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On tangent sphere bundles with contact pseudo-metric structures

Narges Ghaffarzadeh, Morteza Faghfouri

Abstract

In this paper, we introduce a contact pseudo-metric structure on a tangent sphere bundle $T_\varepsilon M$. we prove that the tangent sphere bundle $T_{\varepsilon}M$ is $(κ, μ)$-contact pseudo-metric manifold if and only if the manifold $M$ is of constant sectional curvature. Also, we prove that this structure on the tangent sphere bundle is $K$-contact iff the base manifold has constant curvature $\varepsilon$.

On tangent sphere bundles with contact pseudo-metric structures

Abstract

In this paper, we introduce a contact pseudo-metric structure on a tangent sphere bundle . we prove that the tangent sphere bundle is -contact pseudo-metric manifold if and only if the manifold is of constant sectional curvature. Also, we prove that this structure on the tangent sphere bundle is -contact iff the base manifold has constant curvature .

Paper Structure

This paper contains 4 sections, 6 theorems, 41 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The curvature of the unit tangent sphere bundle with pseudo-metric
  4. The contact pseudo-metric structure of the unit tangent sphere bundle

Key Result

Proposition 2.1

If the index of $g$ is $\nu$ then the index of the Sasaki pseudo-metric $Tg$ is $2\nu$. Let $\tilde{\nabla}$ be the Levi-Civita connection of $Tg$. It is easy to check that for $X,Y\in\Gamma(TM)$ and $(x,u)\in TM$(see Kowalski for more details):

Theorems & Definitions (11)

  • Proposition 2.1
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • theorem 1
  • proof
  • theorem 2
  • proof
  • theorem 3
  • ...and 1 more