Journal of the Mathematical Society of Japan

Darboux curves on surfaces I

Ronaldo GARCIA, Rémi LANGEVIN, and Paweł WALCZAK

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Abstract

In 1872, G. Darboux defined a family of curves on surfaces of $\mathbb{R}^3$ which are preserved by the action of the Möbius group and share many properties with geodesics. Here, we characterize these curves under the view point of Lorentz geometry and prove that they are geodesics in a 3-dimensional sub-variety of a quadric $\Lambda^4$ contained in the 5-dimensional Lorentz space $\mathbb{R}^5_1$ naturally associated to the surface. We construct a new conformal object: the Darboux plane-field $\mathcal{D}$ and give a condition depending on the conformal principal curvatures of the surface which guarantees its integrability. We show that $\mathcal{D}$ is integrable when the surface is a special canal.

Article information

Source
J. Math. Soc. Japan, Volume 69, Number 1 (2017), 1-24.

Dates
First available in Project Euclid: 18 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1484730016

Digital Object Identifier
doi:10.2969/jmsj/06910001

Mathematical Reviews number (MathSciNet)
MR3597545

Zentralblatt MATH identifier
1391.53011

Subjects
Primary: 53A30: Conformal differential geometry
Secondary: 53C12: Foliations (differential geometric aspects) [See also 57R30, 57R32] 53C50: Lorentz manifolds, manifolds with indefinite metrics 57R30: Foliations; geometric theory

Keywords
Darboux curves conformal geometry space of spheres

Citation

GARCIA, Ronaldo; LANGEVIN, Rémi; WALCZAK, Paweł. Darboux curves on surfaces I. J. Math. Soc. Japan 69 (2017), no. 1, 1--24. doi:10.2969/jmsj/06910001. https://projecteuclid.org/euclid.jmsj/1484730016


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