Manifolds of mappings and shapes

This is an overview article. In his Habilitationsvortrag, Riemann described infinite dimensional manifolds parameterizing functions and shapes of solids. This is taken as an excuse to describe convenient calculus in infinite dimensions which allows for short and transparent proofs of the main facts of the theory of manifolds of smooth mappings. Smooth manifolds of immersions, diffeomorphisms, and shapes, and weak Riemannian metrics on them are treated, culminating in the surprising fact, that geodesic distance can vanish completely for them.

Why Use Sobolev Metrics on the Space of Curves

We study reparametrization invariant Sobolev metrics on spaces of regular curves. We discuss their completeness properties and the resulting usability for applications in shape analysis. In particular, we will argue, that the development of efficient numerical methods for higher order Sobolev type metrics is an extremely desirable goal.

Uniqueness of the Fisher-Rao metric on the space of smooth densities

On a closed manifold of dimension greater than one, every smooth weak Riemannian metric on the space of smooth positive probability densities, that is invariant under the action of the diffeomorphism group, is a multiple of the Fisher--Rao metric.

The exponential law for spaces of test functions and diffeomorphism groups

We prove the exponential law $mathcal A(E times F, G) cong mathcal A(E,mathcal A(F,G))$ (bornological isomorphism) for the following classes $mathcal A$ of test functions: $mathcal B$ (globally bounded derivatives), $W^{infty,p}$ (globally $p$-integrable derivatives), $mathcal S$ (Schwartz space), $mathcal D$ (compact sport, $mathcal B^{[M]}$ (globally Denjoy_Carleman), $W^{[M],p}$ (Sobolev_Denjoy_Carleman), $mathcal S_{[L]}^{[M]}$ (Gelfand_Shilov), and $mathcal D^{[M]}$. Here $E, F, G$ are convenient vector spaces (finite dimensional in the cases of $W^{infty,p}$, $mathcal D$, $W^{[M],p}$, and $mathcal D^{[M]})$, and $M=(M_k)$ is a weakly log-convex weight sequence of moderate growth. As application we give a new simple proof of the fact that the groups of diffeomorphisms $operatorname{Diff} mathcal B$, $operatorname{Diff} W^{infty,p}$, $operatorname{Diff} mathcal S$, and $operatorname{Diff}mathcal D$ are $C^infty$ Lie groups, and $operatorname{Diff} mathcal B^{{M}}$, $operatorname{Diff}W^{{M},p}$, $operatorname{Diff} mathcal S_{{L}}^{{M}}$, and $operatorname{Diff}mathcal D^{[M]}$, for non-quasianalytic $M$, are $C^{{M}}$ Lie groups, where $operatorname{Diff}mathcal A = {operatorname{Id} +f : f in mathcal A(mathbb R^n,mathbb R^n), inf_{x in mathbb R^n} det(mathbb I_n+ df(x))>0}$. We also discuss stability under composition.

An exotic zoo of diffeomorphism groups on $mathbb R^n$

Let $C^{[M]}$ be a (local) Denjoy-Carleman class of Beurling or Roumieu type, where the weight sequence $M=(M_k)$ is log-convex and has moderate growth. We prove that the groups ${operatorname{Diff}}mathcal{B}^{[M]}(mathbb{R}^n)$, ${operatorname{Diff}}W^{[M],p}(mathbb{R}^n)$, ${operatorname{Diff}}{mathcal{S}}{}_{[L]}^{[M]}(mathbb{R}^n)$, and ${operatorname{Diff}}mathcal{D}^{[M]}(mathbb{R}^n)$ of $C^{[M]}$-diffeomorphisms on $mathbb{R}^n$ which differ from the identity by a mapping in $mathcal{B}^{[M]}$ (global Denjoy--Carleman), $W^{[M],p}$ (Sobolev-Denjoy-Carleman), ${mathcal{S}}{}_{[L]}^{[M]}$ (Gelfand--Shilov), or $mathcal{D}^{[M]}$ (Denjoy-Carleman with compact support) are $C^{[M]}$-regular Lie groups. As an application we use the $R$-transform to show that the Hunter-Saxton PDE on the real line is well-posed in any of the classes $W^{[M],1}$, ${mathcal{S}}{}_{[L]}^{[M]}$, and $mathcal{D}^{[M]}$. Here we find some surprising groups with continuous left translations and $C^{[M]}$ right translations (called half-Lie groups), which, however, also admit $R$-transforms.

$R$-transforms for Sobolev $H^2$-metrics on spaces of plane curves

We consider spaces of smooth immersed plane curves (modulo translations and/or rotations), equipped with reparameterization invariant weak Riemannian metrics involving second derivatives. This includes the full $H^2$-metric without zero order terms. We find isometries (called $R$-transforms) from some of these spaces into function spaces with simpler weak Riemannian metrics, and we use this to give explicit formulas for geodesics, geodesic distances, and sectional curvatures. We also show how to utilise the isometries to compute geodesics numerically.

A Metric on Shape Space with Explicit Geodesics

This paper studies a specific metric on plane curves that has the property of being isometric to classical manifold (sphere, complex projective, Stiefel, Grassmann) modulo change of parametrization, each of these classical manifolds being associated to specific qualifications of the space of curves (closed-open, modulo rotation etc...) Using these isometries, we are able to explicitely describe the geodesics, first in the parametric case, then by modding out the paremetrization and considering horizontal vectors. We also compute the sectional curvature for these spaces, and show, in particular, that the space of closed curves modulo rotation and change of parameter has positive curvature. Experimental results that explicitly compute minimizing geodesics between two closed curves are finally provided