Second Fundamental Measure

In Chap. 21, we have seen how to define generalized Lipschitz–Killing curvature measures on geometric subsets, which generalize the Lipschitz–Killing curvature measures on smooth surfaces. However, the geometry of a smooth submanifold needs more precise invariants, like principal directions, principal curvatures, and lines of curvatures, which are determined by the full knowledge of the second fundamental tensor. The goal of this chapter is to propose a definition of a second fundamental measure, which can be evaluated on nonsmooth objects using the theory of the normal cycle.