Characterization of warped product manifolds through the [Formula: see text] -curvature tensor with applications to relativity

Abdallah Abdelhameed Syied; Uday Chand De; Nasser Bin Turki; Gabriel-Eduard Vîlcu

doi:10.1016/j.heliyon.2024.e36355

Characterization of warped product manifolds through the $W_{2}$ -curvature tensor with applications to relativity

Heliyon. 2024 Aug 22;10(16):e36355. doi: 10.1016/j.heliyon.2024.e36355. eCollection 2024 Aug 30.

Authors

Abdallah Abdelhameed Syied¹, Uday Chand De², Nasser Bin Turki³, Gabriel-Eduard Vîlcu^{4

5}

Affiliations

¹ Department of Mathematics, Faculty of Science, Zagazig University, P.O. Box 44519, Zagazig, Egypt.
² Department of Pure Mathematics, University of Calcutta, 35, Ballygaunge Circular Road Kolkata 700019, West Bengal, India.
³ Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia.
⁴ "Gheorghe Mihoc-Caius Iacob" Institute of Mathematical Statistics and Applied Mathematics of the Romanian Academy, 050711 Bucharest, Romania.
⁵ National University of Science and Technology Politehnica Bucharest, Faculty of Applied Sciences, Department of Mathematics and Informatics, 313 Splaiul Independenţei, 060042 Bucharest, Romania.

Abstract

We first investigate how the flatness and the symmetry of the $W_{2}$ -tensor impact both the base manifold and the fiber manifold of a warped product manifold. In both cases, we determine the form of the $W_{2}$ -tensor on both the fiber and the base manifolds. Additionally, we establish that the fiber manifold is of constant curvature, while the base manifold is Einstein. It is also proved that a $W_{2}$ -curvature flat GRW space-time is perfect fluid and static. Furthermore, the space-time either represents dark matter era or the state equation has a specific form.

Keywords: W 2 -curvature tensor; Einstein manifolds; GRW space-times; Singly warped product manifolds; primary, 53C25, 53C50; secondary, 53C80, 53B20.