No abstract

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

This article provides an overview of various notions of shape spaces, including the space of parametrized and unparametrized curves, the space of immersions, the diffeomorphism group and the space of Riemannian metrics. We discuss the Riemannian metrics that can be defined thereon, and what is known about the properties of these metrics. We put particular emphasis on the induced geodesic distance, the geodesic equation and its well-posedness, geodesic and metric completeness and properties of the curvature.

Metrics on shape spaces are used to describe deformations that take one shape to another, and to define a distance between shapes. We study a family of metrics on the space of curves, which includes several recently proposed metrics, for which the metrics are characterised by mappings into vector spaces where geodesics can be easily computed. This family consists of Sobolev-type Riemannian metrics of order one on the space Imm(S 1 , R 2 ) of parameterized plane curves and the quotient space Imm(S 1 , R 2 )/ Diff(S 1 ) of unparameterized curves. For the space of open parameterized curves we find an explicit formula for the geodesic distance and show that the sectional curvatures vanish on the space of parameterized open curves and are nonnegative on the space of unparameterized open curves. For one particular metric we provide a numerical algorithm that computes geodesics between unparameterized, closed curves, making use of a constrained formulation that is implemented numerically using the RATTLE algorithm. We illustrate the algorithm with some numerical tests between shapes.2010 Mathematics Subject Classification. 58B20, 58D15, 65D18.

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.