Critical points of higher order for the normal map of immersions in Rd

We study the critical points of the normal map ν:NM→Rk+n, where M is an immersed k-dimensional submanifold of Rk+n, NM is the normal bundle of M and ν(m,u)=m+u if u∈NmM. Usually, the image of these critical points is called the focal set. However, in that set there is a subset where the focusing is highest, as happens in the case of curves in R3 with the curve of the centers of spheres with contact of third order with the curve. We give a definition of r-critical points of a smooth map between manifolds, and apply it to study the 2 and 3-critical points of the normal map in general and the 2-critical points for the case k=n=2 in detail. In the later case we analyze the relation with the strong principal directions of Montaldi (1986) [2].