Equiaffine Darboux Frames for Codimension 2 Submanifolds contained in Hypersurfaces

TL;DR

This paper investigates the singularities of envelopes of tangent spaces for codimension 1 submanifolds within hypersurfaces, introducing equiaffine Darboux frames and analyzing cases with or without certain tangent vector fields.

Contribution

It develops a framework for equiaffine Darboux frames for codimension 2 submanifolds in hypersurfaces, including explicit examples and conditions for the existence of special tangent vector fields.

Findings

Characterization of singularities of tangent space envelopes.

Construction of equiaffine and apolar metrics on submanifolds.

Explicit example of submanifold without a specific tangent vector field.

Abstract

Consider a codimension $1$ submanifold $N^n\subset M^{n+1}$, where $M^{n+1}\subset\mathbb{R}^{n+2}$ 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.