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 $(C,\Sigma)$ can be identified with $C^\infty(C)\times C^\infty(\Sigma)$ and the gauge invariant elastic metrics form a 6-parameter family that we give explicitly.
Categories: math.DG