Geodesics, distance, and the CAT(0) property for the manifold of Riemannian metrics

Details Geodesics, distance, and the CAT(0) property for the manifold of Riemannian metrics Geodesics, distance, and the CAT(0) property for the manifold of Riemannian metrics

2010-11-05

arxiv.org/abs/1011.1521v3

Mathematics

Differential Geometry

35 pages, 3 figures; v3: major revision and expansion with addition of proof of CAT(0) property

Scientific paper

Given a fixed closed manifold M, we exhibit an explicit formula for the distance function of the canonical L^2 Riemannian metric on the manifold of all smooth Riemannian metrics on M. Additionally, we examine the (metric) completion of the manifold of metrics with respect to the L^2 metric and show that there exists a unique minimal path between any two points. This path is also given explicitly. As an application of these formulas, we show that the metric completion of the manifold of metrics is a CAT(0) space.

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