K$ maps $F$ into a
level set of $\,\R_c$.}
\smallskip
(Upon seeing this, the reader may well wonder why it is so
remarkable that $\rh_c$ is closed. After all, if, for every
point ${\bf x}\in\V$, there were an
$H_3$-structure $F$ whose curvature mapping $K$ covered
{\bf x} and whose differential had rank 6, then $\V$ would be
foliated by codimension 1 integral manifolds of $\rh_c$. This
would, of course, imply that $\rh_c\wedge d\,\rh_c\equiv0$,
i.e., that $\rh_c$ was {\sl integrable}. However, this is still a
long way from knowing that $\rh_c$ is closed. An integrable
polynomial 1-form $\theta$ on a vector space $V$ cannot
generally be written (even locally) in the form $\theta = f\,dg$
where $g$ is a polynomial function on $V$.)
\smallskip
Let $r^2 = K^\ast(\r^2_c)$ and $r^3 = K^\ast(\r^3_c)$. Since the
7 components of $\phi$ and $\omega$ form a coframing of $F$,
it follows that there exists a unique vector field $Z$ on $F$
for which we have the identities $\phi(Z)=r^2$ and
$\omega(Z)=r^3$. By (9), (12) and the fact that $LJ=0$, it
follows that $Z(u)$ lies in the kernel of the mapping
$dK_u\colon T_uF\to T_{K(u)}\V$ for all $u\in F$. In
particular, $a$ and $b$ are constant on the integral curves of
$Z$.
\smallskip
Another calculation now reveals a third remarkable identity:
$$\Lie_Z\,\phi = \Lie_Z\,\omega = 0.\leqno(16)$$
(Again, the knowledgeable reader may wonder why this is
remarkable. After all, a general theorem of Cartan [{\bf 1904}]
implies that, locally on $F$, away from the singularities of
$Z$, there exists a multiplier $\mu$ so that the equations (16)
are satisfied with $Z$ replaced by $\mu Z$. However, the
general theory does not provide a method for finding $\mu$
other than integrating a (linear) system of {\sc ODE}. Luckily,
in this case $\mu=1$ happens to work.)
\smallskip
It follows from (16) that the $\V$-valued coframing
$\phi+\omega$ on $F$ is preserved by the flow of $Z$. Of
course, this implies that either $Z$ vanishes identically or
else $Z$ never vanishes. (Remember that we are assuming that
$M$ (and hence $F$) is connected.) We shall say that a
torsion-free $H_3$-structure is {\it regular} if $Z$ never
vanishes and {\it singular} if $Z$ is identically zero.
\smallskip
For any regular torsion-free $H_3$-structure $F$ on $M$, the
rank of the linear map $J$ is 6 at every point of $F$. In
particular, for a regular $F$ structure, the curvature mapping
$K\colon F\to\V$ is a submersion into a level set of ${\bf
R}_c$ in $\V\setminus\Sigma_c$. Moreover, the local flow of
the vector field $Z$ fixes both $\phi$ and $\omega$. Thus, we
say that $Z$ is an {\it infinitesimal symmetry\/} of the
coframing $(\phi,\omega)$. Up to constant scalar multiples,
$Z$ is the only infinitesimal symmetry of $(\phi,\omega)$.
This is because any infinitesimal symmetry $Y$ would be
tangent to the (one-dimensional) fibers of $K\colon F\to\V$
and hence be of the form $\mu Z$ for some function $\mu$.
However, it is easy to see that $\mu Z$ is an infinitesimal
symmetry only if $\mu$ is constant.
\smallskip
Of course, the infinitesimal symmetries of $(\phi,\omega)$ on
$F$ are in one-to-one correspondence with the vector fields on
$M$ whose flows preserve the $H_3$-structure $F$, so it
follows that any regular torsion-free $H_3$-structure on $M$
has a one-dimensional local automorphism group.
\smallskip
We are now ready for the following fundamental theorem:
\medskip
{\sl
\noindent{\sc Theorem 3.4:} Let $c$ be any constant and let
${\cal C}\subset\V\setminus\Sigma_c$ be any level set of
$\R_c$ in $\V\setminus\Sigma_c$. Then $\cC$ can be covered
by open subsets $U$ which have the following property: There
exists a principal $\bbR$-bundle $K\colon F_U\to U$ over $U$
and 1-forms $\phi$ and $\omega$ on $F_U$, with values in
$\V_2$ and $\V_3$ respectively, satisfying
\smallskip
\item{$(i)$} $\phi+\omega$ is a $\V$-valued coframing on
$F_U$\par
\item{$(ii)$} The structure equations (2.1--4) are satisfied
with $a = K^\ast(\a)$ and $b=K^\ast(\b)$.
\smallskip
Moreover, the triple $(F_U,\phi,\omega)$ is unique in the sense
that any other $\bbR$-bundle $F'$ over $U$ equipped with a
$\V$-valued coframing $\phi'+\omega'$ satisfying $(i)$ and
$(ii)$ is bundle equivalent to $F_U$ via an equivalence which
identifies the coframings.
}
\medskip
\noindent{\sc Proof:} Let $\ba$, $\bb$, $\br^2$, $\br^3$, and
$\bJ\/$ denote the restrictions of the functions $\a$, $\b$,
$\r^2$, $\r^3$, and $\J_c$, respectively, to ${\cal C}$.
\smallskip
We claim that there exist 1-forms $\bph$ and $\bo$ on ${\cal
C}$ with values in $\V_2$ and $\V_3$ respectively which
satisfy the equation
$$d\ba + d\bb = \bJ\,(\bph + \bo).\leqno(17)$$
\smallskip
To see this, first note that since ${\cal C}$ lies in
$\V\setminus\Sigma_c$, it follows that $\bJ\/$ has rank 6 at
every point of ${\cal C}$. Moreover, by construction, at each
point of ${\cal C}$, the 1-form $d\ba + d\bb$ takes values in
the 6-dimensional subspace of $\V$ which is the image of
$\bJ$. It follows that, as an inhomogeneous system of linear
equations for $(\bph,\bo)$, the system (17) has a
one-parameter family of solutions at each point of ${\cal C}$.
Because $\bJ\/$ has constant rank 6, it follows that (17) can
be solved smoothly for $(\bph,\bo)$.
\smallskip
The kernel of $\bJ\/$ is spanned at each point by the vector
$\br^2+\br^3$, so once one solution $(\bph,\bo)$ to $(17)$ has
been found, any other can be written in the form
$(\bph+\br^2\alpha,\bo+\br^3\alpha)$ for some unique 1-form
$\alpha$ on ${\cal C}$.
\smallskip
It is worth remarking that the equations (17) can be expressed
in the expanded form:
$$\eqalign{
d\,\ba &= - \la\bph,\ba\ra_1 + \la\bb,\bo\ra_2\cr
d\,\bb &= - \la\bph,\bb\ra_1+ (c-\la\ba,\ba\ra_2)\,\bo +
+{\textstyle{1\over12}}\la \la \ba,\ba\ra_0,\bo\ra_2.
\cr}\leqno(17')$$
Using the explicit formulas for $\br^2$ and $\br^3$ in terms
of $\ba$ and $\bb$ derived from (11), it can be calculated that
$$\eqalign{
d\,\br^2 &= - \la\bph,\br^2\ra_1 - 2\ba\,\la\br^3,\bo\ra_3
+{\textstyle{1\over6}}\la\ba,\la\br^3,\bo\ra_1\ra_2\cr
d\,\br^3 &= - \la\bph,\br^3\ra_1 + \la\br^2,\bo\ra_1\,.
\cr}\leqno(18)$$
\smallskip
Now define the 2-forms
$$\eqalign{
\bTh &= d\,\bo + \la\bph,\bo\ra_1\cr
\bPh &= d\,\bph + {\ts{1\over2}}\la\bph,\bph\ra_1
-\ba\,\la\bo,\bo\ra_3
+{\ts{1\over12}}\la\ba,\la\bo,\bo\ra_1\ra_2\,.
\cr}\leqno(19)$$
After some calculation, the exterior derivative of (17) can be
written in the form $0 = \bJ(\bPh+\bTh)$. Of course, since the
kernel of $\bJ$ is spanned by $\br^2+\br^3$, this implies that
there exists a
2-form $\bPs$ so that $\bPh=\br^2\bPs$ and
$\bTh=\br^3\bPs$. Substituting these relations into (19),
differentiating, and using the relations (18), we compute that
$\br^2\,d\,\bPs=\br^3\,d\,\bPs=0$. This implies that
$d\,\bPs=0$.
\smallskip
Now let $U\subset \cC$ be any open set on which there exists
a 1-form $\alpha$ satisfying $d\,\alpha=-\bPs$. Clearly,
$\cC$ can be covered by such open sets. Replacing the pair
$(\bph,\bo)$ by the pair $(\bph+\br^2\alpha,\bo+\br^3\alpha)$,
we may compute that, for this new pair, we have
$\bPh=\bTh=0$, so we suppose this from now on. In other
words, $(\bph,\bo)$ satisfy the structure equations (2.1--4).
\smallskip
Now, let $F_U = \bbR\times U$, let $t$ denote a coordinate on
the first factor, and let $K\colon F_U\to U$ denote projection
onto the second factor. Let $a$, $b$, $r^2$, $r^3$, and $J\/$
denote the functions $\ba$, $\bb$, $\br^2$, $\br^3$, and
$\bJ\/$ regarded as functions on $F_U$ and define
$$(\phi,\omega) = (\bph+\br^2\,dt,\bo+\br^3\,dt).$$
It is now just a matter of calculation to see that
$(F_U,\phi,\omega)$ satisfies all of the conditions of the
Theorem. (This {\sl heavily\/} uses all sorts of pairing
identities as well as the equations (17--19). It is not
obvious.).
\smallskip
Finally, the uniqueness follows from the standard facts about
mappings preserving coframings, see Gardner [{\bf 1989}].
\hfill\square
\medskip
A corollary of Theorem 3.4 is the existence of regular
torsion-free $H_3$-structures:
\medskip
{\sl
\noindent{\sc Corollary 3.5:} For any constant $c_0$ and any
point $v\in\V\setminus\Sigma_{c_0}$, there exists a regular
torsion-free $H_3$-structure $F$ on a neighborhood of
$0\in\bbR^4$ with $c=c_0$ and a frame $u\in F_0$ so that
$K(u)=v$. Moreover, $F$ is locally unique up to
diffeomorphism.
}
\medskip
\noindent{\sc Proof:} Let $V$ be an open neighborhood of $v$ in
the level set of ${\bf R}_{c_0}$ which contains $v$ and to
which Theorem 3.4 applies, and let $(F_V,\phi,\omega)$ be the
corresponding triple. Let $U\subset F_V$ be an open
neighborhood of a point $u\in F_V$ which satisfies $K(u)=v$.
\smallskip
By the structure equations, the rank 4 Pfaffian system $I$
generated by the components of $\omega$ is completely
integrable on $F_V$. It follows that, by shrinking $U$ if
necessary, we may suppose that there exists a submersion
$\pi\colon U\to\bbR^4$ satisfying $\pi(u)=0$ and whose fibers
are connected and constitute the leaves of $I$ restricted to
$U$.
\smallskip
We claim that there exists a unique immersion $\tau\colon
U\to\F$, where $\F$ is the $\GL(\V_3)$ coframe bundle of
$\bbR^4$ which pulls back the tautological $\V_3$-valued
1-form on $\F$ to become $\omega$ on $U$. Moreover, by
shrinking $U$, we may even suppose that $\tau$ is an
embedding and that the image is an open subset of an
$H_3$-structure $F$ on a neighborhood of $0\in\bbR^4$.
Finally, the mapping $\tau$ pulls back the intrinsic connection
form on $F$ to become $\phi$ and the constant $c$ to be $c_0$.
The argument for these claims is exactly analogous to the one
in the last paragraph of the proof of Proposition 3.1, so we
will not repeat it.
\smallskip
Since the forms $\phi$ and $\omega$ satisfy (2.1--4), it
follows that $F$ is torsion-free. Using $\tau$ to identify $U$
with an open set in $F$, it follows that $K(u)=v$ as desired. By
construction, $K(F)\subset V\subset \V\setminus
\Sigma_{c_0}$, so $F$ is regular.
\smallskip
Finally, in order to prove local uniqueness, we appeal to
Theorem 3.4 again. If $F'$ were another torsion-free
$H_3$-structure on a neighborhood of $0\in\bbR^4$ which
contained a point $u'\in F'_0$ with $K'(u')=v$ and had $c=c_0$,
then locally both $K\colon F\to \V$ and $K'\colon F'\to \V$
would be submersions into the level set of ${\bf R}_{c_0}$
which contained $v$ and hence would, by the uniqueness aspect
of Theorem 3.4, be locally coframe equivalent. This clearly
implies that there exists a local diffeomorphism on a
neighborhood of $0\in\bbR^4$ which carries $F'$ to $F$ (see
Gardner [{\bf 1989}]).\hfill\square
\medskip
Of course, Corollary 3.5 is not the ideal statement. One would
like to be able to prove that each connected component $\cC$
of a level set of ${\bf R}_c$ in $\V\subset\Sigma_c$ is of the
form $K(F_\cC)$ for some torsion-free $H_3$-structure
$F_\cC$ on a connected 4-manifold $M_\cC$ and that this pair
$(M_\cC,F_\cC)$ is unique up to diffeomorphism. If we could
do this, then we could begin to make rigorous sense of the
claim that, up to diffeomorphism, the regular, torsion-free
$H_3$-structures form a moduli space of dimension 2 with
``coordinates'' given by the pair $\left(c,K^\ast({\bf
R}_c)\right)$.
\smallskip
Note that a homothety class of torsion-free $H_3$-structures
in these ``coordinates'' would be described by a curve of the
form $(c^t,k^t)=(t^4c,t^{12}k)$, so the moduli space of regular
homothety classes (and hence, regular torsion-free
connections with holonomy conjugate to $H_3$) would be
one-dimensional.
\smallskip
Unfortunately, because we cannot, as yet, determine the
topology of the components $\cC$, the step in the proof of
Theorem 3.4 where we must assume that the closed 2-form
${\bar\Psi}$ is exact remains a stumbling block. The best we
can do along these lines is state that, if two torsion-free
$H_3$-structures, $F_1$ and $F_2$ are constructed via
Corollary 3.5 using the same constant $c_0$ and points $v_1$
and $v_2$ which lie on the same connected component $\cC$ of
a level set of ${\bf R}_{c_0}$ in $\V\subset\Sigma_{c_0}$,
then they must be ``analytic continuations'' of each other in an
appropriate sense.
\medskip
We now turn our attention to the singular case, i.e., where $Z$
vanishes identically. This is equivalent to the condition that
$K\colon F\to\V$ has its image in $\Sigma_c$. This study
will therefore require a description of the singular locus
$\Sigma_c$. Fortunately, because this locus is invariant under
the action of $\SL(2,\bbR)$ (and because we can use {\sc
Maple}), this description is available.
\smallskip
For $c=0$, it can be shown that $\Sigma_0$ is the union of two
irreducible four-dimensional components, $\Sigma^\pm_0$,
which can be described as follows.
$$\Sigma^\pm_0 =
\left\{ (\pm\v^2 + \u^2,\mp\v^2\u+\ts{1\over3}\u^3)\in\V
\,|\, \u,\v\in\V_1\,\right\}.
\leqno(20)$$
These two components are smooth away from their
two-dimensional intersection, $\Sigma^0_0$, given by
$$\Sigma^0_0 =
\left\{ (\u^2,\ts{1\over3}\u^3)\in\V \,|\,
\u\in\V_1\,\right\}.
$$
Note that $\Sigma^0_0$ itself is smooth except at the point
$(0,0)\in\V$.
\smallskip
Moreover, the matrix $\J_0$ has rank 4 on
$\Sigma^\pm_0\setminus\Sigma^0_0$, rank 2 on
$\Sigma^0_0\setminus\{(0,0)\}$, and (of course) rank 0 at
$(0,0)$.
\medskip
Now, let us assume that $c\ne0$. By (11), it is clear that the
equations $\r^2=\r^3=0$ are satisfied on the locus where
$\p_c$ and $\q$ vanish. Thus, we can define a subset of
$\Sigma_c$ by
$$\Sigma^1_c = \left\{ (\a,\b)\in\V\,|\,
(2c-\la\a,\a\ra_2)\a-\la\b,\b\ra_2
=\la\a,\b\ra_2=0\,\right\}.
\leqno(21)$$
For $c\ne0$, it can be shown that $\Sigma^1_c$ is irreducible
and smooth of dimension 4. Moreover, the matrix $\J_c$ has
rank 4 everywhere on $\Sigma^1_c$.
\smallskip
It turns out that there is one other component $\Sigma^2_c$ of
$\Sigma_c$ and it can be parametrized as follows:
$$\Sigma^2_c =
\left\{ (\v+\u^2,\ts{1\over3}\u^3-\v\u)\in\V\,|\;
\u\in\V_1,\>\v\in\V_2, \;{\rm where}\;
\la\v,\v\ra_2=\ts{2\over3}c\,\right\}.
\leqno(22)$$
It can be shown that $\Sigma^2_c$ is algebraically irreducible
and smooth of dimension 4. (Topologically, $\Sigma^2_c$ is the
product of $\bbR^2$ with an hyperboloid of either two or one
sheets depending on whether $c$ is positive or negative.)
Moreover, the matrix $\J_c$ has rank 4 everywhere on
$\Sigma^2_c$.
\medskip
Note that, as $c\to0$, both $\Sigma^1_c$ and $\Sigma^2_c$
reduce to the union of the two components $\Sigma^\pm_0$ of
$\Sigma_0$.
\medskip
By an analysis similar to the one carried out in the course of
proving Theorem 3.4 and Corollary 3.5, we can arrive at the
following theorem which classifies all of the singular
torsion-free $H_3$-structures. The results of this theorem
are more global than those in the regular case since we have a
good description of the topology of the singular locus
$\Sigma_c$. We omit the (straightforward) proof.
\medskip
{\sl
\noindent{\sc Theorem 3.6:} The following singular
torsion-free $H_3$-structures exist and have the stated
properties. Except as noted otherwise, these belong to distinct
homothety classes. Moreover, every singular torsion-free
$H_3$-structure is locally diffeomorphic to exactly one on
this list.
\smallskip
\item{$(i)$} For each $c_0\ne0$, an $H_3$-structure
$F^1_{c_0}$ on a connected 4-manifold $M^1_{c_0}$ with the
property that $c=c_0$ and whose curvature mapping $K\colon
F^1_{c_0}\to\V$ is a submersion onto $\Sigma^1_{c_0}$. The
symmetry group of $F^1_{c_0}$ has dimension 3. The
structures $F^1_{c_0}$ and $F^1_{c_1}$ belong to the same
homothety class if and only if $c_1$ and $c_0$ have the same
sign.\par
\smallskip
\item{$(ii)$} For each $c_0\ne0$, an $H_3$-structure
$F^2_{c_0}$ on a 4-manifold $M^2_{c_0}$ with the property
that $c=c_0$ and whose curvature mapping $K\colon
F^2_{c_0}\to\V$ is a submersion onto $\Sigma^2_{c_0}$. The
symmetry group of $F^2_{c_0}$ has dimension 3. If $c_0<0$,
then $M^2_{c_0}$ is connected and all of these $F^2_{c_0}$
belong to the same homothety class. If $c_0>0$, then
$M^2_{c_0}$ has two connected components and each of these
components determines a unique homothety class.\par
\smallskip
\item{$(iii)$} For $\epsilon=\pm$, an $H_3$-structure
$F^\epsilon$ on a 4-manifold $M^\epsilon$ with the property
that $c=0$ and whose curvature mapping $K\colon
F^\epsilon\to\V$ is a submersion onto
$\Sigma^\epsilon_0\setminus\Sigma^0_0$. The symmetry
group of $F^\epsilon$ has dimension 3; however, its {\it
conformal\/} symmetry group has dimension 4 and has an open
orbit on $M^\epsilon$. Moreover, each of $F^+$ and $F^-$ is the
unique representative in its homothety class.\par
\smallskip
\item{$(iv)$} An $H_3$-structure $F^0$ on a 4-manifold $M^0$
with the property that $c=0$ and whose curvature mapping
$K\colon F^0\to\V$ is a submersion onto
$\Sigma^0_0\setminus\{(0,0)\}$. The symmetry group of $F^0$
has dimension 5 and acts transitively on $M^0$; however, its
{\it conformal\/} symmetry group has dimension 6. Moreover,
$F^0$ is the unique representative in its homothety class.\par
\smallskip
\item{$(v)$} The standard flat $H_3$-structure $F_0$ on
$\bbR^4$. Its curvature mapping $K\colon F_0\to\V$ is just
$K\equiv(0,0)$. The symmetry group of $F_0$ has dimension 7
and acts transitively on $\bbR^4$; however, its {\it
conformal\/} symmetry group has dimension 8. Moreover,
$F_0$ is the unique representative in its homothety class.\par
}
\medskip
Thus, there exist eight homothety classes of singular
torsion-free $H_3$-structures, the last two of which are
homogeneous. Writing these structures out explicitly is
somewhat tedious, even in case $(iv)$ above, the ``simplest'' of
the non-flat $H_3$-structures. However, as we shall see in
\S5, the manifold $M^0$ is topologically the space of smooth
plane conics in $\bbR\bbP^2$ which pass through a fixed point,
and its $H_3$-structure $F^0$ has a very natural geometric
interpretation.
\bigskip
\filbreak
\centerline{\bf \S4. Path Geometries and Exotic Holonomy}
\bigskip
We now turn to another approach to understanding the
torsion-free $G_3$- and $H_3$-structures on 4-manifolds.
This approach is closely related to the theory of ``path
geometries'' as initiated by \'Elie Cartan and, in a complexified
form to be treated in the next section, to the twistor theory of
Penrose as generalized by Hitchin [{\bf 1982}].
\smallskip
In order to begin the discussion, let us take a closer look at
the $G_3$-representation $\V_3$. As we remarked in the
introduction, $\V_3$ can be regarded as the homogeneous cubic
polynomials in two indeterminates $x$ and $y$. As a result,
$\V_3$ contains a $G_3$-invariant, 2-dimensional cone
$\tilde\C\subset\V_3$ which consists of the polynomials
which are ``perfect cubes'', i.e., are of the form $(ax+by)^3$ for
some linear form $ax+by\in\V_1$. Moreover, $G_3$ is easily
characterized as the subgroup of $\GL(\V_3)$ which preserves
$\tilde\C$. This cone can be made more familiar by noting
that its projectivization $\C\subset\bbP(V_3)$ is a rational
normal curve (sometimes called the ``twisted cubic curve''). It
is not hard to show that, for any rational normal curve
$C\subset\bbP(V)$ where $V$ is a (real) vector space of
dimension 4, there exists an isomorphism $\iota\colon
V\ismto\V_3$ which identifies $C$ with $\C$, and that this
isomorphism is unique up to composition with an element of
$G_3$.
\smallskip
This leads to an alternate description of $G_3$-structures on
4-manifolds which will be important for our second point of
view. If $F$ is a $G_3$-structure on $M^4$, then for each $x\in
M$, there is a well-defined rational normal curve
$C_x\subset\bbP(T_xM)$ which corresponds to $\C$ under any
isomorphism $u\colon T_xM\ismto\V_3$ where $u\in F_x$.
The ``de-projectivized'' cone $\tilde C_x\subset T_xM$ which
corresponds to $\tilde\C\subset\V_3$ will be known as the
``rational normal'' cone (or, sometimes, the ``twisted cubic''
cone) of $F$ at $x$. It is easy to see that the union
$C\subset\bbP(TM)$ of all of these curves is a smooth
submanifold of $\bbP(TM)$ and that the natural projection
$C\to M$ is a smooth submersion. Conversely, given a smooth
subbundle $C\subset\bbP(TM)$ for which the fiber $C_x$ is a
rational normal curve in $\bbP(T_xM)$ for each $x\in M$,
there is a unique $G_3$-structure $F$ which corresponds to
$C$ in the manner just described. Thus, a $G_3$-structure on
$M$ may be regarded as equivalent to a smooth field of
rational normal cones on $M$.
\smallskip
There is another geometric object in $\V_3$ which will be
useful. This is the {\sl quartic cone\/} $\tilde\Q\subset
\V_3$ which is the null cone of the $H_3$-invariant quartic
form $Q$ on $\V_3$ given by $Q(v)=\la\la v,v\ra_2,\la
v,v\ra_2\ra_2$. The null vectors of $Q$ on $\V_3$ are the
cubic polynomials which have a double linear factor. The
``conformal class'' of $Q$ is preserved by $G_3$ and hence both
the ``null cone''of $Q$, i.e., $\tilde\Q$, and its projectivization
$\Q\subset\bbP(\V_3)$ are $G_3$-invariant. The relationship
between $\C$ and $\Q$ is simple: $\C$ is the singular locus of
$\Q$ while $\Q$ is the tangent developable of $\C$. It is not
hard to see that if $p\subset\tilde\Q$ is a linear 2-plane, then
$p$ consists of the multiples $(ax+by)\,\ell^2$ where
$\ell\in\V_1$ is a fixed linear form defined up to a scalar
multiple. Thus, the null planes of $Q$ form a 1-dimensional
rational curve ${\cal N}$ in the Grassmannian $\Gr(2,\V_3)$.
\smallskip
Corresponding to this geometry in $\V_3$, a $G_3$-structure
on $M$ defines a quartic null cone $\tilde\Q_x\subset T_xM$
and a rational curve $N_x\subset \Gr(2,T_xM)$ of null planes
for each $x\in M$. We let $N$ denote the union of the $N_x$ as
$x$ ranges over $M$ and denote the base-point projection by
$\el\colon N\to M$.
\smallskip
It is interesting to compare this to the more familiar case of a
conformal structure of type (2,2) on $M$. In that case, the
space of null 2-planes at each point $x\in M$ forms {\sl two}
disconnected rational one-parameter families, the
$\alpha$-planes and the $\beta$-planes in Penrose's
terminology. The condition for there to exist a null surface of
type $\alpha$ (respectively, of type $\beta$) in $M$ tangent to
each $\alpha$-plane (resp. $\beta$-plane) is that the
conformal structure be {\sl half conformally flat,} i.e., that
one of the two irreducible components of the conformal Weyl
curvature vanish. Our first result will describe an analogous
phenomenon for $G_3$-structures.
\medskip
{\sl
\noindent{\sc Theorem 4.1:} A $G_3$-structure $F$ on a
four-manifold $M$ is torsion-free if and only if for every null
plane $p\in N_x$ there exists a null surface in $M$ which
passes through $x$ and for which $T_xS=p$.
}
\medskip
\noindent{\sc Proof:} Let $F$ be a $G_3$-structure on $M$, and
let $\omega$, $\lambda$, and $\phi$ with values in $\V_3$,
$\V_0$, and $\V_2$ respectively denote the canonical 1-form
and intrinsic connection forms on $F$. From \S2, we know that
the first structure equation can be written in the form
$$d\,\omega = -\lambda\wedge\omega-\la\phi,\omega\ra_1
+\la \tau,\omega\ra_4, \leqno(1)$$
where $\tau$ is a function on $F$ with values in $\V_7$ which
has the equivariance
$${R_g}^\ast(\tau) = \det(g)^{1\over4}g^{-1}\cdot \tau$$
for $g\in G_3$. By definition, the function $\tau$ vanishes if
and only if $F$ is torsion-free.
\smallskip
There is a submersion $\nu\colon F\to N$ defined as follows.
For each $u\in F_x$, let $\nu(u)\in N$ be the subspace of
$T_xM$ which corresponds under $u\colon T_xM\ismto\V_3$
to the subspace of $\V_3$ spanned by $\{x^3,x^2y\}$. Of
course, we have $\pi=\el\circ\nu$. The fibers of $\nu$ are the
orbits in $F$ of the subgroup $P\subset G_3$ which stabilizes
this subspace. It will be useful in what follows to note that
$P= P^0\cup(-P^0)\simeq\bbZ_2\times P^0$ where $P^0$ is the
identity component of $P$. In particular, note that the typical
fiber of $\nu$ is not connected, but consists of two
components.
\smallskip
Let $({\cal I},\Omega)$ denote the differential ideal with
independence condition on $N$ whose integrals are the
canonical lifts of null surfaces. (This is just the restriction of
the canonical contact system on $\Gr(2,TM)$ to $N\subset
\Gr(2,TM)$.) Then $\nu$ pulls this ideal back to $F$ to be the
ideal ${\cal I}^\nu$ generated by $\omega_1$ and $\omega_3$
and the independence condition $\Omega$ pulls back under
$\nu$ to be represented by the 2-form $\Omega^\nu =
\omega_{-1}\wedge\omega_{-3}$. Thus, in order to describe
the integral manifolds of $({\cal I},\Omega)$, it suffices to
study the integral manifolds of $({\cal I}^\nu,\Omega^\nu)$.
\smallskip
From the structure equation (1), we can extract the following
formulas:
$$\left.\eqalign{
d\,\omega_1
&\equiv
4\,\phi_2\wedge\omega_{-1}+
2160\,\tau_5\,\omega_{-3}\wedge\omega_{-1}\cr
d\,\omega_3
&\equiv
\phantom{4\,\phi_2\wedge\omega_{-1}+}\>\>
5040\,\tau_7\,\omega_{-3}\wedge\omega_{-1}\cr
}
\right\}\quad
{\rm modulo}\quad \omega_1,\omega_3.
\leqno(2) $$
It follows from these formulas that no integral elements of
$({\cal I}^\nu,\Omega^\nu)$ exist at points where
$\tau_7\ne0$. Of course, this implies that there are no
integral elements of the original system $({\cal I},\Omega)$ at
any point in $N$ of the form $p=\nu(u)$ where $\tau_7(u)\ne0$.
In particular, if there exists an integral manifold of
$({\cal I},\Omega)$ passing through every point of $N$, then we
must have $\tau_7\equiv0$ on $F$. Since $\V_7$ is
$G_3$-irreducible, this implies that we must have
$\tau\equiv0$, implying that $F$ is torsion-free.
\smallskip
Conversely, suppose that $\tau\equiv0$. Then the structure
equations (2) simplify to
$$\left.\eqalign{
d\,\omega_1&\equiv4\,\phi_2\wedge\omega_{-1}\cr
d\,\omega_3&\equiv\qquad 0\cr}
\right\}\quad
{\rm modulo}\; \omega_1,\omega_3.
\leqno(3) $$
Moreover, the $y^3$-component of $d\,\omega$ yields the
further information that
$$d\,\omega_3\equiv
2\,\phi_2\wedge\omega_1\quad{\rm modulo}\quad\omega_3.
\leqno(4)$$
It follows that the system $({\cal I},\Omega)$ has
one-dimensional Cauchy characteristics and that ${\cal I}$ is
locally equivalent to the second order contact system for
curves in the plane. Thus, for each $p\in N$, there exists an
integral manifold $\tilde S$ of $({\cal I},\Omega)$ passing
through $p$. The projection of $\tilde S$ to $M$ yields the
desired null surface $S$. \hfill\square
\medskip
Geometrically, the Cauchy characteristics of ${\cal I}$ can be
described as the {\sl $\C$-geodesics} of (the intrinsic
connection of) $F$. To see this, note that the Cartan system of
${\cal I}^\nu$ is generated by the system $C({\cal I}) = {\rm
span}\{\omega_3,\omega_1,\omega_{-1},\phi_2\}$. The
connected integral manifolds of this system in $F$ project to
$M$ to become the $\varphi$-geodesics whose tangent vectors
at each point belong to the twisted cubic cone, hence the name
$\C$-geodesics. In particular, note that every null surface can
be regarded (locally) as a one-parameter family of
$\C$-geodesics. The system ${\cal I}$ can thus be regarded as
representing the conditions for selecting a one-parameter
family of $\C$-geodesics in such a way that the resulting
surface is null. Thus, the integral manifolds of $({\cal
I},\Omega)$ are constructible by {\sc ode} methods alone. The
main point is that the Cartan-K\"ahler Theorem is not needed,
so our constructions do not require the assumption of real
analyticity.
\smallskip
From now on, let us assume that $F$ is a torsion-free
$G_3$-structure on $M$. For our purposes, the most
interesting null surfaces in $M$ will be those which are {\sl
totally geodesic}. These surfaces are the integrals of an
augmented system $({\cal I}_+,\Omega)$ on $N$ whose
pull-back under $\nu$ is the system
$({\cal I}^\nu_+,\Omega^\nu)$ where
${\cal I}^\nu_+$ is generated by the 1-forms $\omega_1$,
$\omega_3$, and $\phi_2$.
\medskip
{\sl
\noindent{\sc Proposition 4.2:} If $F$ is a torsion-free
$G_3$-structure on $M$, then the system ${\cal I}_+$ is a
Frobenius system of rank 3. In particular, there exists a
3-parameter family of totally geodesic null surfaces in $M$.
}
\medskip
\noindent{\sc Proof:} Since $F$ is torsion-free, we may
assume that the structure equations (2.6--7) hold. Examining
the $xy^2$- and $y^3$- components of $d\,\omega$ and the
$x^2$-component of $d\,\phi$, we get the equations
$$d\,\omega_1\equiv d\,\omega_3\equiv d\,\phi_2\equiv0
\quad{\rm modulo}\quad\omega_1\,\omega_3,\phi_2.$$
Of course this implies the Proposition. \hfill\square
\medskip
{\sc Remark:} As the reader is probably aware, the only
surprising aspect of this proof is that $d\,\phi_2\equiv0
\>{\rm modulo}\>\omega_1\,\omega_3,\phi_2$.
A na\"\i ve expectation would be that the curvature term in
the formula for $d\,\phi_2$ would make the formula read
$$d\,\phi_2\equiv g\,\omega_{-3}\wedge\omega_{-1}
\quad{\rm modulo}\quad\omega_1\,\omega_3,\phi_2$$
for some function $g$ which is a linear combination of
curvature coefficients. However, this coefficient must be zero
since the ``conservation of index sum'' principle would say that
$g$ would be assigned an index of 6 as a coefficient of some
$\V_k$-valued function on $F$, while we have seen that the
curvature takes values in $\V_2\oplus\V_4$, a space which
has no terms of index 6.
\medskip
From now on, we will refer to a connected, totally geodesic
null surface of a torsion-free $G_3$-structure $F$ on $M$ as a
{\sl sheet\/} of $F$. The sheets of $F$ are in one-to-one
correspondence with the leaves of the Frobenius system ${\cal
I_+}$ on $N$. Since ${\cal I_+}$ is a rank 3 Pfaffian system,
its space of leaves has the structure of a (not necessarily
Hausdorff) three-manifold.
\smallskip
We shall say that $F$ is {\sl amenable} if the space of leaves
of ${\cal I_+}$ on $N$ is Hausdorff. It can be shown that if $F$
is a torsion-free $G_3$-structure on $M$, and $x\in M$ is any
point, then there exists an $x$-neighborhood $U\subset M$ so
that the restricted $G_3$-structure $F_{U}$ is amenable. This
is done by first showing that the canonical flat $G_3$-
structure on $\V_3$ is amenable and then showing that if
${\rm exp}_x\colon T_xM\to M$ is the (locally defined)
exponential mapping of the connection $\varphi$, then $U =
{\rm exp}_x(\tilde U)$ is amenable for any sufficiently small
convex neighborhood $\tilde U$ of $0\in T_xM$. (Essentially
this is true because such a $U$ can be compared ``closely
enough'' with the flat case. Details will be left to the
interested reader.) In the amenable case, we let $Y$ denote
the space of sheets and let $\er\colon N\to Y$ denote the
projection.
\medskip
{\sl
\noindent{\sc Proposition 4.3:} If $F$ is an amenable,
torsion-free $G_3$-structure on $M$, then
$$\matrix{
&&N\hskip-.5em&&\cr
&\raise.5em\hbox{$\el$}\hskip-.5em\swarrow\hskip-.5em&
&\searrow\hskip-.5em\raise.5em\hbox{$\er$}\hskip-1em&\cr
M\hskip-1em&&&&Y\cr }
\leqno(5) $$
is a non-degenerate double fibration. Moreover,
there exists a contact structure on $Y$ for which each of the
curves $R_x=\er(\el^{-1}(x))$ for $x\in M$ is a (closed) contact
curve. Finally, if we let $Y^{(2)}$ denote the manifold of
2-jets of contact curves in $Y$, and let $\er^{(2)}\colon N\to
Y^{(2)}$ be the mapping which sends $p\in N$ to the 2-jet of
$R_{\el(p)}$ at $\er(p)$, then $\er^{(2)}$ is a local
diffeomorphism.
}
\medskip
Before we begin the proof, note that we have the following
consequences of the structure equations on $F\/$:
$$\eqalign{
d\,{\omega_3}_{\phantom{-}}
&\equiv\quad2\,\phi_2\wedge{\omega_1}_{\phantom{-}}
\quad{\rm modulo}\quad\omega_3,\cr
d\,{\omega_1}_{\phantom{-}}
&\equiv\quad4\,\phi_2\wedge\omega_{-1}\quad{\rm
modulo}\quad\omega_3,\omega_1,\cr
d\,\omega_{-1}
&\equiv\quad6\,\phi_2\wedge\omega_{-3}\quad{\rm
modulo}\quad\omega_3,\omega_1,\omega_{-1},\cr
d\,\omega_{-3}
&\equiv\quad\hskip 2 em 0\hskip 2 em\quad{\rm
modulo}\quad\omega_3,\omega_1,
\omega_{-1},\omega_{-3}.\cr
}\leqno(6)
$$
It follows from these equations that the orbits of $P^0\subset
G_3$ are the Cauchy characteristic leaves of the Pfaffian
system on $F$ spanned by
$\left\{\omega_3,\omega_1,\omega_{-1}\right\}$. However,
this system is also invariant under right action by $-1\in
G_3$, so this Pfaffian system is the pull-back under $\nu$ of a
well defined Pfaffian system $I$ of rank 3 on $N=F/P$. Note
that the first (respectively, second) derived system of $I$,
denoted $I^{[1]}$ (respectively, $I^{[2]}$), pulls back via $\nu$
to $F$ to be spanned by the form(s)
$\left\{\omega_3,\omega_1\right\}$ (respectively,
$\left\{\omega_3\right\}$).
\medskip
\noindent{\sc Proof:}
First, we show that the product mapping $\el\times\er\colon
N\to M\times Y$ is an immersion. By construction, the
semi-basic forms for the projection $\pi=\el\circ\nu\colon
F\to M$ are spanned by the 1-forms $\left\{\omega_{-3},
\omega_{-1}, \omega_{1},\omega_{3}\right\}$; the semi-basic
forms for the projection $\er\circ\nu\colon F\to Y$ are
spanned by the 1-forms $\left\{\phi_2, \omega_{1},
\omega_{3}\right\}$; and the semi-basic forms for the
projection $\nu\colon F\to N$ are spanned by the forms
$\left\{\omega_{-3},\omega_{-1},\phi_2,
\omega_{1},\omega_{3}\right\}$. Since this latter collection
is contained in the union of the first two, it follows that
$\el\times\er$ is an immersion. (Note that we have {\sl
not\/} proved that $\el\times\er$ is an embedding, which is
the usual double fibration axiom. This stronger statement may
well fail.)
\smallskip
Now, the intersection of the first two spans is the Pfaffian
system spanned by the forms $\left\{\omega_{1},\omega_{3}
\right\}$. By the first two of the equations (6), it follows
that this system does not contain any Frobenius sub-system.
Thus, the double fibration is non-degenerate.
\smallskip
Now the first equation of (6) shows that the Cartan system of
$I^{[2]}$ is a rank 3 Pfaffian system which consists of the
semi-basic forms for the projection $\er\colon N\to Y$. Since,
by construction, the fibers of $\er$ are connected, a standard
result (see BCG$^3$ [{\bf 1990}]), then implies that there
exists a rank 1 Pfaffian system $(I^{[2]})^\flat$ on $Y$ which
pulls back to $N$ via $\er$ to become the Pfaffian system
$I^{[2]}$. The first equation of (6) then implies that
$(I^{[2]})^\flat$ defines a contact structure on $Y$. We will
denote by $L\subset T^\ast Y$ the line bundle whose space of
sections is equal to $(I^{[2]})^\flat$.
\smallskip
Now for each $x\in M$, the curve $R_x$ is a contact curve for
this contact structure (i.e., is an integral curve of
$(I^{[2]})^\flat$). To see this, note that the fiber
$N_x=\el^{-1}(x)\subset N$ is a smooth, connected, compact
curve and its inverse image under $\nu$ is
$\pi^{-1}(x) = F_x$. Since $\omega$ clearly vanishes when
pulled back to $F_x$, it follows that $N_x$ is an integral curve
of $I$ and hence that $R_x = \er(N_x)$ is an integral curve of
$L$.
\smallskip
Let $\sigma_2\colon Y^{(2)}\to Y$ be the
5-dimensional bundle of 2-jets of contact curves in $Y$ and
let $L^{(2)}\subset T^\ast Y^{(2)}$ be the canonical contact
Pfaffian system. By a standard result in the theory of Pfaffian
systems, namely the Goursat Normal Form Theorem (see
BCG$^3$ [{\bf 1990}], particularly Chapter II), the structure
equations (6) imply that there is a unique mapping
${{\er^\prime}}\colon N\to Y^{(2)}$ which satisfies
$\er=\sigma_2\circ{\er^\prime}$ and which pulls $L^{(2)}$
back to be $I$. The map ${\er^\prime}$ can be defined as
follows: Let $p\in N$ be any point, and let $\gamma\subset
N$ be any integral curve of $I$ which passes through $p$ and is
transverse to the fibers of $\er$. Then ${\er^\prime}(p)$ is
defined to be the 2-jet of the curve $\er(\gamma)$ at $\er(p)$.
(Although we have invoked the Goursat Theorem, the
well-definition of this mapping can be checked directly by the
interested reader. See the discussion below on {\sc ode}.) It
follows from the general theory that ${\er^\prime}$ is a local
diffeomorphism. By construction, the fiber of $\el$ through
any $p\in N$ is an integral curve of $I$, and hence
${\er^\prime}$ is equal to $\er^{(2)}$ as defined in the
proposition. \hfill\square
\medskip
It is worth remarking that, in fact, $L$ is the {\sl unique}
contact structure on $Y$ with respect to which each of the
curves $R_x$ is a contact curve. This follows since our
discussion has shown that the curves of the form $R_x$ which
pass through a given point $y\in Y$ serve to ``fill out'' an open
set in the space of 2-jets of contact curves passing through
$y$. Thus, there is only one 2-plane $E_y\subset T_yY$ which
contains all of the tangent vectors to these curves, and hence
its annihilator $E^\perp_y\subset T^\ast_yY$ is unique and
must be $L_y$.
\medskip
We have shown how to associate to each torsion-free
amenable $G_3$-structure $F$ a contact
3-manifold endowed with a ``non-degenerate'' 4-parameter
family of contact curves. It is instructive to see what this
data looks like in local coordinates. The coordinate form of
the Goursat Normal Form Theorem asserts that every point
$p\in N$ has a neighborhood $U$ on which there exist local
coordinates $\psi=(x,y^0,y^1,y^2,y^3)$ so that if we define
$\theta^i = dy^i - y^{i+1}\,dx$ for $i=0,1,2$, then $I^{[2]}$ is
spanned by $\theta^0$ while $I^{[1]}$ is spanned by
$\left\{\theta^0,\theta^1\right\}$ and $I^{[0]}=I$ is spanned by
$\left\{\theta^0,\theta^1,\theta^2\right\}$. The fibration
$\er$ can then be represented locally in these coordinates by
$(x,y^0,y^1,y^2,y^3)\mapsto(x,y^0,y^1)$. Moreover, after
possibly shrinking $U$, the coordinate system can be chosen so
that $x$ restricts to each fiber of $\el$ to become a local
coordinate. It then follows that the $\el$-fibers are the
integral curves of the rank 4 Pfaffian system spanned by the
1-forms $\left\{\theta^0,\theta^1,\theta^2,
dy^3-\Phi\circ\psi\,dx\right\}$ where $\Phi$ is some smooth
function on $\psi(U)\subset\bbR^5$.
\smallskip
It follows that the (smooth) integral curves of $I$ on which
$dx$ is non-zero can be written locally in the form $y^i =
f^{(i)}(x)$ for some (smooth) function $f$, and that the fibers
of $\el$ then correspond to the functions $f$ which satisfy the
fourth order {\sc ode}
$$f^{(4)}=\Phi(x,f,f',f'',f''').\leqno(7)$$
From this point of view, it is clear that the set of curves of
the form $R_x$ which pass through a given point $s\in Y$ is a
two-parameter family which ``fills out'' an open set in the set
of 2-jets of contact curves passing through $s$.
\smallskip
Thus, our discussion so far has shown how to associate a
fourth order {\sc ode} to any amenable torsion-free
$G_3$-structure. This association is almost canonical. The
only non-canonical aspect of our construction is the choice of
local contact coordinates on $N$. However, by the usual
prolongation procedure, these coordinates are determined by
first three of these coordinates, namely $(x,y^0,y^1)$ which
are local coordinates in $Y$. Thus, the {\sc ode} (7) is uniquely
determined up to an action of the contact pseudo-group in
dimension 3. In the classical language (compare Cartan [{\bf
1941}] and Chern [{\bf 1940}]), one says that the equation (7)
is determined up to contact equivalence.
\medskip
For the remainder of this section, we shall discuss the
feasibility of reversing this procedure. In particular, we shall
show that it is possible to reconstruct (at least locally) the
original torsion-free $G_3$-structure from the data of the
fourth order equation. Moreover we shall determine which
(contact equivalence classes of) fourth order equations arise
from torsion-free $G_3$-structures by our construction.
\smallskip
This process may be viewed as part of the program of
``geometrizing'' ordinary differential equations, as proposed by
Cartan [{\bf 1938}]. We refer the interested reader to the
works of Chern and Cartan listed in the references for further
information about the ``geometrization'' of lower order {\sc
ode}.
\smallskip
In order to formulate this reconstruction problem more
precisely, we introduce a few notions from the theory of
pseudo-groups. In $\bbR^5$ with coordinates
$(x,y^0,y^1,y^2,y^3)$, we introduce the pseudo-group
$\Gamma\subset{\rm Diff}_{loc}(\bbR^5)$ which consists of
the local diffeomorphisms of $\bbR^5$ which preserve the
rank 3 Pfaffian system $I_0$ generated by the three 1-forms
$$\theta^0=dy^0-y^1\,dx,\quad\theta^1=dy^1-y^2\,dx,
\quad\theta^2=dy^2-y^3\,dx.$$
Given a 5-manifold $Z$, a {\sl $\Gamma$-structure} on $Z$ is,
by definition, a maximal atlas {\teneur A} of coordinate charts
whose transition diffeomorphisms lie in $\Gamma$.
Associated to such a $\Gamma$-structure on $Z$, there is a
well-defined rank 3 Pfaffian system $I$ which is mapped onto
$I_0$ by any local coordinate chart in {\teneur A}. Of course,
the system $I$ in turn serves to define the $\Gamma$-atlas
{\teneur A} as the set of coordinate charts on $Z$ which
pull-back $I_0$ to $I$, so we may regard the specification of
$I$ as equivalent to the specification of {\teneur A}.
\smallskip
Since $\Gamma$ also preserves the derived systems
$I_0^{[1]}$ and $I_0^{[2]}$ spanned by $\{\theta^0,\theta^1\}$
and $\{\theta^0\}$ respectively, it follows that these also
correspond to well-defined systems $I^{[1]}$ and $I^{[2]}$ on
$Z$ of ranks 2 and 1 respectively. Moreover, $\Gamma$
preserves the Cauchy foliations of $I_0^{[1]}$ and $I_0^{[2]}$
and hence there exist well-defined foliations of codimensions
4 and 3 respectively on any $Z^5$ endowed with a
$\Gamma$-structure. For our purposes, the Cauchy foliation of
$I^{[2]}$ will be the most important. This is the foliation
which maps under any coordinate chart in {\teneur A} to the
simultaneous level sets of the functions $x$, $y^0$, and~$y^1$.
Generalizing our previous case of ${\cal I}_+$ on $N$, we shall
say that a $\Gamma$-structure on $Z$ is {\it amenable} if the
space $Y^3$ of Cauchy leaves of $I^{[2]}$ is Hausdorff.
\smallskip
An integral curve $\gamma$ of $I$ is said to be {\it
admissible\/} if it is transverse to the Cauchy leaves of
$I^{[2]}$. In an {\teneur A}-chart, this means that, when
$\gamma$ is written in the form $\gamma(t) =
\left((x(t),y^0(t),y^1(t),y^2(t),y^3(t)\right)$, the ``reduced''
curve $\bar\gamma(t) = \left( (x(t),y^0(t),y^1(t) \right)$ is
also an immersed curve.
\smallskip
As an example, note that the contact system $I_0$ on the third
order jet space $J^3(\bbR,\bbR)$ endows $J^3(\bbR,\bbR)$
with a canonical $\Gamma$-structure and the solution curves
of a fourth order {\sc ode} describe a foliation of
$J^3(\bbR,\bbR)$ by admissible integral curves of $I_0$. As
another example, the one that concerns us most in this paper,
note that the manifold $N$ constructed from any torsion-free
$G_3$-structure possesses a natural $\Gamma$-structure
defined by the Pfaffian system $I$ as well as a foliation by
admissible integral curves of $I$.
\smallskip
In the case where $Z$ has an amenable $\Gamma$-structure, a
foliation of $Z$ by admissible integral curves of $I$ has a
natural interpretation as a four-parameter family of contact
curves in the space $Y$. This latter family of curves
determines a {\sl contact path geometry\/} on $Y$ in Cartan's
sense. We do not want to limit ourselves to the amenable
case, so we will refer to a 5-manifold $Z$ endowed with a
$\Gamma$-structure and a foliation by admissible integral
curves as a {\sl generalized contact path geometry}.
\medskip
In the remainder of this section, we will describe some of the
local invariants of a generalized contact path geometry. This
description will be applied in the next section to relate a
certain ``twistor space'' to the geometry of torsion-free
$G_3$-structures.
\smallskip
Let $G\subset \GL(5,\bbR)$ denote the subgroup which
consists of Jacobian matrices of elements of $\Gamma$
relative to the coframe
$$\underline\vartheta=
\left(\matrix{\underline\vartheta^0\cr
\underline\vartheta^1\cr
\underline\vartheta^2\cr
\underline\vartheta^3\cr
\underline\vartheta^4\cr}\right)
=
\left(\matrix{6\,\theta^0\cr
6\,\theta^1\cr
3\,\theta^2\cr
dy^3\cr
{\ts{1\over2}}\,dx\cr}\right).\leqno(8)$$
(The choice of constants is cosmetic, but it does help us
compare the structure equations we are about to derive with
the ones we have already derived.) It can be shown that the Lie
algebra of $G$ is the subalgebra $\eug \subset \eugl(5,\bbR)$
consisting of those matrices of the form
$$\left(
\matrix{
m+3f_0 & 0 & 0 & 0 & 0 \cr
a_1+6f_1 & m+f_0 & 0 & 0 & 0 \cr
a_2+2a_4 & a_1+4f_1& m-f_0 & 0 & 0 \cr
a_3 & a_2+a_4 & a_1+2f_1 & m-3f_0 & a_0 \cr
a_6 & a_5 & 0 & 0 & 2f_0 \cr
}
\right)\leqno(9)$$
where $m$, $f_0$, $f_1$, and $a_i$ for $ 0 \le i \le 6 $ are
arbitrary real numbers. Moreover, $G$ consists of two
connected components: the identity component $G^o\subset
\GL(5,\bbR)$, which is a closed subgroup, and the coset
$h\,G^o$ where $h={\rm diag}(-1,-1,-1,-1,\phantom{-}1)$.
(The choice of basis for $\eug$ implicit in the entry labeling in
(9) is also partly for cosmetic reasons and partly for ease of
reference in an argument to be presented below.)
\smallskip
It follows that, associated to every $\Gamma$-structure
{\teneur A} on $Z^5$, there is a unique (first order)
$G$-structure $F\subset \F$ (where $\F$ is the
$\bbR^5$-coframe bundle of $Z$) with the property that
$\phi^\ast(\underline\vartheta)$ is a section of $F$ over
$U\subset Z$ whenever $\phi\colon U\to\bbR^5$ is a local
coordinate system belonging to {\teneur A}. Of course, the
$G$-structure $F$ in turn determines the $\Gamma$-structure
{\teneur A}, so these may be regarded as equivalent.
\smallskip
We define a closed subgroup $P\subset G$ by the condition that
$P = P^o\>\cup\>h\,P^o$ where $P^o$ is the connected Lie
sub-group of $G$ whose Lie algebra $\eup\subset \eug$
consists of the matrices of the form (9) where all of the $a_i$
have been set to zero.
\medskip
{\sl
\noindent{\sc Proposition 4.4:} Let $Z$ be a 5-manifold
endowed with a $\Gamma$-structure and let $F$ be the
corresponding $G$-structure. Let $\cL$ be a foliation of $Z$
by admissible integral curves of $I$. Then there exists a
unique $P$-structure $F_\cL\subset F$ which has the
following properties:
\smallskip
\item{$(i)$} If $\sigma\colon U\to F_\cL$ is any local section
where $U\subset Z$ is open, then
$\sigma=(\sigma^0,\sigma^1,\ldots,\sigma^4)$ has the
property that the leaves of $\cL\cap U$ are integral curves of
the forms $\sigma^0,\sigma^1,\sigma^2,\sigma^3$.\par
\smallskip
\item{$(ii)$} {The structure equations of $F_\cL$ take the
form
$$\def\k{\kappa}\def\l{\lambda}\def\v{\vartheta}
\left(
\matrix{d\,\v^0\cr d\,\v^1\cr d\,\v^2\cr d\,\v^3\cr d\,\v^4\cr}
\right)
=
-\left(
\matrix{
\l+3\,\k_0 & -2\,\v^4 & 0 & 0 & 0 \cr
6\,\k_1 & \l+\k_0 & -4\,\v^4 & 0 & 0 \cr
0 & 4\,\k_1 & \l-\k_0 & -6\,\v^4 & 0 \cr
0 & 0 & 2\,\k_1 & \l-3\,\k_0 & 0 \cr
0 & 0 & 0 & 0 & 2\,\k_0 \cr
}
\right)\wedge
\left(\matrix{\v^0\cr \v^1\cr \v^2\cr \v^3\cr \v^4\cr}\right)
+
\left(\matrix{0 \cr 0 \cr T^2 \cr T^3 \cr T^4 \cr }\right)$$
for some 1-forms $\kappa_0$, $\kappa_1$, and $\lambda$, and
where the $T^i$ are 2-form expressions in the $\vartheta^i$
which satisfy
$$\def\v{\vartheta}
\eqalign{
T^2 &= 2A\,\v^4\wedge\v^0\cr
T^3 &= \v^4\wedge(A\,\v^1+B\,\v^0)
+\v^0\wedge(C\,\v^2+D\,\v^1)\cr
T^4&\equiv\qquad0\qquad{\rm modulo}\quad \v^0,\v^1\cr
}
$$
for some functions $A$, $B$, $C$, and $D$ on $F_\cL$.
}
\smallskip
\noindent Moreover, the 1-forms $\kappa_0$, $\kappa_1$, and
$\lambda$ and the functions $A$, $B$, $C$, and $D$ which
make these structure equations hold are unique.
}
\medskip
\noindent{\sc Proof:} The proof is a calculation which is a
little long but which offers no difficulty (particularly to {\sc
Maple}) and we will only give an outline here.
\smallskip
First, restrict to the sub-bundle $F_0$ of $F$ which satisfies
$(i)$. The sub-bundle $F_0$ is a principal bundle for the
subgroup $G_0\subset G$ whose Lie algebra $\eug_0$ is the
set of matrices of the form (9) for which $a_0=0$ and which
satisfies $G_0 = G_0^o\cup h\,G_0^o$ where $G_0^o$ is the
identity component of $G_0$. The structure reduction by
Cartan's method of equivalence then proceeds in three stages:
\smallskip
The intrinsic torsion of the $G_0$-structure $F_0$ takes
values in a single two dimensional $G_0$-orbit in the Spencer
cohomology group $H^{0,2}(\eug_0)$. There is a point $\tau_0$
on this orbit whose stabilizer $G_1\subset G_0$ is the
subgroup whose Lie algebra $\eug_1$ is the set of matrices of
the form (9) for which $a_0=a_1=a_2=0$ and which satisfies
$G_1 = G_1^o\cup h\,G_1^o$ where $G_1^o$ is the identity
component of $G_1$. We define $F_1\subset F_0$ to be the
inverse image of $\tau_0$ under the intrinsic torsion map of
$F_0$. Then $F_1$ is a principal $G_1$-bundle.
\smallskip
The intrinsic torsion of the $G_1$-structure $F_1$ takes
values in a single three dimensional $G_1$-orbit in the
Spencer cohomology group $H^{0,2}(\eug_1)$. There is a point
$\tau_1$ on this orbit whose stabilizer $G_2\subset G_1$ is
the subgroup whose Lie algebra $\eug_2$ is the set of
matrices of the form (9) for which
$a_0=a_1=a_2=a_3=a_4=a_5=0$ and which satisfies $G_2 =
G_2^o\cup h\,G_2^o$ where $G_2^o$ is the identity component
of $G_2$. We define $F_2\subset F_1$ to be the inverse image
of $\tau_1$ under the intrinsic torsion map of $F_1$. Then
$F_2$ is a principal $G_2$-bundle.
\smallskip
The intrinsic torsion of the $G_2$-structure $F_2$ need not
take values in a single $G_2$-orbit in $H^{0,2}(\eug_2)$.
However, there is a transitive affine action of $G_2$ on
$\bbR$ and an affine mapping $H^{0,2}(\eug_2)\to\bbR$ which
is $G_2$-equivariant and for which the stabilizer of
$0\in\bbR$ is the subgroup $P\subset G_2$ as defined above.
Thus, we set $F_\cL$ equal to the inverse image of $0\in\bbR$
under the reduced intrinsic torsion mapping of $F_2$.
\smallskip
Since, as is easily computed, $\eup^{(1)} = 0$, the connection
described in the statement of the Proposition is unique.
Further details, including that the torsion has the stated form,
will be left to the reader.\hfill\square
\medskip
The reader may wonder why we have stopped the reduction at
the third stage. The reason is that, while all of the reductions
so far have been {\it ageneric\/}, i.e., accomplished without
using any genericity assumptions, any further reductions {\sl
would\/} require some sort of genericity hypotheses. (In the
terminology of Gardner [{\bf 1989}], one says that the method
of equivalence ``branches'' at this point.)
\smallskip
To see why, note that the case where all of the $T^i$ vanish
does occur: The flat $G_3$-structure on $\bbR^4$ gives rise to
a generalized path geometry on $N= \bbR^4 \times \bbR
\bbP^1$ in which all of the $T^i$ vanish. In this case, there is
no canonical reduction of this $F_\cL$, for in fact, the
automorphism group of the flat $G_3$-structure acts
transitively on $F_\cL$.
\smallskip
Any reduction of $F_\cL$ in the general case must therefore be
based on some genericity assumption such as $A\ne0$. We do
not wish to make genericity assumptions since they are
awkward for the twistor theory in the next section.
Fortunately, for our purposes, genericity assumptions will not
be needed anyway.
\medskip
Now, using the structure equations in Proposition 4.4, it is
easy to show that the semi-basic symmetric differential
forms $A\,(\vartheta^4)^3$ and $C\,\vartheta^4\circ
\vartheta^0$ are $P$-invariant and therefore there exist a
well defined symmetric cubic form $\cA$ and a well defined
symmetric quadratic form $\cC$ on $Z$ which pull-back to
$F_\cL$ to become $A\,(\vartheta^4)^3$ and
$C\,\vartheta^4\circ \vartheta^0$ respectively. We shall refer
to $\cA$ and $\cC$ as {\sl primary invariants\/} of the
foliation $\cL$.
\smallskip
If $\cA$ vanishes identically, then the semi-basic quartic
differential form $B\,(\vartheta^4)^4$ is $P$-invariant and
therefore there exists a well defined symmetric quartic form
$\cB$ on $Z$ which pulls-back to $F_\cL$ to become
$B\,(\vartheta^4)^4$. Similarly, if $\cC$ vanishes, then
$D\,\vartheta^4\circ (\vartheta^0)^2$ is $P$-invariant and
therefore there exists a well defined symmetric cubic form
$\cD$ on $Z$ which pulls-back to $F_\cL$ to become
$D\,\vartheta^4\circ (\vartheta^0)^2$. We shall refer to $\cB$
and $\cD$ (when either exists) as {\sl secondary invariants\/}
of the foliation $\cL$.
\smallskip
For the reader who is comparing our treatment with the theory
of contact equivalence of second (respectively, third) order
{\sc ode} as developed in Cartan [{\bf 1938}] (respectively,
Chern [{\bf 1940}]), the functions $A$ and $C$ represent what
the classical papers call ``relative invariants'' and are
analogous to the W\"unschmann invariant encountered in the
third order theory. In more modern language, ``relative
invariants'' are just sections of certain natural line bundles
associated with the geometry. As we shall see below, the
``line bundle'' interpretation is a useful viewpoint.
\smallskip
The invariants $A$, $B$, $C$, and~$D$ can be written out
explicitly for the generalized contact path geometry defined by
the equations (7) as {\sl polynomials} in the derivatives of
$\Phi$ of order less than or equal to three. However, the
explicit expressions are quite complicated. It is useless to
write them out here.
\medskip
{\sl
\noindent{\sc Theorem 4.5:} Let $Z$ be a 5-manifold with $F$
and $\cL$ as in Proposition 4.4. The primary and secondary
invariants of the foliation $\cL$ vanish if and only if the
$P$-structure $F_\cL$ on $Z$ is locally equivalent to the
$P$-structure $F$ on $N$ associated to a torsion-free
$G_3$-structure on some 4-manifold $M$.
}
\medskip
In order to make sense of this theorem, it will be necessary to
first explain how the group $P$ is being identified in the two
cases. In our original definition of $P$, it was the subgroup of
$G_3=\GL(2,\bbR)$ which preserved the subspace
$W_3\subset \V_3$ spanned by $\{x^3,x^2y\}$. Of course, this
subgroup also acts on $\V_2$, and it is easy to see that it
preserves the subspace $W_2\subset \V_2$ spanned by
$\{x^2,xy\}$. If we identify $\bbR^5$ with the vector space
$\V_3\oplus(\V_2/W_2)$ by using the basis
$(x^3,x^2y,xy^2,y^3, y^2\,{\rm mod}\,W_2)$, then $P$ acts on
$\bbR^5$. It is easy to see that embedding $P$ into
$\GL(5,\bbR)$ this way yields the subgroup of $G$ that we
defined in the course of our structure reduction in the proof of
Proposition 4.4.
\medskip
\noindent{\sc Proof:}
First, let us start with a torsion-free $G_3$-structure on $M$
and construct the bundle $F$ together with its connection
forms $\omega$, $\lambda$, and $\phi$. The first structure
equation is $d\,\omega=-\lambda\wedge\omega -
\la\phi,\omega\ra_1$ and can be written explicitly in the
form
$$\def\w{\omega}\def\l{\lambda}\def\f{\phi}
\left(
\matrix{
d\,{\w_3}_{\phantom{-}}\cr d\,{\w_1}_{\phantom{-}}\cr
d\,\w_{-1}\cr d\,\w_{-3}\cr }
\right)
=
-\left(
\matrix{
\l+3\,\f_0 & -2\,\f_2 & 0 & 0 \cr
6\,\f_{-2} & \l+\f_0 & -4\,\f_2 & 0 \cr
0 & 4\,\f_{-2} & \l-\f_0 & -6\,\f_2 \cr
0 & 0 & 2\,\f_{-2} & \l-3\,\f_0\cr }
\right)\wedge
\left(\matrix{ {\w_3}_{\phantom{-}}\cr
{\w_1}_{\phantom{-}}\cr \w_{-1}\cr \w_{-3}\cr }
\right)\leqno(10)$$
while the $y^2$-component of the second structure equation
implies that
$$d\,\phi_2 \equiv -2\,\phi_0\wedge\phi_2
\qquad{\rm modulo}\qquad \omega_3,\omega_1.
\leqno(11)$$
By considering the coframing correspondence
$$(\omega_3,\omega_1,\omega_{-1},\omega_{-3},
\phi_2,\phi_0,\phi_{-2},\lambda)
\longleftrightarrow
(\vartheta^0,\vartheta^1,\vartheta^2,\vartheta^3,
\vartheta^4,\kappa_0,\kappa_1,\lambda),
\leqno(12)$$
it becomes clear that the bundle $\nu\colon F\to N$ is the
$P$-reduction $F_\cL$ of the $G$-structure over $N$ defined
by the system $I$ relative to the foliation $\cL$ of $N$ which
consists of the fibers of $\el\colon N\to M$. Moreover, the
structure equations on $F$ clearly imply that the primary and
secondary invariants vanish for this $P$-structure on $N$.
This establishes the theorem in one direction.
\smallskip
To go in the other direction, let us suppose that the primary
and secondary invariants of $\cL$ vanish. Then on $F_\cL$, we
have $T^2 = T^3 = 0$. It follows that the forms
$$\omega=\vartheta^3\,x^3 +\vartheta^2\,x^2y +
\vartheta^1\,xy^2 +\vartheta^0\,y^3 $$
and
$$\phi = \kappa_1\,x^2 + \kappa_0\,xy + \vartheta^4\,y^2$$
together with $\lambda$ satisfy the first structure equation
for a torsion-free $G_3$-structure. It is now straightforward
to show that, if $U\subset Z$ is an open subset on which the
foliation $\cL$ is amenable, then there is an induced torsion-
free $G_3$-structure $F^\ast$ on the leaf space $M^\ast =
U/\cL$ and that there is a canonical immersion of $F_\cL$
restricted to $U$ into $F^\ast$ which is a local
diffeomorphism and which pulls the canonical forms on
$F^\ast$ back to $F_\cL$ so that the correspondence (12)
becomes an identity. Further details will be left to the reader.
\hfill\square
\bigskip
\bigskip
\centerline{\bf \S5. Twistor Theory and Exotic Holonomy}
\bigskip
In view of Theorem 4.5, one method of constructing
torsion-free $G_3$-structures is to construct a generalized
contact path geometry whose primary and secondary invariants
vanish. Doing this directly is not easy. In terms of the fourth
order {\sc ode} defined by (4.7), these conditions are
equivalent to a (non-involutive) system of non-linear, third
order {\sc pde} for the function $\Phi$. It is not even clear
that there are any non-trivial solutions.
\smallskip
However, for the same geometric problem in the holomorphic
category, there is another approach, based on the Kodaira
deformation theory of complex manifolds and submanifolds,
which is suggested by Penrose's non-linear graviton
construction and its generalization by LeBrun [{\bf 1983}] and
Hitchin [{\bf 1982}]. Basically, the idea of this theory is to
consider the deformation space of a rational curve (i.e., a
compact Riemann surface of genus zero) in a complex manifold
whose normal bundle is a sum of non-negative line bundles.
This deformation space is a smooth manifold consisting of the
``nearby'' rational curves and it inherits a ``geometry'' whose
properties depend on the Grothendieck type of the normal
bundle of the original rational curve.
\smallskip
We shall not attempt a recounting of the full theory, but
instead refer the reader to Hitchin's paper. Also, we shall not
always give full details in the argument below since they are
generally quite analogous to those in the examples treated in
that paper.
\medskip
Our motivation for moving into the holomorphic setting can be
explained as follows: If $F$ is a real-analytic torsion-free
$G_3$-structure on a real 4-manifold $M$, then, at least
locally, $M$ can be embedded as a real slice of a complex
4-manifold $\bcM$ and $F$ can be extended to a
$\cG_3$-structure $\bcF$ where $\cG_3$ is the connected
holomorphic Lie subgroup of ${\rm Aut}(\V_3\otimes\bbC)$
whose Lie algebra is the complexification of $\eug_3$. The
structure equations of this ``holomorphicized'' structure are
analogous to those in the real category in every way. In
particular, in the amenable case (which can always be arranged
locally by passing to a suitably small neighborhood), the
holomorphic version of Proposition 4.3 holds where the double
fibration (4.5) is replaced by a ``complexified'' version in
which the general fiber of the mapping $\el$ is a copy of
$\bbP^1$. The sheet space $\bcY$ then becomes a complex
contact 3-fold and the corresponding complexified curves
$\bcR_x$ are rational contact curves in the complex contact
3-fold $\bcY$.
\smallskip
Thus, the points of $\bcM$ can be thought of as rational
contact curves in $\bcY$. It is not true that $\bcM$
constitutes the complete deformation space of ``nearby''
rational curves, as in the cases treated previously by Penrose
and Hitchin, but, as we shall see, it does constitute the
complete deformation space of ``nearby'' rational {\sl
contact\/} curves.
\smallskip
We will need some information about the deformations of an
unramified rational contact curve in a general complex contact
3-fold. For the sake of avoiding any possible confusion, let us
note that, by definition, an {\sl unramified rational curve\/} in
a complex manifold $\bcY$ is an equivalence class of
holomorphic immersions $\phi\colon \bbP^1\to\bcY$ where the
equivalence relation is reparametrization in the domain, i.e.,
$\phi_1\sim\phi_2$ iff $\phi_1=\phi_2\circ\psi$ where
$\psi\colon \bbP^1\to\bbP^1$ is a linear fractional
transformation. It is well-known that such an equivalence
class is determined completely by its image $C\subset\bcY$,
so, by abuse of language, we shall speak of $C$ as a rational
curve. To say that $C$ is embedded means that $C$ is the
image of a {\sl one-to-one\/} holomorphic immersion
$\phi\colon \bbP^1\to\bcY$.
\smallskip
The following proposition is derivable by standard
``twistorial'' techniques, but we give a more
deformation-theoretic proof since we will need the
information that this proof provides about local coordinates in
the moduli space.
\medskip
{\sl
\noindent{\sc Proposition 5.1:} Let $\bcY$ denote a complex
contact 3-fold where $L\subset T^\ast\bcY$ denotes the
holomorphic line bundle which defines the contact structure on
$\bcY$. Suppose that $C\subset \bcY$ is an unramified
rational contact curve in $\bcY$ and that the restriction of $L$
to $C$ is isomorphic to $\cO(-k-1)$ for some integer $k\ge0$.
Then then the normal bundle of $C$ in $\bcY$ is isomorphic to
$\cO(k)\oplus\cO(k)$. In particular, the moduli space $\bcZ$
of unramified rational curves in $\bcY$ is smooth and of
complex dimension $2k+2$ near $C$. Moreover, the subspace
$\bcM_k\subset \bcZ$ which consists of contact curves to
which the contact bundle $L$ restricts to be isomorphic to
$\cO(-k-1)$ is a smooth submanifold of $\bcZ$ of complex
dimension $k+2$.
}
\medskip
\noindent{\sc Proof:} First, we will recall a few facts about
complex contact geometry.
\smallskip
If $L\subset T^\ast\bcY$ is the line bundle determining the
contact structure, then $L^2\simeq K_\bcY$. To see this, note
that the mapping $D\colon\cO(L)\to\cO(K_\bcY)$ defined by
$D(\theta)=\theta\wedge d\,\theta$ for any local holomorphic
section $\theta$ of $L$ satisfies $D(f\theta)=f^2D(\theta)$ for
any local holomorphic function $f$ and thus induces a
well-defined sheaf mapping ${\cal D}\colon L^2\to K_\bcY$.
This is an isomorphism since the hypothesis that $L$ defines a
contact structure on $\bcY$ implies that $D(\theta)$ is
non-zero wherever $\theta$ is non-zero. Now, if
$L^\perp\subset T\bcY$ is the rank two vector bundle
annihilated by the sections of $L$, then standard arguments
combined with the above isomorphism imply that
$\Lambda^2(L^\perp)\simeq L^\ast$. Note also that we have a
canonical short exact sequence of vector bundles
\def\lto{\longrightarrow}
$$0\lto L^\perp\lto T\lto L^\ast\lto0\leqno(1)$$
where $T$ is the holomorphic tangent bundle of $\bcY$.
\medskip
Now, by hypothesis, $C\simeq \bbP^1$ is unramified, so the
natural sequence
$$0\lto \tau\lto T_{| C}\lto N_C\lto0$$
is exact where $\tau\simeq\cO(2)$ is the tangent bundle of
$C$ and $N_C$ is the normal bundle of the immersion. (We
caution the reader that, in algebraic geometry, the normal
bundle is often defined as a certain sheaf theoretic quotient.
This alternate definition can disagree with $N_C$ as we have
defined it if $C$ is not embedded.) Moreover, since $C$ is a
contact curve, we have an exact inclusion $0\to\tau\to
L^\perp_{|C}$. It follows that
$$ \tau\otimes(L^\perp_{|C}/\tau)\simeq
\Lambda^2(L^\perp_{| C})\simeq
L^\ast_{|C}\simeq \cO(k+1).$$
Thus, we must have $L^\perp_{|C}/\tau\simeq
\tau^\ast\otimes \cO(k+1)\simeq \cO(k-1)$. From (1) we get
the exactness of the sequence
$$0\lto L^\perp_{|C}/\tau\lto N_C\lto
L^\ast_{|C}\lto0\leqno(2)$$
and this implies that the normal bundle of $C$ fits into an
exact sequence of the form
$$0\lto \cO(k-1)\lto N_C\lto \cO(k+1)\lto0.$$
By standard arguments, it then follows that $N_C$ is
isomorphic to either $\cO(k-1)\oplus\cO(k+1)$ or
$\cO(k)\oplus\cO(k)$ according to whether the exact sequence
(2) does or does not split. In either case, since $k\ge0$ by
hypothesis, we have
$$\eqalign{h^0(N_C)&=2k+2,\cr
h^1(N_C)&=\quad0.\cr}
\leqno(3)$$
If $C$ is embedded, then Kodaira's Main Theorem in Kodaira
[{\bf 1962}], hereinafter referred to as KMT, implies that the
moduli space $\bcZ$ of rational curves in $\bcY$ is smooth and
of dimension $2k+2$ near $C$. The case where $C$ is not
embedded is easily reduced to the embedded case by
``separating the crossings of a tubular neighborhood of $C$''.
More precisely, let $\nu\colon N_C\to\bcY$ be the normal
exponential mapping with respect to any smooth metric on
$\bcY$ and let $\tilde\bcY\subset N_C$ denote a neighborhood
of the zero section which maps locally diffeomorphically into
$\bcY$. Give $\tilde\bcY$ the holomorphic contact structure
which makes $\nu$ a local contact biholomorphism. Note that
the zero section in $\tilde\bcY$ is an embedded rational
contact curve $\tilde C$. If $\tilde\bcZ$ denotes the space of
unramified rational curves in $\bcY$, then $\tilde\bcZ$ is
smooth of complex dimension $2k+2$ near $\tilde C$ and $\nu$
clearly induces a biholomorphism of a neighborhood of $\tilde
C$ in $\tilde\bcZ$ with a neighborhood of $C$ in $Z$.
\smallskip
In view of this, we shall assume for the remainder of the proof
that $C$ is embedded.
\smallskip
We will now identify the subspace of $\bcZ$ near $C$ which
consists of contact curves. To do this, we first observe that
for any curve $C'\subset \bcY$, there is a natural pairing
$L_{|C'}\times\tau_{C'}\to\bbC$ which is the restriction to the
given bundles of the natural pairing of co-vectors with
vectors. We may regard this pairing as a section $\sigma_{C'}$
of the bundle $L^\ast_{|C'}\times\tau^\ast_{C'}$ and note that
this section vanishes if and only if $C'$ is a contact curve. In
the case where $C'$ is a rational curve and
$L_{|C'}\simeq\cO(-k-1)$, it follows that $\sigma_{C'}$ is a
section of a line bundle isomorphic to $\cO(k-1)$ and hence
vanishes identically if and only if it vanishes to order $k$ at
some point on $C'$.
\smallskip
The rest of our argument requires some slight notational
changes in the case $k=0$, so from now on, we will assume
that $k\geq 1$, leaving for the reader the task of making those
changes needed to make the argument go through for $k=0$. (If
$k=0$, then $\sigma_{C'}\equiv0$ anyway since it is a section
of a line bundle of negative degree. Thus, it is clear that
$\bcM_0 = \bcZ$ in a neighborhood of such a $C$.)
\smallskip
Let $p\in C$ be fixed. Since $C$ is embedded, it follows that
we may suppose that there is a neighborhood $U$ of $p$ so that
$C\cap U$ consists of a single, analytically irreducible branch
of $C$. Moreover, by using a slight extension of the
Pfaff-Darboux theorem in the analytic category, we see that it
is possible to choose a $p$-centered holomorphic coordinate
system $(x,y,z)$ on a $p$-neighborhood $U\subset\bcY$ in
which $U\cap C$ is described by the equations $y=z=0$ and so
that the 1-form $\theta=dy - z\,dx\,$ is a section of $L_{|U}$.
(The point is that we can choose the Pfaff coordinates so that
$C\cap U$ is described in such a simple fashion. To see that
this can be done, first choose the $p$-centered coordinate
system $(x,y,z)$ so that $\theta=dy - z\,dx\,$ is a
section of $L_{|U}$ and so that $dx$ is a non-vanishing 1-form
when pulled back to $C\cap U$. Then, since $C$ is a contact
curve, it is easy to see that, by shrinking $U$ if necessary, we
may suppose that there exists a holomorphic function $f$ on a
neighborhood of $0\in \bbC$ so that $C\cap U$ is described by
the equations $y=f(x)$ and $z=f'(x)$. Replacing the coordinate
system $(x,y,z)$ by $\bigl(x,y-f(x),z-f'(x)\bigr)$ (and again
possibly shrinking $U$), we get the desired new coordinate
system.)
\smallskip
By Kodaira's description of the moduli space $\bcZ$ near $C$,
there exist holomorphic functions $y_j,z_j$ for $j\ge0$ on a
$C$-neighborhood $\cU\subset\bcZ$ so that, for $C'\in\cU$,
we have $L_{|C'}\simeq\cO(-k-1)$ and the defining equations
of $U\cap C'$ take the form
$$\eqalign{y&=y_0(C')+y_1(C')x+\cdots+y_j(C')x^j+\cdots\cr
z&=z_0(C')+z_1(C')x+\cdots+z_j(C')x^j+\cdots\cr}
\leqno(4)$$
\smallskip
We want to show that it is possible to select a specific set of
$(2k+2)$ of the functions $y_j,z_j$ in such a way that, after
possibly shrinking $\cU$, this set forms a $C$-centered
coordinate system on $\cU$. In order to do this, we need to
use our information about the normal bundle of $C$. By KMT, if
$\{\nu_\alpha |\;1\le\alpha\le{2k+2}\;\}$ is any basis for the
global sections of $N_C$, then there exists a $C$-centered
local coordinate system $(t^1,t^2,\ldots,t^{2k+2})$ on $\cU$ so
that, with respect to the local coordinate trivialization of
$N_C$ over $C\cap U$ determined by $(x,y,z)$, we have
$$\nu_\alpha =
\biggl(\sum_{j\ge0}{{\partial y_j}\over{\partial
t^\alpha}}(C)\;x^j\biggr)
\left[ {{\partial}\over{\partial y}}\right]
+\biggl(\sum_{j\ge0}{{\partial z_j}\over{\partial
t^\alpha}}(C)\;x^j\biggr)
\left[{{\partial}\over{\partial z}}\right]\leqno(5)$$
for all $\alpha$, where $[\ ]$ represents reduction modulo the
sub-bundle spanned by the vector field $\partial/\partial x$
which spans $\tau$.
\smallskip
Now let us first suppose that $N_C\simeq\cO(k)\oplus\cO(k)$.
Then it is clear that there is a basis $\{\nu_\alpha\,|
\,1\le\alpha\le{2k+2}\,\}$ for the global sections of $N_C$
which is expressed in the coordinate trivialization given above
in the form
$$\eqalign{\nu_{j+1\phantom{+k}} &=
x^j \left[{{\partial}\over{\partial y}}\right]
+O(x^{k+1})\cr
\nu_{j+k+2} &=
x^j \left[{{\partial}\over{\partial z}}\right]
+O(x^{k+1})\cr}\qquad {\rm for}\quad0\le j\le k\leqno(6)$$
where the symbol $O(x^{k+1})$ denotes terms which vanish to
order $(k+1)$ at $x=0$. Comparing (5) and (6), we see that the
Jacobian matrix of the functions $y_0,y_1,\ldots,y_k,
z_0,z_1,\ldots, z_k$ with respect to the coordinate system
$(t^1,\ldots,t^{2k+2})$ is the identity matrix at $C$. By the
Implicit Function Theorem, it is possible to shrink $\cU$ so
that $(y_0,y_1,\ldots,y_k, z_0,z_1,\ldots, z_k)$ forms a
$C$-centered holomorphic coordinate system on $\cU$.
\smallskip
Alternatively, let us suppose that
$N_C\simeq\cO(k-1)\oplus\cO(k+1)$. Then the sequence $(2)$
is canonically split, i.e., there exists a unique holomorphic line
bundle $N'_C\subset N_C$ which projects isomorphically onto
$L^\ast_{C}$ in the sequence $(2)$. Moreover, $N'_C$ must be
everywhere transverse to $L^\perp_{|C}/\tau$. Since
$L^\perp_{|C}\cap U$ is clearly spanned by the vector fields
$\{\partial/\partial x,\partial/\partial z\}$, it follows that
there exists a holomorphic function $f(x)$ for $x$ near
$0\in\bbC$, so that the expression $[\partial/\partial y +
f(x)\,\partial/\partial z]$ is a local section of $N'_C$ over
$C\cap U$. Now, replacing the coordinate system $(x,y,z)$ by
the system $(x,g(x)y,g(x)z+g'(x)y)$ where $g$ is the
holomorphic function which satisfies $g(0)=1$ and
$g'(x)+f(x)g(x)=0$, we retain all of our earlier normalizations
and get a new coordinate system in which the coordinate
trivialization of $N'_C$ over $C\cap U$ is spanned by
$[\partial/\partial y]$. It is now easy to choose a basis
$\{\nu_\alpha\,|\,1\le\alpha\le{2k+2} \}$ for the global
sections of $N_C$ which are expressed in the local coordinate
trivialization in the form
$$\eqalign{\nu_{j+1\phantom{+3}} &=
x^j \left[{{\partial}\over{\partial y}}\right]
+O(x^{k+2})
\qquad {\rm for}\quad0\le j\le{k+1},\cr
\nu_{j+k+3} &=
x^j \left[{{\partial}\over{\partial z}}\right]
+O(x^{k\phantom{+2}})
\qquad {\rm for}\quad0\le j\le{k-1},\cr}$$
Again, an application of the Implicit Function Theorem shows
that it is possible to shrink $\cU$ so that the functions
$y_0,y_1,\ldots,y_{k+1}, z_0,z_1,\ldots, z_{k-1}$ form a
$C$-centered holomorphic coordinate system on $\cU$.
\smallskip
This leads us to the following important observation: {\sl In
either case, the functions $y_0,y_1,\ldots,y_k, z_0,z_1,\ldots,
z_{k-1}$ have linearly independent differentials on a
neighborhood of $C\in\cU$.}
\smallskip
Since the bundle $L$ has been trivialized over $U$, it follows
that for $C'\in\cU$,the section $\sigma_{C'}$ has the local
expression
$$\eqalign{\sigma_{C'} &= (dy - z\,dx)_{|C'}\cr
&= \left(
(y_1(C')-z_0(C'))
+ \cdots + (k\,y_k(C')-z_{k-1}(C'))x^{k-1} + \cdots
\right)\,dx.\cr}
\leqno(6)$$
Because the functions $y_0,y_1,\ldots,y_k, z_0,z_1,\ldots,
z_{k-1}$ have linearly independent differentials at $C$, the
locus $\bcM_\cU$ defined by the equations $z_{j-1} - j\,y_j =
0$ for $1\le j \le k$ is a smooth submanifold of $\bcZ$ of
codimension $k$ in a neighborhood of $C$. Now note that any
$C'$ in $\bcM_\cU$ has $\sigma_{C'}$ vanishing to order $k$ at
at least one point. Since, for $C'\in\cU$, we know that
$\sigma_{C'}$ takes values in a line bundle isomorphic to
$\cO(k-1)$, it follows that any $C'\in \bcM_\cU$ must have
$\sigma_{C'}\equiv0$ and hence is a contact curve. It follows
that $\bcM_\cU$ is equal to $\bcM_k\cap \cU$ and is smooth
of dimension $k+2$.
\smallskip
To finish the proof, we must rule out the possibility that the
sequence (2) splits. By KMT, if the functions
$y_0,y_1,\ldots,y_k,z_0,z_1,\ldots, z_k$ have linearly
independent differentials at $C\in\cU$, then $N_C$ is
isomorphic to $\cO(k)\oplus\cO(k)$. Thus, we proceed to show
this independence.
\smallskip
We have seen that $\bcM_k\cap\cU$ is defined as an analytic
subvariety of $\cU$ by the vanishing of the (independent)
functions $f_{j-1} = z_{j-1} - j\,y_j$ for $1\le j\le k$.
Moreover, since $dy-z\,dx$ vanishes identically on any contact
curve, it follows that $f_k = z_k - (k+1)\,y_{k+1}$ also
vanishes on $\bcM_k\cap\cU$. Since the functions
$f_0,\ldots,f_{k-1}$ are a regular defining ideal for $\bcM_k$
near $C$, the function $f_k$ must be in the ideal defined by
these functions so
$$f_k = a^0\,f_0+a^1\,f_1+\cdots+a^{k-1}\,f_{k-1}$$
for some local holomorphic functions $a^j$. It follows that
{\it at $C$}, we have a relation of the form
$$dz_k = (k+1)\,dy_{k+1} + a^0\,(dz_0-dy_1)+\cdots+
a^{k-1}\,(dz_{k-1}-k\,dy_k)$$
which implies that {\it at $C$}, we have
$$dz_k\equiv(k+1)\,dy_{k+1}\quad{\rm modulo}\quad
dy_0, dy_1,\ldots,dy_k,dz_0, dz_1,\ldots, dz_{k-1}$$
Thus, the functions $y_0,y_1,\ldots,y_k,z_0,z_1,\ldots,z_k$
have linearly independent differentials at $C$ if and only if the
functions $y_0,y_1,\ldots,y_{k+1},z_0,z_1,\ldots,z_{k-1}$
have linearly independent differentials at $C$. Since, by our
above arguments, at least one of these two sets of functions
must have linearly independent differentials at $C$, it follows
that they both do. In particular, the first set does, as we
wished to show. \hfill\square
\medskip
In this paper, the important case is $k=2$ for then we have the
following theorem:
\medskip
{\sl
\noindent{\sc Theorem 5.2:} Let $\bcY$ be a contact 3-fold and
let $\bcM$ be the moduli space consisting of those rational
contact curves $C\subset \bcY$ which satisfy
$L_{|C}\simeq\cO(-3)$. Then $\bcM$ is a smooth manifold of
dimension 4. If we let $\bcN\subset\bcM\times\bcY$ be the
incidence submanifold consisting of pairs $(C,p)\in
\bcM\times\bcY$ such that $p\in C$, and let
$\el\colon\bcN\to\bcM$ and $\er\colon\bcN\to\bcY$ denote
the projections onto the first and second factors, then the
diagram
$$
\matrix{
&&\bcN\hskip-.5em&&\cr
&\raise.5em\hbox{$\el$}\hskip-.5em\swarrow\hskip-.5em&
&\searrow\hskip-.5em\raise.5em\hbox{$\er$}\hskip-1em&\cr
\bcM\hskip-1em&&&&\bcY\cr }
$$
is a non-degenerate double fibration which is a generalized
contact path geometry. Moreover, the primary and secondary
invariants of this geometry vanish, so that there is induced on
$\bcM$ a torsion-free $\cG_3$-structure.
\smallskip
Moreover, every real-analytic torsion-free $G_3$-structure on
a 4-manifold $M$ is locally a real slice of a torsion-free
$\cG_3$-structure which arises in the above manner.
}
\medskip
\noindent{\sc Proof:} The smoothness of $\bcM$ has already
been demonstrated. In order to show the non-degeneracy of the
double fibration, we resort to local coordinates.
\smallskip
FIrst, we show that we can take local coordinates of a
particularly simple kind. Let $(C,p)\in \bcN$ be fixed. Let
$U\subset \bcY$ be a $p$-neighborhood on which there exist
$p$-centered coordinates $(x,y,z)$ as in the proof of
Proposition 5.1. In particular, the bundle $L_{|U}$ has
$\theta=dy - z\,dx$ as a trivializing section and $C\cap U$ is
described by the equations $y=z=0$. Let $\cU\subset \bcM$ be
a $C$-neighborhood with the property that $C'\cap U \ne
\emptyset$ for all $C'\in\cU$ and so that there exist functions
$y_j,z_j$ so that the defining equations of $C'\cap U$ are
given by (4) as before. Since the normal bundle of $C$ in
$\bcY$ is $\cO(2)\oplus\cO(2)$, it follows that the functions
$y_0,y_1,y_2,z_0,z_1,z_2$ contain four functions which form
a coordinate system on a neighborhood of $C\in\bcM$.
However, $\bcM$ is defined as a locus in $\bcZ$ by the
equations $z_0=y_1=z_1-2y_2=0$, so it follows that the set
$y_0,y_1,y_2,z_2$ must be a local coordinate system on $\cU$
once $\cU$ has been shrunk sufficiently. Since we also have
$z_{j-1} = j\,y_j$ for all $j\ge1$, it follows that we can
equally well take $y_0,y_1,y_2,y_3$ as our local coordinate
system near $C$, so we do so.
\smallskip
If we let $F$ denote the function of $(x,y_0,y_1,y_2,y_3)$
which is represented by the series on the right hand side of the
first equation in (4), then it follows that $\bcN\cap(\cU\times
U)$ is defined in $(\cU\times U)$ by the equations
$$y-F=z-{{\partial F}/{\partial x}}=0.$$
By shrinking $\cU$ and $U$ if necessary, we may suppose that
the functions $x$, $p^0=F$, $p^1={{\partial F}/{\partial x}}$,
$p^2={{\partial^2 F}/{\partial x^2}}$, and $p^3={{\partial^3
F}/{\partial x^3}}$ form a local coordinate system
$\psi=(x,p^0,p^1,p^2,p^3)$ on $\bcN\cap(\cU\times U)$. In
particular, there is a function $\Phi$ defined on a neighborhood
of $0\in\bbC^5$ for which we have
$${{\partial^4 F}/{\partial x^4}}=\Phi(x,p^0,p^1,p^2,p^3).$$
It follows easily that the semi-basic forms for the projection
$\el$ are spanned by $\{dp^0-p^1\,dx,\,dp^1-p^2\,\,dx,dp^2-
p^3\,dx,\,dp^3-\Phi\circ\psi\,dx\}$ while the semi-basic
forms for the projection $\er$ are spanned by
$\{dx,\,dp^0,\,dp^1\}$.
\smallskip
Since the point $(C,p)\in\bcN$ was chosen arbitrarily, it
immediately follows that the double fibration is
non-degenerate and represents a generalized contact path
geometry where the $\Gamma$-structure is defined locally by
the Pfaffian system $\{dp^0-p^1\,dx,\,dp^1-p^2\,dx,\,dp^2 -
p^3\,dx\}$ and the fibers of $\el$ give the foliation by
admissible integral curves of this system.
\smallskip
Now, we must show that the primary and secondary invariants
of the foliation by $\el$-fibers vanish on $\bcN$. First, by the
structure equations and the remarks in \S4, we know that
$\cA$ is a holomorphic section of a line bundle on $\bcN$ and
restricts to each fiber of $\el$ to be a section of the third
power of the canonical bundle of the fiber. Since these fibers
are rational curves, there are no non-zero holomorphic
sections of any positive power of the canonical bundle. Thus,
$\cA = 0$. The secondary invariant $\cB$ is now
well-defined and restricts to each fiber of $\el$ to be a
holomorphic section of the fourth power of the canonical
bundle. Thus, $\cB = 0$.
\smallskip
Let ${\hat L}$ denote the pull-back of the line bundle $L$ to
$\bcN$. Note that $\cC$ is a holomorphic section of a line
bundle on $\bcN$ and that this line bundle restricts to each
fiber of $\el$ to be a holomorphic section of the tensor
product of ${\hat L}$ and the canonical bundle of the fiber.
Now ${\hat L}$ restricts to each fiber of $\el$ to be
isomorphic to $\cO(-3)$, so it follows that $\cC$ is a section
of a line bundle isomorphic to $\cO(-5)$. Thus, $\cC = 0$. In
particular, $\cD$ is well-defined. Finally, arguing as above,
$\cD$ restricts to each fiber of $\el$ to be a section of a line
bundle isomorphic to $\cO(-7)$. Thus, $\cD = 0$.
\smallskip
Of course, by the holomorphic version of Theorem 4.5, we now
have the desired torsion-free $\cG_3$-structure on $\bcM$.
\smallskip
The proof of the final statement of the theorem, that every
real-analytic torsion-free $G_3$-structure is a real slice of a
torsion-free $\cG_3$-structure arising in the above manner,
proceeds as expected, except for one point which we will now
explain.
\smallskip
If we suppose that the torsion-free $G_3$-structure $F$ on
$M$ is real analytic in appropriate local coordinate systems,
then by restricting to sufficiently small open sets in $M$, we
may suppose that $M$ is the real slice of a complex manifold
$\bcM$ endowed with a real structure (i.e., an
anti-holomorphic involution whose fixed point set is $M$) and
that $F$ is the real slice of an appropriate $\cG_3$-structure
$\bcF$ on $\bcM$. It easily follows that $\bcF$ is
torsion-free and, moreover, by restricting to an even smaller
open set in $M$ if necessary, we may assume that $\bcF$ is
amenable, so that we get a double fibration as in the
statement of the theorem. {\sl Note that we only have to
localize in $M$.} In particular, it follows that $\bcY$ is a
complex contact 3-fold and that each of the curves $\bcR_x$
for $x\in\bcM$ is a rational contact curve.
\smallskip
To finish the proof, we need to see why the restriction of the
contact bundle $L$ to $\bcR_x$ is isomorphic to $\cO(-3)$.
However, this follows directly from the complex versions of
the structure equations $(ii)$ of Proposition 4.4. \hfill \square
\medskip
Of course, it is possible to describe the $\cG_3$-structure on
$\bcM$ directly in terms of the incidence geometry of the
family of rational curves in $\bcY$. For example, the set of
curves in $\bcM$ which meet a given curve $C\in\bcM$
constitutes the set of points which are ``null-separated'' from
$C$ in the sense determined by the conformal quartic form on
$\bcM$ which determines the $\cG_3$-structure. The set of
curves in $\bcM$ which are tangent to a given curve
$C\in\bcM$ constitutes the set of points which are
``null-separated'' from $C$ in the sense determined by the field
of rational normal cones on $\bcM$ which determines the
$\cG_3$-structure (see \S4). In particular, the geodesics in
$\bcM$ whose tangent vectors are ``perfect cubes'' are in
one-to-one correspondence with the 1-parameter families of
rational curves in $\bcY$ which pass through a given point in a
given direction.
\smallskip
A direct proof that this $\cG_3$-structure is torsion-free can
be based on the holomorphic analogue of Theorem 4.1, using the
fact that the 2-parameter subfamily of curves which pass
through a fixed point in $\bcY$ describes a null surface in
$\bcM$.
\medskip
We now come to the interesting question of how useful
Theorem 5.2 is in proving existence, constructing examples, or
discussing the generality of torsion-free $\cG_3$-structures
on complex 4-folds $\bcM$.
\medskip
First, we show that Theorem 5.2 is not vacuous. We will
describe a class of complex contact 3-folds which contain
rational contact curves to which $L$ restricts to become
isomorphic to $\cO(-3)$.
\smallskip
Let $\bcS$ be a complex surface and let $\bcY=\bbP(T\bcS)$ be
the projectivized holomorphic tangent bundle of $\bcS$. It is
easy to show that $\bcY$ has a canonical contact line bundle
$L\subset T^\ast\bcY$ defined as follows: Let $\ell\subset
T_s\bcS$ be a line and define $L_\ell\subset T^\ast_\ell\bcY$
to be the line $\pi^\ast(\ell^\perp)$ where
$\pi\colon\bcY\to\bcS$ is the base-point projection and
$\ell^\perp\subset T^\ast_s\bcS$ is the annihilator of $\ell$.
\smallskip
Any curve $C\subset \bcS$ has a canonical lift to $\bcY$ as a
contact curve; this lift being defined by sending $p\in C$ to
$T_p C$. (If $C$ is ramified, this lifting extends
holomorphically across the ramification points. In fact, it
``resolves'' simple cusps.) Conversely, it is easy to show that
every irreducible contact curve in $\bcY$ is either a fiber of
$\pi$ or else is the canonical lift of an irreducible curve in
$\bcS$.
\smallskip
From our description, it is clear that if $C\subset \bcS$ is an
unramified rational curve with normal bundle $\cO(k+1)$, then,
as a curve in $\bcY$, it satisfies $L_{|C}\simeq\cO(-k-1)$. If,
in addition, $k\ge0$, then Proposition 5.1 shows that $C$
belongs to a $(k+2)$-parameter family of rational contact
curves. Of course, the projection of these curves back into
$\bcS$ agrees with the moduli space of curves near $C$ in
$\bcS$.
\smallskip
Taking $k=2$ in this construction, we obtain examples of the
desired contact 3-folds.
\medskip
A specific example is obtained by letting $\bcS=\cO(3)$, i.e.,
regarding the bundle $\cO(3)$ over $\bbP^1$ as a surface.
Then, taking $C$ to be the zero-section of $\cO(3)$, we get a
rational curve $C$ with the normal bundle $\cO(3)$. However,
this example is not too interesting because the moduli space
$\bcM$ of nearby contact curves in this case is just the space
of global sections of $\cO(3)$ itself, which forms a vector
space of dimension 4. The corresponding $\cG_3$-structure is
easily seen to be the flat one on $\bbC^4$.
\smallskip
A more interesting class of examples is to take a rational
curve $C$ in $\bbP^2$ of degree $d\ge2$ whose only
singularities are $D=(d-1)(d-2)/2$ distinct nodes. (This is
true for the generic rational curve of degree $d$ in $\bbP^2$.)
The nearby rational curves form a $(3d-1)$-parameter family.
Select a set $S$ consisting of $s$ smooth points on $C$ and a
set $N$ consisting of $n$ nodes on $C$ where $s+2n = 3d-5$.
Let $\bcS$ be the surface got from $\bbP^2$ by blowing up the
points in $S\cup N$. Then the normal bundle of $C$ as a curve
in $\bcS$ is isomorphic to $\cO(3)$. It follows that the
moduli space of nearby rational curves in $\bcS$ carries a
canonical torsion-free $\cG_3$-structure.
\smallskip
The specific case of a non-singular conic in $\bbP^2$ with one
point blown up yields the 4-parameter family of conics
passing through a single point. Thus, this family of curves can
be described in the form
$$(x+p_3)y + p_2\,x^2 + p_1\,x + p_0 = 0.$$
The corresponding fourth order differential equation is easily
seen to be $y^{(4)} = {4\over3}(y''')^2/y''$. It is not difficult to
show that the corresponding $\cG_3$-structure is not flat. In
fact, the holonomy is $\cH_3$ and the $\cH_3$-structure is
(the complexification of) the unique non-flat homogeneous
structure $F^0$ discussed in \S3.
\smallskip
It is an interesting question as to how general a
$\cG_3$-structure can be constructed by varying $d$ and the
sets $S$ and $N$. Not all of these are distinct since, for
example, the conic with one point blown up and the nodal cubic
with its node and two smooth points blown up give rise to
isomorphic $\cG_3$-structures. More generally, any two such
4-parameter families of rational curves which differ by a
Cremona transformation of $\bbP^2$ will give rise to
isomorphic $\cG_3$-structures.
\medskip
Finally, once one example of a rational contact curve $C$ with
normal bundle $N_C\simeq\cO(2)\oplus\cO(2)$ in a contact
manifold $\bcY$ has been constructed, the Spencer-Kodaira
theory of deformation of pseudo-group structures may be
applied. Thus, one expects to be able to construct more
examples by deforming a neighborhood of $C$ in $\bcY$ as a
complex contact manifold. In fact, Kodaira [{\bf 1960}] shows
that the tangent space to the moduli of nearby complex contact
structures on a tubular neighborhood $U$ of $C$ in $\bcY$ is
given by $H^1(U,\Theta_{c})$ where $\Theta_{c}\subset
\Theta$ is the sheaf of contact vector fields, i.e., the vector
fields whose infinitesimal flows preserve the bundle $L$.
\smallskip
Now, on any complex contact manifold $\bcY$ with contact line
bundle $L\subset T^\ast\bcY$, the sheaf of contact vector
fields is isomorphic over $\bbC$ to the sheaf of sections of
$L^\ast$. The reason for this is that the sheaf of
$\cO$-modules
$$0\lto L^\perp\lto T\lto L^\ast\lto0\leqno(7)$$
has a $\bbC$-splitting given by a first order differential
operator $D_0\colon L^\ast\to T$ with the property that a
vector field $X$ on $U\subset\bcY$ is a local contact vector
field if and only if $X = D_0([X])$ where $[\,]$ denotes
reduction modulo $L^\perp$.
\smallskip
Since $L^\ast$ restricts to $C$ to be isomorphic to $\cO(3)$,
it follows that, if $U$ is a sufficiently small tubular
neighborhood of $C$ in $\bcY$, then $H^1(U,\Theta_{c})$ will be
``large''. Thus, one expects there to be many non-trivial local
contact deformations of this neighborhood.
\smallskip
It is not hard to see that under small deformations of $U$, the
4-parameter family of rational contact curves persists.
Essentially, this is because the normal bundle of $C$ is
positive, see Kodaira [{\bf 1963}]. However, some care must
be taken in this argument since we want {\sl contact} curves,
so just Kodaira stability is not quite sufficient.
\medskip
We shall not go into any detail about this deformation theory
here because we have already seen by other methods in \S3
that the general local real analytic solutions in the real
category depend on four arbitrary analytic functions of three
variables. It would be interesting to see whether there is a
direct relation between the description of the generality of
local solutions via the Cartan-K\"ahler theory and the
description of the same ``moduli space'' by Kodaira-Spencer
deformation theory.
\bigskip
\centerline{\bf \S6. Epilogue}
\medskip
The function of this short last section is to collect together
some of the open problems and questions which this
investigation has raised.
\medskip
The fundamental underlying problem from \S1 is, of course,
the classification of the possible holonomy groups of affine
symmetric connections. In particular, what are the remaining
exotic affine holonomies? Perhaps if we knew some of the
methods which enter into the proof of Berger's Theorem 4, we
could determine a finite inclusive list and then examine each
one separately.
\medskip
The methods of \S3 give no information about degenerate
torsion-free $G_3$-structures. Since the symmetry group of a
non-degenerate torsion-free $G_3$-structure is easily seen to
be discrete, it follows that Theorem 3.3 can say nothing about
possible homogeneous examples. Indeed, we do not know if any
homogeneous examples exist other than the flat structure or
the unique homogeneous example with holonomy $H_3$.
\smallskip
Also, the issues of completeness of the intrinsic connection
and/or global amenability are completely ignored in our
treatment. Even in the case where the holonomy is $H_3$, we
do not know anything about the global nature of the
$H_3$-structures. In particular, note that Theorem 3.4 does
not give any global information.
\smallskip
For example, is it reasonable to conjecture that each
connected component of a level set of ${\bf R}_c$ in
$\V\setminus \Sigma_c$ corresponds to a connected smooth
4-manifold endowed with a complete $H_3$-structure?
While it seems reasonable that one should be able to ``piece
together'' the local solutions corresponding to open sets in the
level set to get a connected 4-manifold, the resulting object
might be neither Hausdorff nor smooth. In fact, it seems quite
likely that solutions may develop orbifold-type singularities
(or worse) under this patching process. One source of trouble
for this approach is that each such local solution has a local
1-parameter symmetry group (which may even have fixed
points), so there is no {\it a priori} canonical way to identify
solutions on ``overlaps''.
\smallskip
One obstruction to globalizing the proof of Theorem 3.4 is that,
at a crucial point in the argument, a closed 2-form
${\bar\Psi}$ is assumed to be exact. Of course, this cannot be
done globally if ${\bar\Psi}$ represents a non-trivial element
in the second deRham cohomology group of the stratum.
Unfortunately, due to the complexity of the polynomial
function ${\bf R}_c$, it does not appear to be easy to deduce
anything about the topology of its level sets. Thus,
determining the deRham class of ${\bar\Psi}$ seems to be
difficult.
\smallskip
Nothing is known about the torsion-free $H_3$-structures
which have a three-parameter symmetry group. These
correspond to the irreducible components of the singular locus
$\Sigma_c$. For $c\ne0$, there are two of these components.
While one of these components is rational, the other does not
seem to be (although we have no proof of this).
\medskip
In \S4, a geometric interpretation of the primary and
secondary invariants of generalized contact path geometries is
entirely lacking. We only know an interpretation of the {\sl
vanishing} of those invariants. Since this geometry is
essentially equivalent to the geometry of fourth order {\sc
ode} in the plane up to contact equivalence, some
interpretation of these invariants should be possible in terms
of classical {\sc ode}.
\smallskip
A sample question is: What are the conditions on a fourth
order {\sc ode} which characterize the Euler-Lagrange
equations of second order functionals for curves in the plane?
These conditions, which are analogous to ``Helmholtz
conditions with multipliers,'' should be expressible in terms of
the invariants we have derived.
\smallskip
Of course, this particular question should be easy to answer by
the method of equivalence. It may even be possible to do this
by comparing our calculations with Cartan's calculations in
Cartan [{\bf 1936}]. Nevertheless this serves to illustrate
what sort of problems there are to consider.
\medskip
In \S5, we have given no discussion of how one might
explicitly incorporate reality conditions into the construction
so that one can see the relationship between the real analytic
and complex settings explicitly. The reality condition is, of
course, important for computation and indicates interesting
differences between this theory and the analogous theory of
Penrose for the non-linear graviton.
\smallskip
Our construction associates a complex twistor space to each
of the local real analytic $G_3$-holonomy affine connections
while Penrose's construction associates a complex twistor
space to each self-dual or anti-self-dual solution to Einstein's
equations.
On the other hand, Penrose's construction works for
pseudo-metrics of two distinct signatures, (4,0) and (2,2). Our
construction works for only one of the real forms of $\cG_3$.
In fact, the other real forms of $\cG_3$ in this representation
cannot occur as the affine holonomy of a symmetric connection
as they do not satisfy Berger's two conditions.
\medskip
One consequence of Proposition 5.1 is that a holomorphic
torsion-free $\cG_3$-structure on a complex 4-manifold
$\bcM$ induces a local embedding of $\bcM$ into a certain
complex 6-manifold, $\bcZ$. This embedding is constructed by
interpreting the points of $\bcM$ as rational contact curves in
a contact 3-fold $\bcY$ and then regarding $\bcZ$ as the full
space of rational curves in $\bcY$ with normal bundle
$\cO(2)\oplus\cO(2)$.
\smallskip
Since the tangent space at each point of $\bcZ$ is isomorphic
to $\bbC^2\otimes H^0\left(\cO(2)\right)$, it is apparent that
$\bcZ$ has a canonical $\GL(2,\bbC)\otimes
\SO(3,\bbC)$-structure. For twistorial reasons, it seems
likely that the intrinsic torsion of this structure vanishes. If
so, it is also quite likely that $\GL(2,\bbC)\otimes
\SO(3,\bbC)\subset \GL(6,\bbC)$ is yet another exotic affine
holonomy group. The relationship between the geometry on
$\bcZ$ and that on $\bcM$ is far from understood.
\medskip
Finally there is the question of characterizing the complex
contact manifolds which correspond to the finite dimensional
family of affine connections with holonomy $\cH_3$. These
should be some special complex manifolds which have at least
a one-parameter symmetry group. Other than the homogeneous
case that we discussed at the end of \S3 and \S5, none of the
other examples are known. In particular, the complex contact
manifolds corresponding to the two components of the singular
locus $\Sigma_c$ should have 3-parameter symmetry groups
and may be ``almost homogeneous'', but so far, they have
resisted explication.
\bigskip
\noindent{\sc Department of Mathematics
\hfill March 1990}\par
\noindent{\sc Duke University}\par
\noindent{\sc Durham, NC 27706 \hfill
bryant@math.duke.edu}\par
\bigskip
\vfill\eject
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\bye