On a Class of Complete and Projectively Flat Finsler Metrics

TL;DR

This paper classifies certain complete, locally projectively flat Finsler metrics of $(eta,eta)$-type on manifolds, showing they are related to spherical geometries and exploring their geodesic and curvature properties.

Contribution

It provides a classification of non-Randers $(eta,eta)$-Finsler manifolds that are complete and projectively flat, linking them to spherical geometries and analyzing their geometric features.

Findings

Non-trivial class is homeomorphic to $S^n$ with metrics related to spherical Riemannian manifolds.

Derived geometric properties of geodesics and scalar flag curvature on $S^n$.

Special results for metrics of general square type.

Abstract

An $(\alpha,\beta)$-manifold $(M,F)$ is a Finsler manifold with the Finsler metric $F$ being defined by a Riemannian metric $\alpha$ and $1$-form $\beta$ on the manifold $M$. In this paper, we classify $n$-dimensional $(\alpha,\beta)$-manifolds (non-Randers type) which are positively complete and locally projectively flat. We show that the non-trivial class is that $M$ is homeomorphic to the $n$-sphere $S^n$ and $(S^n,F)$ is projectively related to a standard spherical Riemannian manifold, and then we obtain some special geometric properties on the geodesics and scalar flag curvature of $F$ on $S^n$, especially when $F$ is a metric of general square type.