Geodesic distance for right invariant Sobolev metrics of fractional order on the diffeomorphism group

Abstract

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.

Metric registration of curves and surfaces using optimal control

“…In the context of shape analysis of submanifolds a similar result has been found by Michor and Mumford for the L 2 -metric on the space of unparametrized submanifolds and on the diffeomorphism group, see [53]. These results have been later extended to spaces of parametrized submanifolds and to fractional-order metrics on diffeomorphism groups, see [8,12]. For Sobolev metrics of order s ≥ 1 on spaces of submanifolds the following theorem shows that this ill-behavior cannot appear, which renders this class of metrics relevant for applications in shape analysis.…”

“…As mentioned earlier, d Var defined by (12) and (13) is only a pseudo-distance on S unless the RKHS W associated with the kernel is such that the mapping [µ] is injective. Unfortunately, it can be seen that, in the case of oriented varifold representation (11), this is actually impossible for the full space of unparametrized immersed curves, no matter the choice of W (see for example [7] for a counterexample with curves).…”

Overview of the Geometries of Shape Spaces and Diffeomorphism Groups

“…This naturally raises the question, what happens for the H s -metric with 0 < s < 1. For M = S 1 a complete answer is provided in [6], whereas for more general manifolds N a partial answer was given in the articles [6,8]. …”

Fractional Sobolev metrics on spaces of immersions

“…The variational formulations allow one to study analytical properties of the PDEs in relation to geometric properties of the underlying infinite-dimensional Riemannian manifold [51,48,4,5,13,34]. Most importantly, local well-posedness of the PDE, including smooth dependence on initial conditions, is closely related to smoothness of the geodesic spray on Sobolev completions of the configuration space [23].…”