The Riemannian L^2 topology on the manifold of Riemannian metrics

TL;DR

This paper explores the topology induced by the L^2 Riemannian metric on the manifold of all Riemannian metrics over a closed manifold, revealing an L^1-type topology and its properties.

Contribution

It characterizes the topology and completion of the manifold of metrics under the L^2 metric, providing new insights into its structure and convergence criteria.

Findings

The L^2 topology induces an L^1-type topology on the manifold.

The completion of the manifold under the L^2 metric is characterized.

A practical criterion for convergence in the L^2 metric is provided.

Abstract

We study the manifold of all Riemannian metrics over a closed, finite-dimensional manifold. In particular, we investigate the topology on the manifold of metrics induced by the distance function of the L^2 Riemannian metric - so called because it induces an L^2 topology on each tangent space. It turns out that this topology on the tangent spaces gives rise to an L^1-type topology on the manifold of metrics itself. We study this new topology and its completion, which agrees homeomorphically with the completion of the L^2 metric. We also give a user-friendly criterion for convergence (with respect to the L^2 metric) in the manifold of metrics.