Symplectic Geometry and Hilbert's Fourth Problem

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. Using this duality, 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.