准共形曲率张量对翘曲乘积流形的影响

Axioms Pub Date : 2024-07-26 DOI:10.3390/axioms13080500
Bang-Yen Chen, S. Shenawy, U.c. De, Alaa Rabie, Nasser Bin Turki
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引用次数: 0

摘要

这项研究探讨了准共形平坦、准共形对称或无发散准共形曲率张量的存在对单翘积流形的因子流形的影响。准共形平坦翘积流形表现出三种不同的情况:一种情况是基流形具有恒定曲率,而在另外两种情况下,基流形是准爱因斯坦流形。或者,纤维流形在两种情况下具有恒定曲率,而在一种情况下是爱因斯坦流形。准共形对称翘积流形有三种不同的情况:第一种情况是基流形是 Ricci 对称的,而纤维是爱因斯坦的;第二种情况是基流形是 Cartan 对称的,而纤维具有恒定曲率;最后一种情况是纤维是 Cartan 对称的,而基流形的 Ricci 张量是 Codazzi 类型的。最后,还提供了单翘积流形的条件,即允许无发散的准共形曲率张量,以确保因子流形的黎曼曲率张量是谐调的。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
The Impact of Quasi-Conformal Curvature Tensor on Warped Product Manifolds
This work investigates the effects on the factor manifolds of a singly warped product manifold resulting from the presence of a quasi-conformally flat, quasi-conformally symmetric, or divergence-free quasi-conformal curvature tensor. Quasi-conformally flat warped product manifolds exhibit three distinct scenarios: in one scenario, the base manifold has a constant curvature, while in the other two scenarios, it is quasi-Einstein. Alternatively, the fiber manifold has a constant curvature in two scenarios and is Einstein in one scenario. Quasi-conformally symmetric warped product manifolds present three distinct cases: in the first scenario, the base manifold is Ricci-symmetric and the fiber is Einstein; in the second case, the base manifold is Cartan-symmetric and the fiber has constant curvature; and in the last case, the fiber is Cartan-symmetric, and the Ricci tensor of the base manifold is of Codazzi type. Finally, conditions are provided for singly warped product manifolds that admit a divergence-free quasi-conformal curvature tensor to ensure that the Riemann curvature tensors of the factor manifolds are harmonic.
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