## Journal of the Mathematical Society of Japan

- J. Math. Soc. Japan
- Volume 69, Number 4 (2017), 1331-1352.

### Equiaffine Darboux frames for codimension 2 submanifolds contained in hypersurfaces

Marcos CRAIZER, Marcelo J. SAIA, and Luis F. SÁNCHEZ

#### 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 does not admit a vector field with the above property.

#### Article information

**Source**

J. Math. Soc. Japan, Volume 69, Number 4 (2017), 1331-1352.

**Dates**

First available in Project Euclid: 25 October 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.jmsj/1508918560

**Digital Object Identifier**

doi:10.2969/jmsj/06941331

**Mathematical Reviews number (MathSciNet)**

MR3716497

**Zentralblatt MATH identifier**

06821643

**Subjects**

Primary: 53A15: Affine differential geometry

**Keywords**

Darboux frames developable tangent surfaces visual contours Transon planes equiaffine metrics

#### Citation

CRAIZER, Marcos; SAIA, Marcelo J.; SÁNCHEZ, Luis F. Equiaffine Darboux frames for codimension 2 submanifolds contained in hypersurfaces. J. Math. Soc. Japan 69 (2017), no. 4, 1331--1352. doi:10.2969/jmsj/06941331. https://projecteuclid.org/euclid.jmsj/1508918560