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

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.