On almost-Riemannian surfaces
Séminaire de théorie spectrale et géométrie, Volume 29 (2010-2011), pp. 15-49.

An almost-Riemannian structure on a surface is a generalized Riemannian structure whose local orthonormal frames are given by Lie bracket generating pairs of vector fields that can become collinear. The distribution generated locally by orthonormal frames has maximal rank at almost every point of the surface, but in general it has rank 1 on a nonempty set which is generically a smooth curve. In this paper we provide a short introduction to 2-dimensional almost-Riemannian geometry highlighting its novelties with respect to Riemannian geometry. We present some results that investigate topological, metric and geometric aspects of almost-Riemannian surfaces from a local and global point of view.

Une structure presque riemannienne sur une surface est une structure riemannienne généralisée où les repères orthonormaux locaux sont donnés par des paires de champs de vecteurs qui peuvent être parallèles, mais dont l’algèbre de Lie engendrée a dimension 2 en tout point. En presque tout point de la surface, la distribution engendrée localement par ces repères a rang maximal, mais en général il existe un lieu, génériquement une courbe lisse, où la distribution a rang 1. Dans cet article on fournit une courte introduction à la géométrie presque-riemannienne de dimension 2, en soulignant les phénomènes nouveaux par rapport à la géométrie riemannienne. On présente quelques résultats décrivant des aspect topologiques, métriques et géométriques des surfaces presque riemanniennes d’un point de vue local et global.

DOI: 10.5802/tsg.284
Classification: 49J15,  53CXX,  34K35
Keywords: almost-Riemannian geometry, geodesics, Grushin plane, Lipschitz classification, Pontryagin maximum principle, Gauss-Bonnet formula.
Ghezzi, Roberta 1

1 Department of Mathematical Sciences and Centre of Computational and Integrative Biology, Rutgers University Camden - Camden, 311 N 5th Street, Camden, NJ 08102, USA.
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Ghezzi, Roberta. On almost-Riemannian surfaces. Séminaire de théorie spectrale et géométrie, Volume 29 (2010-2011), pp. 15-49. doi : 10.5802/tsg.284. http://www.numdam.org/articles/10.5802/tsg.284/

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