# High-order angles in almost-Riemannian geometry

Ugo Boscain^{[1]}; Mario Sigalotti^{[2]}

- [1] SISSA-ISAS Via Beirut 2-4, 34014 Trieste (Italy) and Université de Bourgogne LE2i, CNRS UMR5158 9, avenue Alain Savary BP 47870 21078 DIJON cedex (France)
- [2] Institut Élie Cartan, UMR 7502 INRIA/Nancy-Université/CNRS POB 239 54506 Vandœuvre-lès-Nancy (France)

Séminaire de théorie spectrale et géométrie (2006-2007)

- Volume: 25, page 41-54
- ISSN: 1624-5458

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topBoscain, Ugo, and Sigalotti, Mario. "High-order angles in almost-Riemannian geometry." Séminaire de théorie spectrale et géométrie 25 (2006-2007): 41-54. <http://eudml.org/doc/11229>.

@article{Boscain2006-2007,

abstract = {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-$\{\mathcal\{C\}\}^2$ 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.},

affiliation = {SISSA-ISAS Via Beirut 2-4, 34014 Trieste (Italy) and Université de Bourgogne LE2i, CNRS UMR5158 9, avenue Alain Savary BP 47870 21078 DIJON cedex (France); Institut Élie Cartan, UMR 7502 INRIA/Nancy-Université/CNRS POB 239 54506 Vandœuvre-lès-Nancy (France)},

author = {Boscain, Ugo, Sigalotti, Mario},

journal = {Séminaire de théorie spectrale et géométrie},

keywords = {Gauss-Bonnet formula; singularities},

language = {eng},

pages = {41-54},

publisher = {Institut Fourier},

title = {High-order angles in almost-Riemannian geometry},

url = {http://eudml.org/doc/11229},

volume = {25},

year = {2006-2007},

}

TY - JOUR

AU - Boscain, Ugo

AU - Sigalotti, Mario

TI - High-order angles in almost-Riemannian geometry

JO - Séminaire de théorie spectrale et géométrie

PY - 2006-2007

PB - Institut Fourier

VL - 25

SP - 41

EP - 54

AB - 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-${\mathcal{C}}^2$ 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.

LA - eng

KW - Gauss-Bonnet formula; singularities

UR - http://eudml.org/doc/11229

ER -

## References

top- 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. Zbl1198.49041MR2379474
- A.A. Agrachev, Yu.L. Sachkov, Control Theory from the Geometric Viewpoint, Encyclopedia of Mathematical Sciences, 87, Springer, 2004. Zbl1062.93001MR2062547
- 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. Zbl0862.53031MR1421822
- 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. Zbl1093.49001
- 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. Zbl0552.35032MR753153
- 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. MR279436
- 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. Zbl0238.47038MR283630
- 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. Zbl0117.31702MR166037

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