On Sprays of Scalar Curvature and Metrizability

TL;DR

This paper introduces the concept of sprays with constant curvature, explores conditions for their Finsler metrizability, and characterizes the local structure of certain projectively flat Berwald sprays with isotropic or constant curvature.

Contribution

It defines sprays of constant curvature, establishes criteria for Finsler metrizability, and characterizes the local structure of projectively flat Berwald sprays with specific curvature properties.

Findings

Sprays of isotropic curvature are not necessarily of constant curvature in dimension n≥3.

Complete conditions for Finsler metrizability of sprays of isotropic or constant curvature are provided.

The local structure of certain projectively flat Berwald sprays with isotropic or constant curvature is characterized.

Abstract

Every Finsler metric naturally induces a spray but not so for the converse. The notion for sprays of scalar (resp. isotropic) curvature has been known as a generalization for Finsler metrics of scalar (resp. isotropic) flag curvature. In this paper, a new notion, sprays of constant curvature, is introduced and especially it shows that a spray of isotropic curvature is not necessarily of constant curvature even in dimension $n\ge3$. Further, complete conditions are given for sprays of isotropic (resp. constant) curvature to be Finsler-metrizabile. As applications of such a result, the local structure is determined for locally projectively flat Berwald sprays of isotropic (resp. constant) curvature which are Finsler-metrizable, and some more sprays of isotropic curvature are discussed for their metrizability. Besides, the metrizability problem is also investigated for sprays of scalar curvature under certain curvature conditions.