Einstein Finsler Metrics and Killing Vector Fields on Riemannian Manifolds

TL;DR

This paper constructs a new class of Finsler metrics on Riemannian manifolds using Killing forms, characterizes Einstein metrics within this class, and provides explicit examples on the 3-sphere with various Ricci curvatures.

Contribution

It introduces a novel method to generate Finsler metrics from Killing forms and identifies Einstein metrics among them, including explicit examples on S^3.

Findings

Constructed Einstein Finsler metrics on S^3 with specific Ricci curvatures.

Identified conditions characterizing Einstein metrics in this class.

Provided examples where Ricci curvature is constant but flag curvature varies.

Abstract

In this paper, we use a Killing form on a Riemannian manifold to construct a class of Finsler metrics. We find equations that characterize Einstein metrics among this class. In particular, we construct a family of Einstein metrics on $S^3$ with ${\rm Ric} = 2 F^2$, ${\rm Ric}=0$ and ${\rm Ric}=- 2 F^2$, respectively. This family of metrics provide an important class of Finsler metrics in dimension three, whose Ricci curvature is a constant, but the flag curvature is not.