Almost Local Metrics on Shape Space of Hypersurfaces inn-Space

Abstract

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.

Overview of the Geometries of Shape Spaces and Diffeomorphism Groups

“…Then geodesics of very long translations will go via a strong shrinking part and growing part, and almost all of the translation will be done with the shrunken version of the shape. This behavior, which also occurs for the class of conformal metrics, is described in [12]. 6 Sobolev type metrics on shape space Sobolev-type inner metrics on the space Imm(M, R d ) of immersions are metrics of the form…”

“…A reason for this might be that the distance between the two boundary shapes is not sufficiently large. In the article [12] it has been shown that this behavior carries over to the case of higher-dimensional surfaces, c.f. Fig.…”

“…More general almost local metrics on the space of plane curves were considered in [96] and they have been generalized to hypersurfaces in higher dimensions in [3,12,13].…”

“…3.1]. The proof was generalized to the space of hypersurfaces in higher dimensions in [12,Thm. 8.7].…”

“…For plane curves and conformal metrics the curvature has been calculated in [115] and for Ψ (c) = Φ( c , κ c ) in [96]. Similarly for higher dimensional surfaces the curvature has been calculated for Ψ (q) = Φ(Vol q , H q ) in [12].…”