Syied Abdallah Abdelhameed, De Uday Chand, Turki Nasser Bin, Vîlcu Gabriel-Eduard
Department of Mathematics, Faculty of Science, Zagazig University, P.O. Box 44519, Zagazig, Egypt.
Department of Pure Mathematics, University of Calcutta, 35, Ballygaunge Circular Road Kolkata 700019, West Bengal, India.
Heliyon. 2024 Aug 22;10(16):e36355. doi: 10.1016/j.heliyon.2024.e36355. eCollection 2024 Aug 30.
We first investigate how the flatness and the symmetry of the -tensor impact both the base manifold and the fiber manifold of a warped product manifold. In both cases, we determine the form of the -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 -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.
我们首先研究(\varPhi)-张量的平坦性和对称性如何影响扭曲积流形的底流形和纤维流形。在这两种情况下，我们确定了纤维流形和底流形上(\varPhi)-张量的形式。此外，我们确定纤维流形具有常曲率，而底流形是爱因斯坦流形。还证明了一个(\varPhi)-曲率平坦的广义 Robertson-Walker（GRW）时空是完美流体且静态的。此外，该时空要么代表暗物质时代，要么状态方程具有特定形式。
