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

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

2010-01-14

arxiv.org/abs/1001.2532v2

Ann. Global Anal. Geom. 39 (2011), no. 2, 131-163

Mathematics

Differential Geometry

31 pages; v2: minor corrections, published version

Scientific paper

10.1007/s10455-010-9227-z

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.

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