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Conformal vector fields on compact connected homogeneous Finsler manifolds

Ming Xu

Abstract

Let $(M,F)$ be a compact connected homogeneous non-Riemannian Finsler manifold with $\dim M>1$. We prove that any conformal vector field on $(M,F)$ is a Killing vector field. Further more, we prove that $ρF$ is a homogeneous Finsler metric on $M$ if and only if $ρ$ is a positive constant function.

Conformal vector fields on compact connected homogeneous Finsler manifolds

Abstract

Let be a compact connected homogeneous non-Riemannian Finsler manifold with . We prove that any conformal vector field on is a Killing vector field. Further more, we prove that is a homogeneous Finsler metric on if and only if is a positive constant function.
Paper Structure (8 sections, 11 theorems, 7 equations)

This paper contains 8 sections, 11 theorems, 7 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Minkowski norm and Finsler metric
  4. Finslerian conformality and isometry
  5. Conformality for a homogeneous Finsler manifold
  6. Average process and group of conformal transformations
  7. Conformal fields on a compact homogeneous Finsler manifold
  8. Proof of the main results

Key Result

Theorem 1.1

Let $(M,g)$ be a compact connected homogeneous Riemannian manifold with $\dim M>1$. Suppose that $C_0(M,g)\neq I_0(M,g)$, then $(M,g)$ is isometric to a sphere with constant curvature.

Theorems & Definitions (11)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Lemma 4.1
  • Lemma 4.2
  • Theorem 4.3
  • ...and 1 more