Shape spaces of nonlinear flags

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