Shape Spaces of Nonlinear Flags

Abstract

The shape space considered in this article consists of surfaces embedded in $\mathbb {R}^3$, that are decorated with curves. It is a special case of the Fréchet manifolds of nonlinear flags, i.e. nested submanifolds of a fixed type. The gauge invariant elastic metric on the shape space of surfaces involves the mean curvature and the normal deformation, i.e. the sum and the difference of the principal curvatures $\kappa _1,\kappa _2$. The proposed gauge invariant elastic metrics on the space of surfaces decorated with curves involve, in addition, the geodesic and normal curvatures $\kappa _g,\kappa _n$ of the curve on the surface, as well as the geodesic torsion $\tau _g$.

More precisely, we show that, with the help of the Euclidean metric, the tangent space at $(C,\varSigma )$ can be identified with $C^\infty (C)\times C^\infty (\varSigma )$ and the gauge invariant elastic metrics form a 6-parameter family:

$$\begin{aligned} \mathcal G_{(C,\varSigma )}(h_1,h_2)&= a_1\int _C(h_1\kappa _g+{h_2}|_C\kappa _n)^2d\ell&+ a_2\int _{\varSigma }(h_2)^2(\kappa _1-\kappa _2)^2dA\\&+b_1\int _C(D_sh_1-{h_2}|_C\tau _g)^2d\ell&+ b_2\int _{\varSigma } (h_2)^2(\kappa _1+\kappa _2)^2dA\\&+c_1\int _C(D_s({h_2}|_C)+h_1\tau _g)^2d\ell&+ c_2\int _{\varSigma } |\nabla h_2|^2 dA, \end{aligned}$$

where $h_1\in C^\infty (C),h_2\in C^\infty (\varSigma )$.

Alice Barbora Tumpach is supported by FWF grant I 5015-N, Institut CNRS Pauli, and University of Lille. The first and the third authors are supported by a grant of the Romanian Ministry of Education and Research, CNCS-UEFISCDI, project number PN-III-P4-ID-PCE-2020-2888, within PNCDI III.