High-order angles in almost-Riemannian geometry
Séminaire de théorie spectrale et géométrie, Volume 25  (2006-2007), p. 41-54

Let X and Y be two smooth vector fields on a two-dimensional manifold M. If X and Y are everywhere linearly independent, then they define a Riemannian metric on M (the metric for which they are orthonormal) and they give to M the structure of metric space. If X and Y become linearly dependent somewhere on M, then the corresponding Riemannian metric has singularities, but under generic conditions the metric structure is still well defined. Metric structures that can be defined locally in this way are called almost-Riemannian structures. The main result of the paper is a generalization to almost-Riemannian structures of the Gauss-Bonnet formula for domains with piecewise-𝒞 2 boundary. The main feature of such formula is the presence of terms that play the role of high-order angles at the intersection points with the set of singularities.

@article{TSG_2006-2007__25__41_0,
     author = {Boscain, Ugo and Sigalotti, Mario},
     title = {High-order angles in almost-Riemannian geometry},
     journal = {S\'eminaire de th\'eorie spectrale et g\'eom\'etrie},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {25},
     year = {2006-2007},
     pages = {41-54},
     doi = {10.5802/tsg.246},
     mrnumber = {2478807},
     zbl = {1159.53320},
     language = {en},
     url = {http://www.numdam.org/item/TSG_2006-2007__25__41_0}
}
Boscain, Ugo; Sigalotti, Mario. High-order angles in almost-Riemannian geometry. Séminaire de théorie spectrale et géométrie, Volume 25 (2006-2007) , pp. 41-54. doi : 10.5802/tsg.246. http://www.numdam.org/item/TSG_2006-2007__25__41_0/

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