Two-Dimensional Almost-Riemannian Structures with Tangency Points

Andrei Agrachev; Ugo Boscain; Grégoire Charlot; Roberta Ghezzi; Mario Sigalotti

hal-00410127, version 1

Two-Dimensional Almost-Riemannian Structures with Tangency Points

Andrei Agrachev ¹, Ugo Boscain ², Grégoire Charlot ³, Roberta Ghezzi ¹, Mario Sigalotti () ^4, 5

Annales de l'Institut Henri Poincare (C) Non Linear Analysis 27, 3 (2010) 793-307

Abstract: Two-dimensional almost-Riemannian structures are generalized Riemannian structures on surfaces for which a local orthonormal frame is given by a Lie bracket generating pair of vector ﬁelds that can become collinear. We study the relation between the topological invariants of an almost-Riemannian structure on a compact oriented surface and the rank-two vector bundle over the surface which deﬁnes the structure. We analyse the generic case including the presence of tangency points, i.e. points where two generators of the distribution and their Lie bracket are linearly dependent. The main result of the paper provides a classiﬁcation of oriented almost-Riemannian structures on compact oriented surfaces in terms of the Euler number of the vector bundle corresponding to the structure. Moreover, we present a Gauss-Bonnet formula for almost-Riemannian structures with tangency points.

1: Scuola Internazionale Superiore di Studi Avanzati / International School for Advanced Studies (SISSA / ISAS)
Scuola Internazionale Superiore di Studi Avanzati/International School for Advanced Studies (SISSA/ISAS)
2: Centre de Mathématiques Appliquées - Ecole Polytechnique (CMAP)
Polytechnique - X – CNRS : UMR7641
3: Institut Fourier (IF)
CNRS : UMR5582 – Université Joseph Fourier - Grenoble I
4: Institut Elie Cartan Nancy (IECN)
CNRS : UMR7502 – INRIA – Université Henri Poincaré - Nancy I – Université Nancy II – Institut National Polytechnique de Lorraine (INPL)
5: CORIDA (INRIA Nancy - Grand Est / IECN / LMAM)
INRIA – CNRS : UMR7502 – Université de Lorraine

Collaboration : CORIDA

hal-00410127, version 1
http://hal.archives-ouvertes.fr/hal-00410127
oai:hal.archives-ouvertes.fr:hal-00410127
From: Mario Sigalotti <>
Submitted on: Monday, 17 August 2009 19:09:22
Updated on: Friday, 29 April 2011 14:55:58

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arXiv: 0908.2564

DOI: 10.1016/j.anihpc.2009.11.011

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%% hal-00410127, version 1
%% http://hal.archives-ouvertes.fr/hal-00410127
@article{agrachev:hal-00410127,
    hal_id = {hal-00410127},
    url = {http://hal.archives-ouvertes.fr/hal-00410127},
    title = {{Two-Dimensional Almost-Riemannian Structures with Tangency Points}},
    author = {Agrachev, Andrei and Boscain, Ugo and Charlot, Gr{\'e}goire and Ghezzi, Roberta and Sigalotti, Mario},
    abstract = {{Two-dimensional almost-Riemannian structures are generalized Riemannian structures on surfaces for which a local orthonormal frame is given by a Lie bracket generating pair of vector fields that can become collinear. We study the relation between the topological invariants of an almost-Riemannian structure on a compact oriented surface and the rank-two vector bundle over the surface which defines the structure. We analyse the generic case including the presence of tangency points, i.e. points where two generators of the distribution and their Lie bracket are linearly dependent. The main result of the paper provides a classification of oriented almost-Riemannian structures on compact oriented surfaces in terms of the Euler number of the vector bundle corresponding to the structure. Moreover, we present a Gauss-Bonnet formula for almost-Riemannian structures with tangency points.}},
    language = {English},
    affiliation = {Scuola Internazionale Superiore di Studi Avanzati / International School for Advanced Studies - SISSA / ISAS , Centre de Math{\'e}matiques Appliqu{\'e}es - Ecole Polytechnique - CMAP , Institut Fourier - IF , Institut Elie Cartan Nancy - IECN , CORIDA - INRIA Nancy - Grand Est / IECN / LMAM},
    pages = {793-307},
    journal = {Annales de l'Institut Henri Poincare (C) Non Linear Analysis},
    volume = {27},
    number = {3 },
    note = {IF\_PREPUB IF\_PREPUB },
    audience = {international },
    collaboration = {CORIDA },
    doi = {10.1016/j.anihpc.2009.11.011 },
    year = {2010},
    pdf = {http://hal.archives-ouvertes.fr/hal-00410127/PDF/euler4.pdf},
}

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