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...
