High-order angles in almost-Riemannian geometry

Boscain, 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 -