Abstract. Consider a codimension 1 submanifold N n ⊂ M n+1 , where M n+1 ⊂ R n+2 is a hypersurface. The envelope of tangent spaces of M along N generalizes the concept of tangent developable surface of a surface along a curve. In this paper, we study the singularities of these envelopes.There are some important examples of submanifolds that admit a vector field tangent to M and transversal to N whose derivative in any direction of N is contained in N . When this is the case, one can construct transversal plane bundles and affine metrics on N with the desirable properties of being equiaffine and apolar. Moreover, this transversal bundle coincides with the classical notion of Transon plane. But we also give an explicit example of a submanifold that do not admit a vector field with the above property.Mathematics Subject Classification (2010). 53A15.

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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 condition 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 A . We prove that the g-Laplacian of the position vector belongs to A 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.Mathematics Subject Classification (2010). 53A15.

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