an:06787772
Zbl 1376.49062
Ghezzi, Roberta
On almost-Riemannian surfaces
EN
Actes de S??minaire de Th??orie Spectrale et G??om??trie. Ann??e 2010--2011. St. Martin d'H??res: Universit?? de Grenoble I, Institut Fourier. S??minaire de Th??orie Spectrale et G??om??trie 29, 15-49 (2011).
2011
a
49Q99 53C17 34K35 49J15
almost-Riemannian geometry; geodesics; Grushin plane; Lipschitz classification; Pontryagin maximum principle; Gauss-Bonnet formula; control theory
Summary: 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.
For the entire collection see [Zbl 1356.35008].