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Invariant Einstein Kropina metrics on Lie groups and homogeneous spaces

Masoumeh Hosseini, Hamid Reza Salimi Moghaddam

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.

Invariant Einstein Kropina metrics on Lie groups and homogeneous 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 is given. Then, we classify all left invariant Einstein Kropina metrics on simply connected -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.

Paper Structure

This paper contains 3 sections, 9 theorems, 8 equations, 1 table.

Table of Contents

  1. Introduction
  2. left invariant Einstein Kropina metrics on Lie groups
  3. Einstein Kropina Metrics on Homogeneous Spaces

Key Result

Theorem 2.1

Let $G$ be an $n$-dimensional Lie group equipped with an Einstein left invariant Riemannian metric $\textbf{h}$ with the Einstein scalar $\sigma$. Suppose that $W$ is a right invariant vector field on $G$ such that $\|W\|_{\textbf{h}}=1$. Then the corresponding Kropina metric to the navigation data

Theorems & Definitions (21)

  • Theorem 2.1
  • proof
  • Corollary 2.2
  • proof
  • Corollary 2.3
  • Theorem 2.4
  • proof
  • Remark 2.5
  • Example 2.6
  • Lemma 2.7
  • ...and 11 more