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M. Bauer, P. harms:
"Metrics on spaces of surfaces where horizontality equals normality";
Differential Geometry and its Applications, 39 (2015), 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