Journal of the Mathematical Society of Japan

Darboux curves on surfaces I

Ronaldo GARCIA, Rémi LANGEVIN, and Paweł WALCZAK

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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.

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

First available in Project Euclid: 18 January 2017

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Zentralblatt MATH identifier

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

Darboux curves conformal geometry space of spheres


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.

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