The $L^2$-completion of the space of Riemannian metrics is CAT$(0)$: a shorter proof

TL;DR

This paper provides a simplified proof that the $L^2$-completion of the space of Riemannian metrics on a compact manifold is a CAT(0) space, revealing its structure as an $L^2$-mapping space into a CAT(0) space.

Contribution

It offers a shorter, more accessible proof of the CAT(0) property of the $L^2$-completion of Riemannian metrics, extending previous results.

Findings

The $L^2$-completion is CAT(0).

The completion is isometric to an $L^2$-mapping space.

Simplified proof of a known geometric property.

Abstract

We reprove in an easier way a result of Brian Clarke: the completion of the space of Riemannian metrics of a compact, orientable smooth manifold with respect to the $L^2$-distance is CAT$(0)$. In particular we show that this completion is isometric to the space of $L^2$-maps from a standard probability space to a fixed CAT$(0)$ space.