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Symplectic geometry and Hilbert's fourth problem (new version)
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 RP^{n} whose geodesics are projective lines and a class of symplectic forms on the Grassmannian of 2-planes in R^{n+1}.
Anti-self-dual symplectic forms and integral geometry
Although the title seems unrelated to Finsler geometry, one of the aims of this paper is to establish a twistor-like correspondence between anti-self-dual symplectic forms on the Grassmannian of oriented 2-planes in R^{4} and Finsler metrics on RP^{3} whose geodesics are projective lines.
This paper appeared in in "Analysis, Geometry, Number Theory, The Mathematics of Leon Ehrenpreis", Eds. S. Bernahu, E.L. Grinberg, M. Knopp, G. Mendoza, and E.T. Quinto, Contemp. Math. AMS. (1999), 15--25. Reprints can be obtained from the author. |
Contact topology, taut immersions. and Hilbert's fourth problem
In this paper, the author uses symplectic and contact geometry to study the geometry of curves and wave fronts in Finsler surfaces all of whose geodesics are straight lines or great circles. The main theme is that Lagrangian intersection theory can replace Morse theory in the study of taut manifolds. As a result, the theory of taut immersions developed by Banchoff, Kuiper, Chern, Thorbergsson, and others can be generalized to a large class of Finsler spaces and to submanifolds admitting certain types of singularities (wave fronts). This paper appared in "Differential and Symplectic Topology of Knots and Curves". S. Tabachnikov (Ed.). Adv. in Math. Sciences. AMS, 1999. Reprints can be obtained from the author.
Dual mixed volumes and isosystolic inequalities
In this paper the author extends the theory of dual mixed volumes and uses it to prove several isosystolic inequalities for Finsler metrics and Hamiltonian systems. In turn, these inequalities are applied to study projective Finsler metrics on projective spaces. Two sample results are: Theorem. Let L be a reversible Finsler metric on the n-dimensional sphere that is conformal to a projective metric. If L has constant flag curvature, then it is Riemannian. Theorem. Projective subspaces of a (not necessarily reversible) projective metric on n-dimensional projective space are minimal for the Holmes-Thompson area functional. This paper supersedes Isosystolic inequalities for Finsler metrics and applications which the author has "retired from circulation". |