Also, you might want to look at the list of my earlier writings.
This article is an exposition of four loosely related remarks on the geometry of Finsler manifolds with constant positive flag curvature.
The first remark is that there is a canonical Kahler structure on the space of geodesics of such a manifold.
The second remark is that there is a natural way to construct a (not necessarily complete) Finsler n-manifold of constant positive flag curvature out of a hypersurface in suitably general position in complex projective n-space.
The third remark is that there is a description of the Finsler metrics of constant curvature on the 2-sphere in terms of a Riemannian metric and 1-form on the space of its geodesics. In particular, this allows one to use any (Riemannian) Zoll metric of positive Gauss curvature on the 2-sphere to construct a global Finsler metric of constant positive curvature on the 2-sphere.
The fourth remark concerns the generality of the space of (local)
Finsler metrics of constant positive flag curvature in dimension n+1>2 .
It is shown that such metrics depend on n(n+1) arbitrary functions of
n+1 variables and that such metrics naturally correspond to certain
torsion-free S^1 x GL(n,R)-structures on 2n-manifolds. As a by-product,
it is found that these groups do occur as the holonomy of torsion-free
affine connections in dimension 2n, a hitherto unsuspected phenomenon.
The problem of immersing a simply connected surface with a prescribed shape operator is discussed. From classical and more recent work, it is known that, aside from some special degenerate cases, such as when the shape operator can be realized by a surface with one family of principal curves being geodesic, the space of such realizations is a convex set in an affine space of dimension at most 3. The cases where this maximum dimension of realizability is achieved have been classified and it is known that there are two such families of shape operators, one depending essentially on three arbitrary functions of one variable (called Type I in this article) and another depending essentially on two arbitrary functions of one variable (called Type II in this article).
In this
article, these classification results are rederived, with an emphasis on
explicit computability of the space of solutions. It is shown that, for
operators of either type, their realizations by immersions can be
computed by quadrature. Moreover, explicit normal forms for each can be
computed by quadrature together with, in the case of Type I, by solving
a single linear second order ODE in one variable. (Even this last step
can be avoided in most Type I cases.) The space of realizations is
discussed in each case, along with some of their remarkable geometric
properties. Several explicit examples are constructed (mostly already in
the literature) and used to illustrate various features of the problem.
A second order family of special Lagrangian submanifolds of complex m-space
is a family characterized by the satisfaction of a set of pointwise conditions
on the second fundamental form. For example, the set of ruled special Lagrangian
submanifolds of complex 3-space is characterized by a single algebraic equation
on the second fundamental form.
While the `generic' set of such conditions turns out to be incompatible, i.e.,
there are no special Lagrangian submanifolds that satisfy them, there are many
interesting sets of conditions for which the corresponding family is unexpectedly
large. In some cases, these geometrically defined families can be described
explicitly, leading to new examples of special Lagrangian submanifolds.
In other cases, these conditions characterize already known families in a new way.
For example, the examples of Lawlor-Harvey constructed for the solution
of the angle conjecture and recently generalized by Joyce turn out to be a natural
and easily described second order family.
I use local differential geometric techniques to prove that the
algebraic cycles in certain extremal homology classes in Hermitian symmetric
spaces are either rigid (i.e., deformable only by ambient motions) or quasi-rigid
(roughly speaking, foliated by rigid subvarieties in a nontrivial way).
These rigidity results have a number of applications: First, they prove
that many subvarieties in Grassmannians and other Hermitian symmetric spaces
cannot be smoothed (i.e., are not homologous to a smooth subvariety). Second,
they provide characterizations of holomorphic bundles over compact Kahler
manifolds that are generated by their global sections but that have certain
polynomials in their Chern classes vanish (for example, c_{2} = 0,
c_{1}c_{2} - c_{3} = 0, or c_{3} = 0, etc.).
In this article, I classify the hypersurfaces described in the title.
The interesting thing is that it turns out that nearly all of the solutions
are invariant under a 1-parameter group of isometries. The one exception
(up to holomorphic homothety) is a certain hypersurface in C^{2} with very
interesting properties. Though it has no continuous symmetries, it is periodic
with respect to a lattice of type F_{4} and the complex leaves in the lattice
quotient are compact Riemann surfaces of genus 3.
This is a set of notes on the action of certain spin groups on their
spinor spaces, mostly concentrating on the 'medium low' dimensions, Spin(p,q)
where (p,q) is one of (8,0), (9,0), (10,0), (9,1), (11,0), (10,1), and
(10,2).
I wrote these in preparation for doing some work on constructing examples of pseudo-Riemannian manifolds of type (10,1) that have a parallel spinor of algebraically special type. (I needed a treatment of the spinors in these dimensions that used a consistent notation.)
Nothing in these notes is new. I am making them available because they have been referred to in the literature and I have been receiving requests for them.
This manuscript is partly obsolete because it has partly been incorporated
into the above manuscript on parallel spinor
fields.
This is a short remark giving a local description of the metrics
in 10+1 dimensions that possess a parallel null spinor and have the maximal
degree of holonomy among such metrics. I show that the holonomy can be
as large as is allowed by the existence of a parallel spinor. There is
an interesting relation with Spin(7)-metrics (actually 1-parameter families
of such) in the solution. This manuscript is now obsolete since it has
been incorporated into the above manuscript on parallel
spinor fields.
This is a longer, more expository version of the published article
Finsler structures on the 2-sphere satisfying K=1. I do not plan
to submit it for publication, but will instead incorporate it into my Aisenstadt
lectures.
I may not submit this for publication, but instead incorporate it
into my Aisenstadt lectures.