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Summary: SYMPLECTIC GEOMETRY AND HILBERT'S FOURTH
PROBLEM
J.C. ´ALVAREZ PAIVA
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
.
... es quiz´a un error suponer que puedan inventarse
met´aforas. Las verdaderas, las que formulan ´intimas
conexiones entre una imagen y otra, han existido siem-
pre ... .
Jorge Luis Borges.
Contents
1. Introduction 1
2. Finsler manifolds and spaces of geodesics 5
3. Reduction to two-dimensions 9
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