This paper relates the classical integral-geometric solution of Hilbert's fourth problem by Busemann, Pogorelov, among others, with the symplectic "solution" given by myself in the preprint Symplectic geometry and Hilbert's fourth problem . In the process we clarified both the classical and the symplectic construction.
In this appendix I showed how to simplify the constructions of the paper of Fokas and Gelfand by using Cartan's method of moving frames.
My very own model of the hyperbolic plane. Nothing deep, but neat.
Using a classical formula of integral geometry and Birkhoff's minimax principle, I proved the following theorem:
Theorem. If S is a smooth surface with positive Gaussian curvature in Euclidean 3-space, then it carries a closed geodesic whose length is less than or equal to one half the total mean curvature of S.
This paper is a detailed annoucement of the extension of the classical Crofton formulas to all Finsler metrics on affine or projective spaces whose geodesics are straight lines. Symplectic geometry plays there a very important role since it intervenes both in the definition of area for submanifolds of Finsler spaces and as the natural geometric structure on the space of geodesics of a projective Finsler space. Another of the highlights of this paper is our Crofton formula for double fibrations and an important functorial property of the Gelfand transform. These two things allow us to close the gap between the integral geometry of Blaschke school and the integral geometry of the Gelfand school and thus finish the work started by I.M. Gelfand and M. Smirnov in Crofton densities.
This paper is a first attempt at the symplectification of the theory of taut immersions. The idea is that by remplacing Morse theory by Lagrangian intersection theory, the theory of taut immersions becames more geometric and its domain of application is considerably enlarged.
We show that the Crofton formula for Minkowski spaces is equivalent to a an explicit expression for the k-area densities on Finsler manifolds in terms of the Fourier transform of the Finsler metric at each tangent space. An interesting application of the formula is that if a Finsler metric on an open convex domain of RPn is such that line segments are extremals for the arclength functional, then flat domains are extremals for the area functional.
Abstract. There is a natural integral-geometric transform mapping 3-forms on the the dual projective space RP3* to closed anti-self-dual 2-forms on the Grassmannian of oriented 2-planes in R4. I give a simple characterization of those 3-forms that are mapped onto symplectic anti-self-dual 2-forms and show how to use these 2-forms to construct metrics on RP3 whose geodesics are projective lines. A new geometric construction of anti-self-dual symplectic forms is presented.
Abstract. If we are given n real-valued smooth functions on R2n which are in involution then, under some mild hypotheses, the subset of R2n where these functions are linearly independent is not simply connected.
Abstract. The notion of isometric submersion is extended to Finsler manifolds and it is used to construct examples of Finsler metrics on complex and quaternionic projective spaces all of whose geodesics are (geometrical) circles.
This paper presents, at a level suitable for undergraduates, the different solutions to Hilbert's fourth problem in dimension two given by Busemann, Pogorelov, Ambatzumian, and Alexander.
This paper is a survey of the recent progress in the study of Crofton formulas. The new techniques are illustrated by giving ``book proofs" for the Crofton formulas in homogeneous and Minkowski spaces.
This is a long survey paper on the study of volume and area on normed and Finsler spaces. Rather than collecting results, we tried to build a theory that would frame them economically and elegantly. The paper should be interesting to convex geometers, integral geometers, and people working on the local theory of Banach spaces.
This is a short survey, with some new proofs, on what is known about the area and perimeter of the unit disc in a two-dimensional normed space.
In the 1970's, Schaeffer defined the girth of a normed space as the infimum of the lengths of all curves on the unit sphere that join a pair of antipodes. He conjectured that the girth of a normed space equals the girth of its dual. This paper settles the conjecture.
Fernandes and I show that classical integral geometric formulas such as those of Crofton, Cauchy, and Chern can be easily and systematically obtained through the study of Radon-type transforms on double fibrations. The methods also allow us to extend these formulas to non-homogeneous settings where group-theoretic techniques are no longer useful.
Abstract Inspired by Hofer's definition of a metric on the space of compactly supported Hamiltonian maps on a symplectic manifold, this paper exhibits an area-length duality between a class of metric spaces and a class of symplectic manifolds. In particular, it is shown that there is a twistor-like correspondence between Finsler metrics on RPn whose geodesics are projective lines and a class of symplectic forms on the Grassmannian of 2-planes in Rn+1. Symplectic and integral geometry are also used to show that if the geodesics of a Finsler metric on an open convex subset of RPn are projective line segments, then hyperplanes are area-minimizing.
Remark. This paper is a much improved version of the early preprint of mine that led to the collaboration with Gelfand in "Crofton densities, symplectic geometry, and Hilbert's fourth problem".
This is an unorthodox survey of Finsler geometry that stresses what is ignored and much as what is known.
This note is a short introduction to the theory of volumes on normed and Finsler spaces. It is also a quick update to the survey of Álvarez and Thompson, where the reader will find proofs, references, and some of the history of the subject.
I extend E. Lutwak's dual Brunn-Minkowski theory to cotangent bundles and apply it to prove several isosystolic inequalities for Finsler manifolds and Hamiltonian systems.
We show that if the Hausdorff measure is adopted as a notion of volume on Finsler spaces, the totally geodesic submanifolds are not necessarily minimal, filling results such as that of Sergey Ivanov no longer hold, and integral-geometric formulas do not exist.
Abstract. In 1987, Cecil and Chern showed that tautness is invariant under Lie sphere transformations. This note presents a very simple proof of this result and a manifestly invariant definition of tautness for Legendrian submanifolds in the space of cooriented contact elements of the sphere.
Abstract. We study the geometry of an important class of generic curves in the Grassmann manifolds of n-dimensional subspaces and Lagrangian subspaces of R2n under the action of the linear and linear symplectic groups.