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.

DOI : https://doi.org/10.5802/tsg.246
Classification:  49j15,  53c17
@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},
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/

[1] A.A. Agrachev, U. Boscain, M. Sigalotti, A Gauss-Bonnet-like Formula on Two-Dimensional Almost-Riemannian Manifolds, Discrete Contin. Dyn. Syst., 20 (4) 2008, pp. 801-822. | MR 2379474

[2] A.A. Agrachev, Yu.L. Sachkov, Control Theory from the Geometric Viewpoint, Encyclopedia of Mathematical Sciences, 87, Springer, 2004. | MR 2062547 | Zbl 1062.93001

[3] A. Bellaïche, The tangent space in sub-Riemannian geometry, in Sub-Riemannian geometry, edited by A. Bellaïche and J.-J. Risler, pp. 1–78, Progr. Math., 144, Birkhäuser, Basel, 1996. | MR 1421822 | Zbl 0862.53031

[4] U. Boscain, B. Piccoli, A short introduction to optimal control, in Contrôle non linéaire et applications, edited by T. Sari, pp. 19–66, Travaux en cours, Hermann, Paris, 2005. | Zbl 1093.49001

[5] B. Franchi, E. Lanconelli, Hölder regularity theorem for a class of linear nonuniformly elliptic operators with measurable coefficients, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 10 (1983), pp. 523–541. | Numdam | MR 753153 | Zbl 0552.35032

[6] V.V. Grušin, A certain class of hypoelliptic operators (Russian), Mat. Sb. (N.S.), 83 (125) 1970, pp. 456–473. English translation: Math. USSR-Sb., 12 (1970), pp. 458–476. | MR 279436

[7] V.V. Grušin, A certain class of elliptic pseudodifferential operators that are degenerate on a submanifold (Russian), Mat. Sb. (N.S.), 84 (126) 1971, pp. 163–195. English translation: Math. USSR-Sb., 13 (1971), pp. 155–185. | MR 283630 | Zbl 0238.47038

[8] L.S. Pontryagin, V.G. Boltyanskii, R.V. Gamkrelidze, E.F. Mishchenko, The Mathematical Theory of Optimal Processes, Interscience Publishers John Wiley and Sons, Inc, New York-London, 1962. | MR 166037 | Zbl 0117.31702