Open Access
January, 2017 Darboux curves on surfaces I
Ronaldo GARCIA, Rémi LANGEVIN, Paweł WALCZAK
J. Math. Soc. Japan 69(1): 1-24 (January, 2017). DOI: 10.2969/jmsj/06910001

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.

Citation

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Ronaldo GARCIA. Rémi LANGEVIN. Paweł WALCZAK. "Darboux curves on surfaces I." J. Math. Soc. Japan 69 (1) 1 - 24, January, 2017. https://doi.org/10.2969/jmsj/06910001

Information

Published: January, 2017
First available in Project Euclid: 18 January 2017

zbMATH: 1391.53011
MathSciNet: MR3597545
Digital Object Identifier: 10.2969/jmsj/06910001

Subjects:
Primary: 53A30
Secondary: 53C12 , 53C50 , 57R30

Keywords: conformal geometry , Darboux curves , space of spheres

Rights: Copyright © 2017 Mathematical Society of Japan

Vol.69 • No. 1 • January, 2017
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