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Article Dans Une Revue Journal of the Mathematical Society of Japan Année : 2017

Darboux curves on surfaces I


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.

Dates et versions

hal-01493227 , version 1 (21-03-2017)



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