Journal of the Mathematical Society of Japan

Equiaffine Darboux frames for codimension 2 submanifolds contained in hypersurfaces

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

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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

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

First available in Project Euclid: 25 October 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53A15: Affine differential geometry

Darboux frames developable tangent surfaces visual contours Transon planes equiaffine metrics


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.

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