Let $M$ and $N$ be connected manifolds without boundary with $\dim(M) < \dim(N)$, and let $M$ compact. Then shape space in this work is either the manifold of submanifolds of $N$ that are diffeomorphic to $M$, or the orbifold of unparametrized immersions of $M$ in $N$. We investigate the Sobolev Riemannian metrics on shape space: These are induced by metrics of the following form on the space of immersions: $$ G^P_f(h,k) = \int_{M} \g(P^f h, k)\, \vol(f^*\g)$$ where $\g$ is some fixed metric on $N$, $f^*\g$ is the induced metric on $M$, $h,k \in \Gamma(f^*TN)$ are tangent vectors at $f$ to the space of embeddings or immersions, and $P^f$ is a positive, selfadjoint, bijective scalar pseudo differential operator of order $2p$ depending smoothly on $f$. We consider later specifically the operator $P^f=1 + A\Delta^p$, where $\Delta$ is the Bochner-Laplacian on $M$ induced by the metric $f^*\bar g$. For these metrics we compute the geodesic equations both on the space of immersions and on shape space, and also the conserved momenta arising from the obvious symmetries. We also show that the geodesic equation is well-posed on spaces of immersions, and also on diffeomorphism groups. We give examples of numerical solutions.Comment: 52 pages, final version as it will appea

Abstract. We study Sobolev-type metrics of fractional order s ≥ 0 on the group Diffc(M ) of compactly supported diffeomorphisms of a manifold M . We show that for the important special case M = S 1 the geodesic distance on Diffc(S 1 ) vanishes if and only if s ≤ 1 2 . For other manifolds we obtain a partial characterization: the geodesic distance on Diffc(M ) vanishes for M = R × N, s < 1 2 and for M = S 1 × N, s ≤ 1 2 , with N being a compact Riemannian manifold. On the other hand the geodesic distance on Diffc(M ) is positive for dim(M ) = 1, s > 1 2 and dim(M ) ≥ 2, s ≥ 1. For M = R n we discuss the geodesic equations for these metrics. For n = 1 we obtain some well known PDEs of hydrodynamics: Burgers' equation for s = 0, the modified Constantin-Lax-Majda equation for s = 1 2 and the Camassa-Holm equation for s = 1.

Abstract. This paper extends parts of the results from [P.W. Michor and D. Mumford, Appl. Comput. Harmon. Anal., 23 (2007), pp. 74-113] for plane curves to the case of hypersurfaces in R n . Let M be a compact connected oriented n − 1 dimensional manifold without boundary like the sphere or the torus. Then shape space is either the manifold of submanifolds of R n of type M , or the orbifold of immersions from M to R n modulo the group of diffeomorphisms of M . We investigate almost local Riemannian metrics on shape space. These are induced by metrics of the following form on the space of immersions:whereḡ is the Euclidean metric on R n , f * ḡ is the induced metric on M , h, k ∈ C ∞ (M, R n ) are tangent vectors at f to the space of embeddings or immersions, where Φ : R 2 → R >0 is a suitable smooth function, Vol(f ) = M vol(f * ḡ ) is the total hypersurface volume of f (M ), and the trace Tr(L) of the Weingarten mapping is the mean curvature. For these metrics we compute the geodesic equations both on the space of immersions and on shape space, the conserved momenta arising from the obvious symmetries, and the sectional curvature. For special choices of Φ we give complete formulas for the sectional curvature. Numerical experiments illustrate the behavior of these metrics.

Abstract. The Virasoro-Bott group endowed with the right-invariant L 2 -metric (which is a weak Riemannian metric) has the KdV-equation as geodesic equation. We prove that this metric space has vanishing geodesic distance.

On the manifold M(M ) of all Riemannian metrics on a compact manifold M one can consider the natural L 2 -metric as described first by [11]. In this paper we consider variants of this metric which in general are of higher order. We derive the geodesic equations, we show that they are well-posed under some conditions and induce a locally diffeomorphic geodesic exponential mapping. We give a condition when Ricci flow is a gradient flow for one of these metrics.