Equiaffine Darboux Frames for Codimension 2 Submanifolds contained in Hypersurfaces

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Consider a codimension 1 submanifold Nⁿ⊂ Mⁿ⁺¹, where Mⁿ⁺¹ⁿ⁺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.