Some remarks on Finsler manifolds with constant flag curvature

TL;DR

This paper explores various geometric properties and constructions of Finsler manifolds with constant positive flag curvature, revealing new structures, relationships with complex projective hypersurfaces, and the diversity of such metrics.

Contribution

It provides four new insights into Finsler manifolds with constant positive flag curvature, including canonical structures, construction methods, and classification results.

Findings

Existence of a canonical Kähler structure on geodesic spaces.

Construction of Finsler manifolds from complex projective hypersurfaces.

Description of Finsler metrics on the 2-sphere via Zoll metrics.

Abstract

This article is an exposition of four loosely related remarks on the geometry of Finsler manifolds with constant positive flag curvature. <p> The first remark is that there is a canonical Kahler structure on the space of geodesics of such a manifold. <p> 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. <p> 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. <p> 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.