Sobolev metrics on the manifold of all Riemannian metrics

TL;DR

This paper explores higher-order Sobolev metrics on the space of all Riemannian metrics, deriving geodesic equations, analyzing well-posedness, and connecting Ricci flow to gradient flows within this geometric framework.

Contribution

It introduces and analyzes higher-order Sobolev metrics on the manifold of Riemannian metrics, including geodesic equations and conditions linking Ricci flow to gradient flows.

Findings

Derivation of geodesic equations for Sobolev metrics

Conditions for well-posedness of geodesic equations

Identification of when Ricci flow is a gradient flow

Abstract

On the manifold $\Met(M)$ of all Riemannian metrics on a compact manifold $M$ one can consider the natural $L^2$-metric as described first by \cite{Ebin70}. In this paper we consider variants of this metric which in general are of higher order. We derive the geodesic equations, we show that they are well-posed under some conditions and induce a locally diffeomorphic geodesic exponential mapping. We give a condition when Ricci flow is a gradient flow for one of these metrics.