Characterizations of the Lorentzian manifolds admitting a type of semi-symmetric metric connection
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Characterizations of the Lorentzian manifolds admitting a type of semi-symmetric metric connection Sudhakar K. Chaubey1
· Young Jin Suh2 · Uday Chand De3
Received: 29 January 2019 / Revised: 29 January 2019 / Accepted: 16 October 2020 © Springer Nature Switzerland AG 2020
Abstract We set a type of semi-symmetric metric connection on the Lorentzian manifolds. It is proved that a Lorentzian manifold endowed with a semi-symmetric metric ρconnection is a G RW spacetime. We also characterize the Ricci semisymmetric Lorentzian manifold and study the solution of Eisenhart problem of finding the second order parallel (skew-)symmetric tensor on Lorentzian manifolds. Finally, we address physical interpretation of some geometric results of our paper. Keywords Lorentzian manifolds · Symmetric spaces · Semi-symmetric metric connection · G RW Spacetimes · Torse-forming vector field · Different curvature tensors Mathematics Subject Classification 53B30 · 53B50 · 53C15
1 Introduction Let M be an n-dimensional semi-Riemannian manifold. If the torsion tensor T˜ with ˜ defined by T˜ (X , Y ) = ∇˜ X Y − ∇˜ Y X − [X , Y ], respect to a linear connection ∇,
B
Sudhakar K. Chaubey [email protected] Young Jin Suh [email protected] Uday Chand De [email protected]
1
Section of Mathematics, Department of Information Technology, University of Technology and Applied Sciences, P.O. Box 77, 324 Shinas, Oman
2
Department of Mathematics and RIRCM, Kyungpook National University, Daegu 41566, Republic of Korea
3
Department of Pure Mathematics, University of Calcutta 35, Ballygaunge Circular Road, Kolkata, West Bengal 700019, India 0123456789().: V,-vol
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vanishes identically on M, then ∇˜ is called a symmetric connection. Otherwise, it is known as a non-symmetric connection. Based on the different forms of T˜ , we classify the linear connection ∇˜ into different classes, for instance, a semi-symmetric [22] if T˜ (X , Y ) = π(Y )X − π(X )Y
(1.1)
holds for vector fields X and Y on M, where the 1-form π and the associated vector field ρ are connected through a semi-Riemannian metric g by π(X ) = g(X , ρ).
(1.2)
If we replace X by φ X and Y by φY in the right hand side of (1.1), then ∇˜ becomes a ˜ = 0, quarter-symmetric connection. A linear connection ∇˜ is said to be metric if ∇g otherwise it is non-metric. Hayden [23] studied the properties of metric connection (Hayden connection) on a Riemannian manifold in 1932. Pak [34] took the Hayden connection along with equation (1.1) and showed that it is a semi-symmetric metric connection. A systematic study of the semi-symmetric metric connection on a Riemannian manifold was given by Yano [43] in 1970. Recently, Chaubey et al. [7] defined and studied the properties of semi-symmetric metric P-connection on a Riemannian manifold and it has been further investigated in [8]. In [10], authors have studied the properties of Lorentzian manifolds endowed with a quarter-symmetric non-metric ξ -connection. A semi-Riemannian manifold M is said to be semisym
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