Concircular Curvature on Warped Product Manifolds and Applications
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Concircular Curvature on Warped Product Manifolds and Applications Uday Chand De1 · Sameh Shenawy2 · Bülent Ünal3 Received: 26 February 2019 / Revised: 21 November 2019 © Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2019
Abstract This study aims mainly at investigating the effects of concircular flatness and concircular symmetry of a warped product manifold on its fiber and base manifolds. Concircularly flat and concircularly symmetric warped product manifolds are investigated. The divergence-free concircular curvature tensor on warped product manifolds is considered. Finally, we apply some of these results to generalized Robertson–Walker and standard static space-times. Keywords Concircular curvature · Concircularly symmetric manifolds · Concircularly flat manifolds · Warped product manifolds Mathematics Subject Classification Primary 53C21 · 53C25; Secondary 53C50 · 53C80
1 Introduction A transformation which preserves geodesic circles is called a concircular transformation [31]. The geometry which deals with concircular transformation is called
Communicated by Rosihan M. Ali.
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Uday Chand De [email protected] Sameh Shenawy [email protected]; [email protected] Bülent Ünal [email protected]
1
Department of Pure Mathematics, University of Calcutta, 35 Bally-Gaunge Circular Road, Kolkata, West Bengal 700019, India
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Basic Science Department, Modern Academy for Engineering and Technology, Maadi, Egypt
3
Department of Mathematics, Bilkent University, 06800 Bilkent, Ankara, Turkey
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U. C. De et al.
concircular geometry. The concircular curvature tensor C remains invariant under concircular transformation of a (pseudo-)Riemannian manifold M. M is called concircularly flat if its concircular curvature tensor C vanishes at every point. A concircularly flat manifold M is a manifold of constant curvature. Thus, the tensor C measures the deviation of M from constant curvature. (For further details, see [1,31].) In a series of studies, Pokhariyal and Mishra studied the recurrent properties and relativistic significance of concircular curvature tensor, among many others, in Riemannian manifolds [22–25]. Concircularly semi-symmetric K -contact manifolds are considered in [18], and concircularly recurrent Finsler manifolds are studied in [33] . In [19], the authors considered N (k)-contact metric manifolds satisfying C·P = 0, where P denotes the projective curvature tensor. Similarly, a study of (k, μ, ν) −contact metric 3-manifolds satisfying one of the conditions ∇C = 0, C (ζ, X ) · C = 0, R (ζ, X ) · C = 0, where ζ is the Reeb field, is considered in [16]. Perfect fluid space-times with either vanishing or divergence-free concircular curvature tensor are considered in [2]. The authors of [34] considered equitorsion concircular mapping between generalized Riemannian manifolds (in the sense of Eisenhart’s definition) and obtained some invariant curvature tensors. These tensors are generalizations of concircular curvature tensor on Riemannian manifolds. In a
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