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Characterizations of a Lorentzian Manifold with a semi-symmetric metric connection

Uday Chand De, Krishnendu De, Sinem Güler

Abstract

In this article, we characterize a Lorentzian manifold $\mathcal{M}$ with a semi-symmetric metric connection. At first, we consider a semi-symmetric metric connection whose curvature tensor vanishes and establish that if the associated vector field is a unit time-like torse-forming vector field, then $\mathcal{M}$ becomes a perfect fluid spacetime. Moreover, we prove that if $\mathcal{M}$ admits a semi-symmetric metric connection whose Ricci tensor is symmetric and torsion tensor is recurrent, then $\mathcal{M}$ represents a generalized Robertson-Walker spacetime. Also, we show that if the associated vector field of a semi-symmetric metric connection whose curvature tensor vanishes is a $f-$ Ric vector field, then the manifold is Einstein and if the associated vector field is a torqued vector field, then the manifold becomes a perfect fluid spacetime. Finally, we apply this connection to investigate Ricci solitons.

Characterizations of a Lorentzian Manifold with a semi-symmetric metric connection

Abstract

In this article, we characterize a Lorentzian manifold with a semi-symmetric metric connection. At first, we consider a semi-symmetric metric connection whose curvature tensor vanishes and establish that if the associated vector field is a unit time-like torse-forming vector field, then becomes a perfect fluid spacetime. Moreover, we prove that if admits a semi-symmetric metric connection whose Ricci tensor is symmetric and torsion tensor is recurrent, then represents a generalized Robertson-Walker spacetime. Also, we show that if the associated vector field of a semi-symmetric metric connection whose curvature tensor vanishes is a Ric vector field, then the manifold is Einstein and if the associated vector field is a torqued vector field, then the manifold becomes a perfect fluid spacetime. Finally, we apply this connection to investigate Ricci solitons.

Paper Structure

This paper contains 8 sections, 11 theorems, 66 equations.

Table of Contents

  1. Introduction
  2. Semi-symmetric metric connection
  3. Proof of the Theorem \ref{['3a']}
  4. Proof of the Theorem \ref{['4a']}
  5. Proof of the Theorem \ref{['5a']}
  6. Proof of the Theorem \ref{['6a']}
  7. Applications to Ricci solitons
  8. Proof of the Theorem \ref{['7a']}

Key Result

Theorem \oldthetheorem

survey$\mathcal{M}$ is a $GRW$ spacetime if and only if it admits a unit time-like torse-forming vector field: $\nabla_{k}v_{j}=\varphi (g_{ij}+v_{j}v_{i})$, which is also an eigenvector of the Ricci tensor.

Theorems & Definitions (19)

  • Definition 1.1
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Definition 1.2
  • Theorem \oldthetheorem
  • Definition 1.3
  • Definition 1.4
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • ...and 9 more