Geodesics, distance, and the CAT(0) property for the manifold of Riemannian metrics

TL;DR

This paper derives explicit formulas for the distance and minimal paths in the space of Riemannian metrics on a manifold, demonstrating that its metric completion is a CAT(0) space, thus revealing its non-positive curvature properties.

Contribution

It provides explicit formulas for distances and geodesics in the manifold of Riemannian metrics and proves the space's metric completion is a CAT(0) space.

Findings

Explicit distance formula for the L^2 metric on Riemannian metrics

Existence and uniqueness of minimal geodesics between metrics

The metric completion is a CAT(0) space

Abstract

Given a fixed closed manifold M, we exhibit an explicit formula for the distance function of the canonical L^2 Riemannian metric on the manifold of all smooth Riemannian metrics on M. Additionally, we examine the (metric) completion of the manifold of metrics with respect to the L^2 metric and show that there exists a unique minimal path between any two points. This path is also given explicitly. As an application of these formulas, we show that the metric completion of the manifold of metrics is a CAT(0) space.