On Finsler surfaces with certain flag curvatures

TL;DR

This paper characterizes Finsler surfaces with specific flag curvature conditions, providing necessary and sufficient criteria for Landsbergian properties and identifying when such surfaces are Riemannian, with implications for geodesic flow integrability.

Contribution

It offers new classification results for Finsler surfaces satisfying certain flag curvature conditions and establishes conditions under which these surfaces are Riemannian.

Findings

Finsler surfaces with V(K)=-I/F^2 and S(K)=0 are Riemannian.

Surfaces with V(K)=-I*K and S(K)=0 are Riemannian.

Identifies conditions for Landsbergian Finsler surfaces based on Berwald curvature.

Abstract

In the present paper, we find out necessary and sufficient conditions for a Finsler surface $(M,F)$ to be Landsbregian in terms of the Berwald curvature $2$-forms. We study Finsler surfaces which satisfy some flag curvature $K$ conditions, viz., $V(K)=0,\,\,V(K)= -\mathcal{I}/F^2$ and $V(K)=-\mathcal{I}\,K,$ where $\mathcal{I}$ is the Cartan scalar. In order to do so, we investigate some geometric objects associated with the global Berwald distribution $\mathcal{D}:= \operatorname{span}\{S, H, V:=JH\}$ of a $2$-dimensional Finsler metrizable nonflat spray $S$. We obtain some classifications of such surfaces and show that under what hypothesis these surfaces turn to be Riemannian. The existence of a first integral for the geodesic flow in each case has some remarkable consequences concerning rigidity results. We prove that a Finsler surface with $V(K)= -\mathcal{I}/F^2$ and either $S(K)=0$ or $S(\mathcal{J})=0$ is Riemannian. Further, a Finsler surface with $V(K)=-\mathcal{I}\,K$ and $S(K)=0$ is Riemannian.