If reprints are available, I have noted this in the entries below and you can request one by sending me e-mail.
You might also want to consult my preprints page for recent articles that have not yet appeared.
The support from these institutions is hereby gratefully acknowledged. However, that support should not be construed as their agreement with my results and conclusions, for which I am wholly responsible.
An examination of the isometric embedding problem in the overdetermined
range. The essential technical tool is the analysis of the Gauss equations
and the role that they play in rigidity theorems. This is an announcement.
The real paper is The Gauss equations and rigidity
of isometric embeddings.
When can a real hypersurface in complex n-space contain any complex
curves? Since the tangent spaces to such a curve would have to be null
vectors for the Levi form, a necessary condition is that the Levi form
have zeros. The simplest way this can happen in the non-degenerate case
is for the Levi form to have the Lorentzian signature.
In this paper, I show that a Lorentzian CR-manifold M has at most a finite parameter family of holomorphic curves, in fact, at most an n^{2} parameter family if the dimension of M is 2n+1. This maximum is attained, as I show by example. When n=2, the only way it can be reached is for M to be CR-flat. In higher dimensions, where the CR-flat model does not achieve the maximum, it is still unknown whether or not there is more than one local model with the maximal dimension family of holomorphic curves.
The technique used is exterior differential systems together with the Chern-Moser theory in the n=2 case.
Reprints are available.
This is a brief introduction to the ideas of exterior differential
systems. It was later expanded and became the basis of our 1991 book with
Gardner and Goldschmidt of the same title.
A study of the geometry of submanifolds of real 8-space under the
group of motions generated by translations and rotations in the subgroup
Spin(7) instead of the full SO(8). I call real 8-space endowed with this
group
O or octonian space.
The fact that the stabilizer of an oriented 2-plane in Spin(7) is U(3) implies that any oriented 6-manifold in O inherits a U(3)-structure. The first part of the paper studies the generality of the 6-manifolds whose inherited U(3)-structure is symplectic, complex, or Kähler, etc. by applying the theory of exterior differential systems.
I then turn to the study of the standard 6-sphere in O as an almost complex manifold and study the space of what are now called pseudo-holomorphic curves in the 6-sphere. I prove that every compact Riemann surface occurs as a (possibly ramified) pseudo-holomorphic curve in the 6-sphere. I also show that all of the genus zero pseudo-holomorphic curves in the 6-sphere are algebraic as surfaces.
Reprints are available.
This is a short paper in which I use the twistor fibration of of
complex projective 3-space over the 4-sphere to construct, for each compact
Riemann surface, a conformal and minimal immersion of that surface into
the 4-sphere.
The idea is that complex projective 3-space has a natural, SO(5)-invariant holomorphic contact structure and, under the twistor fibration, holomorphic contact curves project conformally to minimal surfaces in the 4-sphere. As a holomorphic contact manifold, complex projective 3-space is birationally equivalent to the projectivized tangent bundle of the complex projective plane. Since every compact Riemann surface occurs as an immersed curve in the complex projective plane, it's just a matter of putting it in general position (so as to avoid the singularities of the birational transformation) in order to get every compact Riemann surface as an embedded contact curve in complex projective 3-space.
Not every minimal surface in the 4-sphere arises this way (although all of the genus 0 ones do), and I unfortunately coined the term super-minimal to refer to the ones that do. There are several reasons to abandon this terminology now and I discourage its use. It would be better if they were to be called isotropic, in consonance with the usage in the theory of harmonic maps.
Reprints are available.
We examine the infinitesimal period relations of a 3-fold with trivial
canonical bundle and show that they represent an exterior differential
system on the higher period domain that is the first prolongation of a
contact system on a `de-prolonged' period domain. This has some interesting
consequences for relations between the so called A-periods and B-periods,
showing that they can be expressed in terms of a single `potential' function.
Of course, in recent years, 3-folds with trivial canonical bundle have become a central part of string theory. The literature on these complex manifolds and their geometry and deformation theory has seen an explosive growth, which I am not competent to catalog. I would suggest that you contact my colleague, D. Morrison (drm@math.duke.edu), if you want pointers to a survey article.
Reprints are available.
The full version, with proofs, of a study of the characteristic
variety of the isometric embedding problem in the overdetermined range.
The main goals are rigidity and finiteness theorems, although we do study
some degenerate situations that generalize Cartan's study of isometric
embeddings of hyperbolic n-space into real (2n-1)-space, where we get some
classification results for the `maximally non-rigid' case.
A study of the characteristic variety of the isometric embedding
problem in the determined dimension. We show that, except for metrics whose
Riemann curvature tensor lies in a a very small set of normal forms, a
3-manifold can be isometrically embedded into real 6-space. The method
is to show that the system can be made suitably hyperbolic so that a version
of the Nash-Moser theorem can be made to apply.
Deane Yang and Johnathan Goodman have since generalized the principal
analytic result from hyperbolic systems to systems of real principal type
and have used this to prove isometric embedding results for 4-manifolds.
In higher dimensions, the characteristic variety tends to have singularities
that no analytic methods are known to handle (in the smooth category).
A study of surface theory in conformal 3-space, with an application
to the extremals of the Willmore functional, which can be thought of as
the conformal area of a surface in this geometry.
Among the results are a proof that every compact extremal of genus 0 is conformally a minimal surface. This relies on a vanishing theorem plus a careful analysis of the singularities of the geometry near the `umbilic' points. Also the critical values of the Willmore functional on 2-spheres are shown to be discrete and the moduli space of the extrema having the first non-minimal critical value is computed.
Since this paper, much has been done. For an update, see Surfaces in Conformal Geometry. Also, Lucas Hsu has kindly compiled a list of errata and has allowed me to include an amstex version of it here.
Reprints are available.
A classification of integrable twistor spaces of various kinds over
Riemannian symmetric spaces. Given an even dimensional Riemannian manifold
N, the bundle J(N) over N of orthogonal complex structures on the tangent
spaces of N has a natural almost complex structure and complex horizonal
plane field. Unless N has constant sectional curvature, however, the almost
complex structure on J(N) will not be integrable. A twistor subspace Z
of J(N) is a sub-bundle that is an almost complex submanifold of J(N) and
that has the additional properties that the induced almost complex structure
on Z is actually integrable and that the horizontal plane field is tangent
to Z.
In this paper, I find all of the twistor subspaces of J(N) when N is a Riemannian symmetric space. These all turn out to be orbits of the isometry group of N and so can be classified by examining the root systems of the simple Lie groups.
In the last section, I construct a different sort of twistor space over each Riemannian symmetric space.
For further developments in this area, consult the works of F. Burstall,
J. Rawnsley, S. Salamon, and J. Wood.
This is essentially a first announcement of the existence of metrics
with the two exceptional holonomies. The full exposition with proofs is
the Annals of Mathematics paper below. However, the main ideas of the proofs
are sketched, so it might not be a bad introduction.
Reprints are available.
I prove that the only minimal surfaces of constant positive Gaussian
curvature in the n-sphere are, up to rigid motion, the Boruvka spheres,
i.e., the 2-dimensional orbits of an irreducible representation of SO(3)
into SO(m) for some m less than or equal to n+1. I rederive Ejiri's classification
of the minimal surfaces of zero Gaussian curvature in the n-sphere and
prove that there are no minimal surfaces of constant negative curvature
in the n-sphere. (Partial results had been obtained by Ejiri.) I also prove
that the only minimal surfaces of constant curvature in the hyperbolic
n-ball are the totally geodesic surfaces. (That the only minimal surfaces
of constant curvature in flat space are the planes is due to Pinl.)
The methods are purely local and depend on analysing the overdetermined system for minimal isometric embedding by organizing the integrability conditions into managable form, so that one can actually differentiate them many times and still have some control over the resulting relations.
The actual results are stronger than these theorems suggest. What I do is classify the harmonic maps with constant energy density from a surface of constant Gauss curvature to the n-sphere. In this form, I have recently generalized these results in On extremals with prescribed Lagrangian densities below.
Reprints are available.
Most Kähler manifolds do not contain any submanifolds that
are simultaneously minimal and Lagrangian since the combination of the
two conditions is equivalent to an overdetermined system of PDE for the
submanifold that is generally incompatible.
However, in case the 2n-manifold M is Kähler-Einstein, the situation is different. I prove that, in this case, the overdetermined system is involutive. In fact, every real-analytic submanifold of dimension n-1 that is sub-Lagrangian (i.e., on which the Kähler form vanishes) lies in a circle of n-manifolds, each of which is minimal and Lagrangian (these will not generally be compact).
Quite recently, these minimal Lagrangian manifolds have become a subject of interest to physicists (in the physics literature this comes under the heading of `BPS states' in string theory). Works in this area in physics can be found by such authors as Vafa, Witten, Yau, and Zaslow. On the mathematical side, R. Schoen and J. Wolfson have worked in this area, not to mention R. Harvey and H. B. Lawson (in the Ricci-flat case).
For some reason, I have been getting many requests for reprints of this
article lately, perhaps because of it being referenced in the physics literature.
However, I never received reprints for this article and it predates my
use of TeX by many years; all I have is a typescript. Since the article
has been published and is easily available, I am not making photocopies
to mail out.
A survey paper that details what was known at the time about Riemannian
metrics with reduced holonomy.
This paper has since been superceded by S. Salamon's 1989 book on Riemannian
geometry and holonomy. However both of these are now inadequate. The results
of D. Joyce show that there exist compact manifolds with holonomy G_{2}
or Spin(7). Considerably more is now known about the geometry of the so-called
quaternionic-Kähler manifolds. The works of Joyce, Salamon, and LeBrun
should be consulted for further information on these subjects.
This paper gives an exposition of a way of computing the Euler-Lagrange
equations and the conservation laws for them that arise from symmetries
in geometrically defined variational problems. The main technical advantage
of this method over the more classical Pontrjagin Maximum Principle is
the way it avoids choosing coordinates that are not needed, but works directly
on the invariant coframing of the group of symmetries.
Some extended examples are computed for Euler elastica in space forms and on surfaces of constant curvature.
Since this paper appeared, David Mumford has shown how to get a complete integration of the equations in the flat case by a very clever use of theta-functions. It would be interesting to see if this would work also in the case of space elastica or for elastica in other space forms.
Reprints are available.
This is a short expository paper discussing the different notions
of equivalence, such as simple equivalence, contact equivalence, and divergence
equivalence for variational problems with one independent variable. I show
that the group of symmetries under one notion of equivalence can be distinct
from the group of symmetries under another notion. I discuss how one can
set up the equivalence problem to compute these different groups of symmetries,
but do not enter into any actual equivalence calculations.
Please note: This paper was typeset directly from my handwritten manuscript
(nearly the last handwritten one I ever produced) and I did not get to
proofread the result before it appeared. Consequently, there are many typos,
particularly in the matrices on the last pages.
This is an exposition, with complete details, of the existence and
generality of metrics with holonomy G_{2} or Spin(7). In the last
section, I also construct explicit examples (which, however, are not complete).
There has been a great deal of progress on the holonomy problem in the intervening years. One source for further information is my 1999 article Recent advances in the theory of holonomy.
As for errata and addenda to the article itself, I am only aware of two: First, McLean has pointed out that SO^{*}(2p) does not satisfy Berger's criteria and so should never have appeared on the list of possible holonomies in the first place. Second, on page 537, I make a couple of remarks about the ideal I in two special cases that are either misleading or wrong. I say that, in the Sp(n)Sp(1) case, the closure of the fundamental 4-form does not imply 1-flatness when n>1, but, in fact, the only cases where the 4-form is closed but the structure is not 1-flat happen when n=2. For all n>2, it's OK. I also say that the ideal I is not involutive when the group is Sp(n) and n>1. However, this is false. It is involutive for all n.
Reprints are available.
I construct a Weierstraß formula for surfaces of mean curvature
one in hyperbolic space and use it to investigate the complete surfaces
of mean curvature one and finite total curvature.
Although it was unknown to me at the time that I wrote this article, Bianchi had long ago pointed out that the local surfaces of mean curvature one in hyperbolic space admit a Weierstraß representation. (See Bianchi's Lezioni... Volume 2, Part 2, pp. 607-613.) While Bianchi's representation is not quite the same as the one I derive, the two are essentially equivalent for local purposes or in the simply connected case. I am grateful to Pedro Roitman for making me aware of Bianchi's work on this problem and supplying me with the above reference. In particular, it is clear that these CMC-1 surfaces in hyperbolic space should certainly not be called "Bryant surfaces".
Using my version of the Weierstraß formula, I show that, though each such surface is locally isometric to a minimal surface in Euclidean 3-space, there are striking differences. For example, even though such a surface is conformally a compact surface punctured at a finite number of points, the total area is not necessarily a rational multiple of Pi and the natural Gauss map need not complete holomorphically across the punctures. I compute some examples, investigate the simply connected case, and derive necessary and sufficient conditions for such a surface with prescribed seond fundamental form near a puncture to be realizable as a punctured disk in hyperbolic space.
Since this paper appeared, Umehara and others have computed more examples and further explored the geometry of these surfaces. The reader might try looking at works of
I do not have any reprints of this article (Asterisque never sent me
any) and the original is a typescript, with no electronic version available.
A survey paper. However, there are some new results. Building on
the results in A duality theorm for Willmore surfaces,
I use the Klein correspondance to determine the moduli space of Willmore
critical spheres for low critical values and also determine the moduli
space of Willmore minima for the real projective plane in 3-space.
Reprints are available.
We study the Sp(n)-invariant calibrations in quaternionic n-space
and, for many of these calibrations, we compute the corresponding space
of calibrated submanifolds.
Reprints are available.
We construct examples of complete metrics with holonomy G_{2}
and Spin(7). Specifically, on the product of the 3-sphere with real 4-space,
we construct a complete SO(4)-invariant metric with holonomy G_{2},
on the bundle of self-dual 2-forms on the complex projective plane, we
construct a complete SU(3)-invariant metric with holonomy G_{2},
and on the positive spin bundle over the 4-sphere, we construct a complete
Spin(5)-invariant metric with holonomy Spin(7).
The method is to look among the group invariant metrics for one that has the right holonomy. The point, in each case, is that the group acts with cohomogeneity one and so the problem is reduced to an ODE problem. These ODE turn out to be managable. In many ways, the construction is reminiscent of Calabi's construction of a complete metric with holonomy Sp(n) on the holomorphic cotangent bundle of complex projective n-space.
It is now known, from Dominic Joyce's work (see MR 99j:53065 and the references contained therein), that compact examples exist in both cases.
Reprints are available.
An austere submanifold of Euclidean space is one such that each
of the quadratic forms in the second fundamental form has its eigenvalues
occuring in oppositely signed pairs. In particular, an austere submanifold
is minimal, but, except in the case of surfaces, austerity is much more
restrictive than minimality. The term austere was coined by Harvey
and Lawson in their fundamental paper Calibrated Geometries and
characterises those submanifolds whose conormal bundle is special Lagrangian,
and hence absolutely minimizing.
The largest known class of examples of austere submanifolds are the complex submanifolds of complex n-space regarded as real submanifolds of Euclidean 2n-space.
In the first part of this paper, I classify the possible second fundamental forms of 3- and 4-dimensional austere submanifolds of Euclidean space and in the remaining parts of the paper, I determine the generality of the 3-dimensional austere submanifolds corresponding to each possible type of second fundamental form.
The classification of the possible austere second fundamental forms in higher dimensions is still unknown and it is also unknown whether or not there exist austere 4-manifolds corresponding to each of the possible algebraic types of austere second fundamental forms found in the first part of the paper. For further progress in the analysis of some examples of austere submanifolds, consult the work of Dajczer and Gromoll, The Weierstrass representation for complete minimal real Kähler submanifolds of codimension two, Inventiones Mathematicae 119 (1995), 235242.
Reprints are available, as is a .dvi
file.
This book covers the basics of Pfaffian systems, normal forms, Cartan-Kähler
theory, involution, the characteristic variety, prolongation, and the Spencer
theory together with its modern developments.
This book is no longer in print, unfortunately, but we are trying to
get it reprinted.
The .dvi file is here,
and the .pdf file is here.
Reprints are available.
In this manuscript, Hsu and I show that, for the generic 2-plane
field D on a manifold of dimension 4 or more, there exist so-called 'rigid'
D-curves, i.e., smooth curves tangent to the plane field D with the property
that they admit no compactly supported smooth variations through D-curves
other than reparametrization. These curves will therefore be abnormal extremals
for any variational problem for D-curves.
We investigate related phenomena, such as locally rigid curves that are not globally rigid, and compute several examples drawn from geometry and mechanics. For example, we analyze the mechanical system of one surface rolling over another without twisting or slipping (the case where the surfaces are a plane and a sphere had already been treated by Brockett and Dai) as well as the geometry of space curves of constant curvature (but variable torsion).
Since our paper, quite a lot of work has appeared about rigid curves and abnormal extremals in the context of sub-Riemannian geometry, particularly, see the recent works of H. Sussman and W. Liu, Agrachev and Sarychev, Milyutin, and Dmitruk.
This paper is available as either a .dvi file or as a .pdf file
Reprints are available.
This is a series of nine elementary lectures on Lie groups and symplectic
geometry that were the basis for a short course in the subject that I gave
in Park City, Utah in 1990 as part of the Regional Geometry Institute Summer
School. These notes cover the usual introductory material in Lie groups
(with some extra material on Lie's method of solving differential equations
with symmetry), Lagrangian mechanics, Noether's Theorem, symplectic manifolds,
the moment map and reduction, and concludes with a brief look at the elliptic
methods that have become so important in symplectic geometry in the last
ten years, largely due to the influence of M. Gromov.
As a result of my using these lectures again as a resource for a graduate
course in Spring 2003 and also as a result of
Eugene Lerman pointing out some problems with Lecture 8 (wherein I
discuss hyperKähler reduction), I have found some serious flaws in
Lecture 8. In particular, the purported Theorems 2 and 5 (Kähler
and hyperKähler reduction) are not true in the generality in which
I stated them in the published version. I apologize for these mistakes
and
thank Eugene for bringing them to my attention.
I have corrected these mistakes, and in doing
so, have found it to be a good idea to modify both Lectures 7 and 8.
I have posted the corrected version here.
Please be aware, though, that these lectures are essentially the same as
the original lectures, i.e., they are meant to
be a very quick and cursory introduction to the subject in the title, not
an exhaustive treatment (of which there are now
several very good ones by experts in symplectic geometry).
This paper is the first of a series in which we explore the relationship
between the geometry of a PDE (in the sense of differential invariant theory)
and its so-called 'characteristic cohomology', a generalization of the
notion of conservation laws that is largely due to Vinogradov. Rather than
work directly with a PDE system, we work with the associated exterior differential
system and formulate the theory in a way that is natural in this context.
We develop some new commutative algebra tools to help deal with the computations
that arise in our treatment of the spectral sequences involved, and explore
the relationship between the characteristic variety of the system and various
vanishing theorems that generalize the famous Vinogradov 2-line theorem.
The .dvi file is here.
The .dvi file is here.
(Includes the .dvi file for Part II below.) Reprints are available.
The .dvi file is here.
(Actually, this is the same as the .dvi file for Part I above.) Reprints
are available.
In this long paper, we apply the ideas from Part I together with
the equivalence method to classify the parabolic PDE in the plane that
admit conservation laws. We show, in particular, that a parabolic PDE that
has more than 3 independent conservation laws is linearizable by a (contact)
change of coordinates and exhibit equations (to our knowledge, the first
known ones) of parabolic equations that have exactly 3 independent conservation
laws. In the final section of the paper, we prove a classification theorem
for parabolic systems that admit non-trivial integrable extensions (i.e.,
'coverings' in Vinogradov's terminology) and give examples of systems that
admit non-trivial coverings but no conservation laws.
My former student, Jeanne Nielsen Clelland (now at the University of Colorado in Boulder), has now generalized many of these results to the case of two independent space variables and is developing the theory very nicely.
The .dvi file is here.
The .dvi file is here.
This manuscript studies some examples of the family of problems
where a Lagrangian is given for maps from one manifold to another and one
is interested in the extremal mappings for which the Lagrangian density
takes a prescribed form. The first problem is the study of when two minimal
graphs can induce the same area function on the domain without differing
by trivial symmetries. The second problem is similar but concerns a different
`area Lagrangian' first investigated by Calabi. The third problem classified
the harmonic maps between spheres (more generally, manifolds of constant
sectional curvature) for which the energy density is a constant multiple
of the volume form. In the first and third cases, the complete solution
is described. In the second case, some information about the solutions
is derived, but the problem is not completely solved.
The .dvi file is here.
I have not received any reprints.
The .dvi file is here.
Reprints are available.
The .dvi file is here.
I have not received any reprints.
In this paper, I study the Finsler structures on the 2-sphere whose
Finsler-Gauss curvature is identically 1. I show that these structures
can be regarded as more classical geometric structures on the `dual' 2-sphere
of geodesics of the structure and use this fact to construct explicit examples
that are not Riemannian.
The .dvi file is here. Reprints are available.
A longer and more expository version of this article is available here.
This paper might be regarded as a sequel to the previous one. In
it, I prove that, up to diffeomorphism, there is a 2-parameter family of
Finsler metrics on the standard 2-sphere whose geodesics are the great
circles and whose Finsler-Gauss curvature is identically 1. Explicit formulas
for these Finsler metrics are established and it is shown that the only
symmetric Finsler metrics with this property are the known Riemannian ones.
The introduction contains a discussion of the relation of these results
with Hilbert's Fourth Problem.
The .dvi file is here.
Reprints are available.
I prove 3 classification results about harmonic morphisms whose
fibers have dimension one. All are valid when the domain is at least of
dimension 4. (The character of this overdetermined problem is very different
when the dimension of the domain is 3 or less.) The first result is a local
classification for such harmonic morphisms with specified target metric,
the second is a finiteness theorem for such harmonic morphisms with specified
domain metric, and the third is a complete classification of such harmonic
morphisms when the domain is a space form of constant sectional curvature.
The methods used are exterior differential systems and the moving frame.
The fundamental results are local, but, because of the rigidity of the
solutions, they allow a complete global classification.
It was recently pointed out by E. Witten that for a D-brane to consistently
wrap a submanifold of some manifold, the normal bundle must admit a Spin^{c}
structure. We examine this constraint in the case of type II string compactifications
with vanishing cosmological constant, and argue that in all such cases,
the normal bundle to a supersymmetric cycle is automatically Spin^{c}.
I point out some very elementary examples of special Lagrangian
tori in certain Calabi-Yau manifolds that occur as hypersurfaces in complex
projective space. All of these are constructed as real slices of smooth
hypersurfaces defined over the reals. This method of constructing special
Lagrangian submanifolds is, of course, well known. What does not appear
to be in the current literature is an explicit description of such examples
in which the special Lagrangian submanifold is a 3-torus.
This is the text of a lecture that I gave June 1999 in the Bourbaki
Seminar. As the title suggests, it surveys recent progress in the description
of the possible holonomy of a torsion-free connection with irreducibly
acting holonomy, both for local structures and global structures.
This is the text of a lecture that I gave at the Institut d'Élie
Cartan in Nancy, France in June 1998 as part of their Journées Élie
Cartan.
The theme is that Cartan's understanding of the role of duality in the homogeneous spaces of the rank 2 Lie groups had a profound effect on his development of the method of equivalence, connections, and path geometry.
Available as a PDF
file (this simplifies matters because of the embedded PostScript figures).
I discuss geometry and normal forms for pseudo-Riemannian metrics
with parallel spinor fields in some interesting dimensions. I also discuss
the interaction of these conditions for parallel spinor fields with the
Einstein equations.
A Kähler metric is said to be Bochner-Kähler if its Bochner
curvature vanishes. This is a nontrivial condition when the complex dimension
of the underlying manifold is at least 2. In this article it will be shown
that, in a certain well-defined sense, the space of Bochner-Kähler
metrics in complex dimension n has real dimension n+1 and a recipe for
an explicit formula for any Bochner-Kähler metric will be given.
It is shown that any Bochner-Kähler metric in complex dimension n has local (real) cohomogeneity at most n. The Bochner-Kähler metrics that can be `analytically continued' to a complete metric, free of singularities, are identified. In particular, it is shown that the only compact Bochner-Kähler manifolds are the discrete quotients of the known symmetric examples. However, there are compact Bochner-Kähler orbifolds that are not locally symmetric. In fact, every weighted projective spacecarries a Bochner-Kähler metric.
The fundamental technique is to construct a canonical infinitesimal
torus action on a Bochner-Kähler metric whose associated momentum
mapping has the orbits of its symmetry pseudo-groupoid as fibers.
Every closed, oriented, real analytic Riemannian 3-manifold can
be isometrically embedded as a special Lagrangian submanifold of a Calabi-Yau
3-fold, even as the real locus of an antiholomorphic, isometric involution.
Every closed, oriented, real analytic Riemannian 4-manifold whose bundle
of self-dual 2-forms is trivial can be isometrically embedded as a coassociative
submanifold in a G_{2}-manifold, even as the fixed locus of an
anti-G_{2} involution.
These results, when coupled with McLean's analysis of the moduli spaces
of such calibrated submanifolds, yield a plentiful supply of examples of
compact calibrated submanifolds with nontrivial deformation spaces.