On the projective Ricci curvature

TL;DR

This paper introduces the concept of projectively Ricci-flat sprays, explores their properties, and provides a global rigidity result for such sprays with nonnegative Ricci curvature, including a characterization of Randers metrics.

Contribution

It defines projectively Ricci-flat sprays, proves a global rigidity theorem for nonnegative Ricci curvature cases, and characterizes projectively Ricci-flat Randers metrics.

Findings

Global rigidity result for projectively Ricci-flat sprays with nonnegative Ricci curvature

Characterization of projectively Ricci-flat Randers metrics

Introduction of the notion of projectively Ricci-flat sprays

Abstract

The notion of the Ricci curvature is defined for sprays on a manifold. With a volume form on a manifold, every spray can be deformed to a projective spray. The Ricci curvature of a projective spray is called the projective Ricci curvature. In this paper, we introduce the notion of projectively Ricci-flat sprays. We establish a global rigidity result for projectively Ricci-flat sprays with nonnegative Ricci curvature. Then we study and characterize projectively Ricci-flat Randers metrics.