Equiaffine Darboux frames for codimension 2 submanifolds contained in hypersurfaces

Abstract

Consider a codimension 1 submanifold Nⁿ ⊂ Mⁿ⁺¹, where Mⁿ⁺¹ ⊂ ℝⁿ⁺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 does not admit a vector field with the above property.