[Paper Review] Metrics on Spaces of Surfaces where Horizontality equals Normality

Martin Bauer, Philipp Harms|arXiv (Cornell University)|Mar 6, 2014

Morphological variations and asymmetry28 references4 citations

TL;DR

This paper characterizes reparametrization-invariant Sobolev-type metrics on the space of immersions where the horizontal bundle (with respect to shape space projection) coincides with the normal bundle (with respect to the ambient metric). By identifying metrics for which horizontal and normal decompositions align, the study simplifies geodesic computation on shape space, and in the case of planar curves, shows that higher metric regularity controls the smoothness of curves in the metric completion.

ABSTRACT

In this article, we study metrics on shape space of surfaces that have a particularly simple horizontal bundle. More specifically, we consider reparametrization invariant Sobolev type metrics $G$ on the space $\operatorname{Imm}(M,N)$ of immersions of a compact manifold $M$ in a Riemannian manifold $(N,\overline{g})$. The tangent space $T_f\operatorname{Imm}(M,N)$ at each immersion $f$ has two natural splittings: one into components that are tangential/normal to the surface $f$ (with respect to $\overline{g}$) and another one into vertical/horizontal components (with respect to the projection onto the shape space $B_i(M,N)=\operatorname{Imm}(M,N)/\operatorname{Diff}(M)$ of unparametrized immersions and with respect to the metric $G$). The first splitting can be easily calculated numerically, while the second splitting is important because it mirrors the geometry of shape space and geodesics thereon. Motivated by facilitating the numerical calculation of geodesics on shape space, we characterise all metrics $G$ such that the two splittings coincide. In the special case of planar curves, we show that the regularity of curves in the metric completion can be controlled by choosing a strong enough metric within this class.

Motivation & Objective

• To identify reparametrization-invariant Sobolev-type metrics on the space of immersions where the horizontal and normal splittings of the tangent space coincide.
• To simplify the numerical computation of geodesics on shape space by exploiting the coincidence of horizontal and normal decompositions.
• To analyze the regularity of curves in the metric completion of shape space for planar curves under such metrics.
• To provide a geometric framework that aligns intrinsic shape space geometry with ambient Riemannian structure.

Proposed method

• The study uses the space of immersions $\operatorname{Imm}(M,N)$ of a compact manifold $M$ into a Riemannian manifold $(N,\overline{g})$ as the underlying configuration space.
• It considers reparametrization-invariant Sobolev-type metrics $G$ on $\operatorname{Imm}(M,N)$, defined via inner products on vector fields along immersions.
• The tangent space at each immersion $f$ is split both tangentially/normal to the image of $f$ and vertically/horizontally with respect to the shape space quotient $\operatorname{Imm}(M,N)/\operatorname{Diff}(M)$.
• The key condition is that the horizontal distribution induced by $G$ coincides with the normal bundle of $f$ in $N$, which simplifies the geometric structure.
• For planar curves, the paper analyzes the metric completion and shows that increasing the Sobolev regularity of $G$ controls the regularity of curves in the completion.

Experimental results

Research questions

• RQ1 Which reparametrization-invariant Sobolev-type metrics on the space of immersions induce a horizontal bundle that coincides with the normal bundle of the immersed surface in the ambient manifold?
• RQ2 How does the coincidence of horizontal and normal splittings simplify the computation of geodesics on shape space?
• RQ3 What is the relationship between the regularity of the metric $G$ and the regularity of curves in the metric completion of the shape space for planar immersions?
• RQ4 Can the geometric structure of shape space be simplified by choosing metrics for which horizontal directions align with normal directions?

Key findings

• The paper fully characterizes all reparametrization-invariant Sobolev-type metrics $G$ for which the horizontal bundle with respect to the shape space quotient coincides with the normal bundle of the immersion in the ambient manifold.
• For such metrics, the horizontal and normal splittings of the tangent space at each immersion are identical, significantly simplifying the geometric and computational structure.
• In the case of planar curves, the metric completion of the shape space contains only curves of regularity determined by the order of the Sobolev metric; higher-order metrics yield smoother curves.
• The coincidence of horizontal and normal components allows for more efficient numerical computation of geodesics on shape space, as the horizontal projection reduces to the normal projection.