Ricci flow of unwarped and warped product manifolds

Details Ricci flow of unwarped and warped product manifolds Ricci flow of unwarped and warped product manifolds

2009-08-10

arxiv.org/abs/0908.1295v2

Int.J.Geom.Meth.Mod.Phys.07:837-856,2010

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

25 pages, 25 figures, some figures replaced

Scientific paper

10.1142/S0219887810004579

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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