Shape Spaces of Nonlinear Flags

The shape space considered in this article consists of surfaces embedded $$\mathbb {R}^3$$ , that are decorated with curves. It is a special case the Fréchet manifolds nonlinear flags, i.e. nested submanifolds fixed type. gauge invariant elastic metric on involves mean curvature and normal deformation, sum difference principal curvatures $$\kappa _1,\kappa _2$$ . proposed metrics curves involve, addition, geodesic _g,\kappa _n$$ curve surface, as well torsion $$\tau _g$$ More precisely, we show that, help Euclidean metric, tangent at $$(C,\varSigma )$$ can be identified $$C^\infty (C)\times C^\infty (\varSigma form 6-parameter family: $$\begin{aligned} \mathcal G_{(C,\varSigma )}(h_1,h_2)&= a_1\int _C(h_1\kappa _g+{h_2}|_C\kappa _n)^2d\ell&+ a_2\int _{\varSigma }(h_2)^2(\kappa _1-\kappa _2)^2dA\\&+b_1\int _C(D_sh_1-{h_2}|_C\tau _g)^2d\ell&+ b_2\int } (h_2)^2(\kappa _1+\kappa _2)^2dA\\&+c_1\int _C(D_s({h_2}|_C)+h_1\tau c_2\int |\nabla h_2|^2 dA, \end{aligned}$$ where $$h_1\in (C),h_2\in