Journal of the Mathematical Society of Japan

Equiaffine Darboux frames for codimension 2 submanifolds contained in hypersurfaces

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

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

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

References

• M. P. do Carmo, Riemannian Geometry, Birkhauser, 1992.
• R. Cipolla and P. J. Giblin, Visual Motion of Curves and Surfaces, Cambridge University Press, 2000.
• J. W. Bruce and P. J. Giblin, Curves and Singularities, Cambridge University Press, 1992.
• S. Izumiya and S. Otani, Flat approximations of surfaces along curves, Demonstratio Mathematica, 48 (2015), 217–241.
• S. Hananoi, N. Ito and S. Izumiya, Spherical Darboux images of curves on surfaces, Beitr. Algebra Geom., 56 (2015), 575–585.
• B. Juttler, Osculating paraboloids of second and third order, Abh. Math. Sem. Univ. Hamburg, 66 (1996), 317–335.
• K. Nomizu and T. Sasaki, Affine differential geometry, Cambridge Tracts in Mathematics, 111, Cambridge University Press, Cambridge, 1994.
• M. J. Saia and L. Sánchez, Affine metrics for locally strictly convex manifolds of codimension 2, Real and Complex Singularities, Cont. Math., 675 (2016), 299–314.
• L. Sánchez, Surfaces in 4-space from the affine differential geometry viewpoint, Ph.D. Thesis, USP São Carlos, 2014.
• A. Transon, Recherches sur la courbure des lignes et des surfaces, J. De Math. Pures e Appliquées, $1^{re}$ série, tome 6, 1841, 191–208.