Curvature and distance function from a manifold

This paper is concerned with the relations between the differential invariants of a smooth manifold embedded in the Euclidean space and the square of the distance function from the manifold. In particular, we are interested in curvature invariants like the mean curvature vector and the second fundamental form. We find that these invariants can be computed in a very simple way using the third order derivatives of the squared distance function. Moreover, we study a general class of functionals depending on the derivatives up to a given order γ of the squared distance function and we find an algorithm for the computation of the Euler equation. Our class of functionals includes as particular cases the well-known area functional (γ = 2), the integral of the square of the quadratic norm of the second fundamental form (γ = 3), and the Willmore functional.