无曲和曲积流形的Ricci流
Ricci flow of unwarped and warped product manifolds
摘要：
我们分析里奇流（规范化/非规范化）的产品流形-unwarped以及翘曲，通过研究通用的例子。首先，我们调查这样的流动的未翘曲的情况下的流形类型$\mathbb S^n\times \mathbb S^m$，$\mathbb S^n\times \mathbb H^m$，$\mathbb H^m\times \mathbb H^n$，也类似的多个产品。我们能够挑出通用的功能，如奇点形成，各向同性在特定值的流动参数和演化特征。随后，受弯曲膜世界和额外维度的启发，我们研究了弯曲时空的里奇流。在这里，我们能够找到一个特殊的情况下，通过变量分离的解析解。对于其他人，我们数值求解方程（向前和向后流动），并得出一些有用的推论的翘曲因子的演变，标量曲率以及在有限值的流动参数的奇异性的发生。我们还研究了初始条件对流的奇异性的依赖性。我们希望我们的结果是有用的，在任何物理/数学背景下，这种产品流形可能会出现。
英文摘要：
We analyse Ricci flow (normalised/un-normalised) of product manifolds --unwarped as well as warped, through a study of generic examples. First, we investigate such flows for the unwarped scenario with manifolds of the type $\mathbb S^n\times \mathbb S^m$, $\mathbb S^n\times \mathbb H^m$, $\mathbb H^m\times \mathbb H^n$ and also, similar multiple products. We are able to single out generic features such as singularity formation, isotropisation at particular values of the flow parameter and evolution characteristics. Subsequently, motivated by warped braneworlds and extra dimensions, we look at Ricci flows of warped spacetimes. Here, we are able to find analytic solutions for a special case by variable separation. For others we numerically solve the equations (for both the forward and backward flow) and draw certain useful inferences about the evolution of the warp factor, the scalar curvature as well the occurence of singularities at finite values of the flow parameter. We also investigate the dependence of the singularities of the flow on the inital conditions. We expect our results to be useful in any physical/mathematical context where such product manifolds may arise.
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