Invariant Einstein Kropina metrics on Lie groups and homogeneous spaces

TL;DR

This paper develops methods to construct and classify Einstein Kropina metrics on Lie groups and homogeneous spaces, providing explicit examples and demonstrating the non-existence on certain spaces.

Contribution

It introduces a new construction method for Einstein Kropina metrics and classifies all such metrics on 3D Lie groups and spheres.

Findings

Constructed Einstein Kropina metrics on SO(n)

Classified all left invariant Einstein Kropina metrics on 3D Lie groups

Proved non-existence of homogeneous non-Riemannian Einstein Kropina metrics on projective spaces

Abstract

In this article, we study Einstein Kropina metrics on Lie groups and homogeneous spaces. We give a method to construct Einstein Kropina metrics on Lie groups. As an example of this method, a family of non-Riemannian Einstein Kropina metrics on the special orthogonal group $SO(n)$ is given. Then, we classify all left invariant Einstein Kropina metrics on simply connected $3$-dimensional real Lie groups. We provide a procedure to build Einstein Kropina metrics on homogeneous spaces. Using this technique, we study invariant Einstein Kropina metrics on spheres. Finally, we show that projective spaces do not admit any homogeneous non-Riemannian Einstein Kropina metrics.