The 𝐿²-completion of the space of Riemannian metrics is CAT(0): A shorter proof

Abstract

We reprove in an easier way a result of Brian Clarke [Math. Z. 273 (2013), pp. 55–93]: the completion of the space of Riemannian metrics of a compact, orientable smooth manifold with respect to the L 2 L^2 -distance is CAT ( 0 ) (0) . In particular we show that this completion is isometric to the space of L 2 L^2 -maps from a standard probability space to a fixed CAT ( 0 ) (0) space.