Affine focal sets of codimension-2 submanifolds contained in hypersurfaces. Issue 5 (22nd April 2018) - Craizer, Marcos; Saia, Marcelo J.; Sánchez, Luis F.
Abstract : In this paper we study the affine focal set, which is the bifurcation set of the affine distance to submanifolds N n contained in hypersurfaces M n +1 of the ( n + 2)-space. We give conditions under which this affine focal set is a regular hypersurface and, for curves in 3-space, we describe its stable singularities. For a given Darboux vector field ξ of the immersion N ⊂ M, one can define the affine metric g and the affine normal plane bundle . We prove that the g -Laplacian of the position vector belongs to if and only if ξ is parallel. For umbilic and normally flat immersions, the affine focal set reduces to a single line. Submanifolds contained in hyperplanes or hyperquadrics are always normally flat. For N contained in a hyperplane L, we show that N ⊂ M is umbilic if and only if N ⊂ L is an affine sphere and the envelope of tangent spaces is a cone. For M hyperquadric, we prove that N ⊂ M is umbilic if and only if N is contained in a hyperplane. The main result of the paper is a general description of the umbilic and normally flat immersions: given a hypersurface f and a point O in the ( n + 1)-space, the immersion ( ν, ν · ( f − O )), where ν is the co-normal of f, is umbilic and normally flat, and conversely, any umbilic and normally flat immersion is of this type.