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# Darboux curves on surfaces I

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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 Mobius 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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https://hal-univ-bourgogne.archives-ouvertes.fr/hal-01493227
Contributeur : Imb - Université de Bourgogne <>
Soumis le : mardi 21 mars 2017 - 10:54:52
Dernière modification le : jeudi 28 janvier 2021 - 10:28:03

### Citation

Ronaldo Garcia, Rémi Langevin, Paweł Walczak. Darboux curves on surfaces I. Journal of the Mathematical Society of Japan, Maruzen Company Ltd, 2017, 69 (1), pp.1 - 24. ⟨10.2969/jmsj/06910001⟩. ⟨hal-01493227⟩

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