Bach flows of product manifolds

TL;DR

This paper studies the behavior of Bach tensor-based geometric flows on product and warped manifolds, providing exact solutions, fixed point conditions, and comparisons with Ricci flow to understand their properties.

Contribution

It offers new exact solutions and fixed point analyses for Bach flows on product and warped manifolds, expanding understanding of higher-order geometric flows.

Findings

Exact solutions for Bach flow on $S^2\times S^2$ and $R^2\times S^2$

Fixed point conditions for general $(2,2)$ manifolds

Reduced flow equations for asymmetrically warped manifolds

Abstract

We investigate various aspects of a geometric flow defined using the Bach tensor. Firstly, using a well-known split of the Bach tensor components for $(2,2)$ unwarped product manifolds, we solve the Bach flow equations for typical examples of product manifolds like $S^2\times S^2$, $R^2\times S^2$. In addition, we obtain the fixed point condition for general $(2,2)$ manifolds and solve it for a restricted case. Next, we consider warped manifolds. For Bach flows on a special class of asymmetrically warped four manifolds, we reduce the flow equations to a first order dynamical system, which is solved exactly to find the flow characteristics. We compare our results for Bach flow with those for Ricci flow and discuss the differences qualitatively. Finally, we conclude by mentioning possible directions for future work.