Critical points of higher order for the normal map of immersions in R^ d

Abstract

We study the critical points of the normal map ν : N M → R k + n , where M is an immersed k-dimensional submanifold of R k + n , N M is the normal bundle of M and ν ( m , u ) = m + u if u ∈ N m M. 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 R 3 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].