Open AccessDarboux curves on surfaces I
Author(s) -
Ronaldo García,
Rémi Langevin,
Paweł Walczak
Publication year - 2017
Publication title -
journal of the mathematical society of japan
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.047
H-Index - 36
eISSN - 1881-1167
pISSN - 0025-5645
DOI - 10.2969/jmsj/06910001
Subject(s) - geodesic , conformal map , surface (topology) , lorentz transformation , mathematics , quadric , integrable system , pure mathematics , field (mathematics) , space (punctuation) , lorentz space , plane (geometry) , mathematical physics , geometry , mathematical analysis , physics , classical mechanics , linguistics , philosophy
International audienceIn 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