Let and be two smooth vector fields on a two-dimensional manifold . If and are everywhere linearly independent, then they define a Riemannian metric on (the metric for which they are orthonormal) and they give to the structure of metric space. If and become linearly dependent somewhere on , 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- 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},
pages = {41--54},
publisher = {Institut Fourier},
address = {Grenoble},
volume = {25},
year = {2006-2007},
doi = {10.5802/tsg.246},
zbl = {1159.53320},
mrnumber = {2478807},
language = {en},
url = {http://www.numdam.org/articles/10.5802/tsg.246/}
}
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 SP - 41 EP - 54 VL - 25 PB - Institut Fourier PP - Grenoble UR - http://www.numdam.org/articles/10.5802/tsg.246/ DO - 10.5802/tsg.246 LA - en ID - TSG_2006-2007__25__41_0 ER -
%0 Journal Article %A Boscain, Ugo %A Sigalotti, Mario %T High-order angles in almost-Riemannian geometry %J Séminaire de théorie spectrale et géométrie %D 2006-2007 %P 41-54 %V 25 %I Institut Fourier %C Grenoble %U http://www.numdam.org/articles/10.5802/tsg.246/ %R 10.5802/tsg.246 %G en %F TSG_2006-2007__25__41_0
Boscain, Ugo; Sigalotti, Mario. High-order angles in almost-Riemannian geometry. Séminaire de théorie spectrale et géométrie, Tome 25 (2006-2007), pp. 41-54. doi : 10.5802/tsg.246. http://www.numdam.org/articles/10.5802/tsg.246/
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