Ricci flow of unwarped and warped product manifolds

TL;DR

This paper studies Ricci flow on both unwarped and warped product manifolds, revealing generic features like singularity formation and isotropisation, with analytic and numerical solutions providing insights into their geometric evolution.

Contribution

It provides the first detailed analysis of Ricci flow on warped product manifolds, including analytic solutions for special cases and numerical studies for more general scenarios.

Findings

Identification of singularity formation during Ricci flow

Analytic solutions for specific warped manifolds

Numerical insights into warp factor and curvature evolution

Abstract

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.