M. Bauer, P. harms:

"Metrics on spaces of surfaces where horizontality equals normality";

Differential Geometry and its Applications,39(2015), S. 166 - 183.

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.

Metrics on Spaces of Surfaces where Horizontality equals Normality (PDF Download Available). Available from: https://www.researchgate.net/publication/260604990_Metrics_on_Spaces_of_Surfaces_where_Horizontality_equals_Normality [accessed Jan 4, 2016].

http://dx.doi.org/10.1016/j.difgeo.2014.12.008

Erstellt aus der Publikationsdatenbank der Technischen Universität Wien.