The Riemannian L 2 topology on the manifold of Riemannian metrics
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.
KeywordsManifold of Riemannian metrics Superspace Manifold of Riemannian structures L2 metric
Mathematics Subject Classification (2000)Primary: 58D17 Secondary: 58B20 51F99
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