Open Access
Translator Disclaimer
October, 2017 Equiaffine Darboux frames for codimension 2 submanifolds contained in hypersurfaces
Marcos CRAIZER, Marcelo J. SAIA, Luis F. SÁNCHEZ
J. Math. Soc. Japan 69(4): 1331-1352 (October, 2017). DOI: 10.2969/jmsj/06941331


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.


Download Citation

Marcos CRAIZER. Marcelo J. SAIA. Luis F. SÁNCHEZ. "Equiaffine Darboux frames for codimension 2 submanifolds contained in hypersurfaces." J. Math. Soc. Japan 69 (4) 1331 - 1352, October, 2017.


Published: October, 2017
First available in Project Euclid: 25 October 2017

zbMATH: 06821643
MathSciNet: MR3716497
Digital Object Identifier: 10.2969/jmsj/06941331

Primary: 53A15

Keywords: Darboux frames , developable tangent surfaces , equiaffine metrics , Transon planes , visual contours

Rights: Copyright © 2017 Mathematical Society of Japan


Vol.69 • No. 4 • October, 2017
Back to Top