Symmetries of Ricci Flows

TL;DR

This paper identifies Lie point symmetries of Ricci flows on n-dimensional manifolds, introduces a method to reuse these symmetries for specific metrics, and derives invariant solutions for particular metric families.

Contribution

It presents a novel method to compute and utilize Lie symmetries of Ricci flows and related Einstein equations for specific metric types.

Findings

Lie point symmetries of Ricci flow are characterized.

A method to reuse symmetries for particular metrics is introduced.

Invariant solutions of Ricci flow are obtained for selected metrics.

Abstract

In the present work we find the Lie point symmetries of the Ricci flow on an $n$-dimensional manifold. and we introduce a method in order to reutilize these symmetries to obtain the Lie point symmetries of particular metrics. We apply this method to retrieve the Lie point symmetries of the Einstein equations -- seen as a "static" Ricci flow -- , and of some particular types of metrics of interest, such as, on warped products of manifolds. Finally, we use the symmetries found to obtain invariant solutions of the Ricci flow for the particular families of metrics considered.