For a compact manifold Mm equipped with a smooth fixed background Riemannian metric ĝ we consider the space MetHs (ĝ) (M) of all Riemannian metrics of Sobolev class Hs for real s > m/2 with respect to ĝ. The L2-metric on MetC∞ (M) was considered by DeWitt, Ebin, Freed and Groisser, Gil-Medrano and Michor, Clarke. Sobolev metrics of integer order on MetC∞ (M) were considered in [M. Bauer, P. Harms, and P. W. Michor: Sobolev metrics on the manifold of all Riemannian metrics. J. Differential Geom., 94(2):187-208, 2013.] In this talk we consider variants of these Sobolev metrics which include Sobolev metrics of any positive real (not integer) order s > m/2. We derive the geodesic equations and show that they are well-posed under some conditions and induce a locally diffeomorphic geodesic exponential mapping.
Based on collaborations with M. Bauer, M. Bruveris, P. Harms.
Peter Michor (Vienna): General Sobolev metrics on the manifold of all Riemannian metrics
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