Abstract

Based on collaborations with M.Bauer, M.Bruveris, P.Harms. For a compact manifold $M^{m$ equipped with a smooth fixed background Riemannian metric $\hat g$ we consider the space $\operatorname{Met}(M)$ of all Riemannian metrics of Sobolev class $H}s$ for real $s>\frac m2$ with respect to $\hat g$. The $L^{2$-metric on $\operatorname{Met}}{C\infty}(M)$ was considered by DeWitt, Ebin, Freed and Groisser, Gil-Medrano and Michor, Clarke. Sobolev metrics of integer order on $\operatorname{Met}_{C^\infty}(M)$ 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 $s$. We derive the geodesic equations and show that they are well-posed under some conditions and induce a locally diffeomorphic geodesic exponential mapping.