High-order angles in almost-Riemannian geometry | Zendy

Ugo Boscain | Zendy; Mario Sigalotti | Zendy

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High-order angles in almost-Riemannian geometry

Author(s) -

Ugo Boscain,

Mario Sigalotti

Publication year - 2007

Publication title -

séminaire de théorie spectrale et géométrie

Language(s) - English

Resource type - Journals

eISSN - 2118-9242

pISSN - 1624-5458

DOI - 10.5802/tsg.246

Subject(s) - fundamental theorem of riemannian geometry , mathematics , riemannian geometry , statistical manifold , levi civita connection , metric (unit) , information geometry , boundary (topology) , riemannian manifold , fisher information metric , pure mathematics , mathematical analysis , generalization , manifold (fluid mechanics) , order (exchange) , geometry , metric space , scalar curvature , injective metric space , curvature , mechanical engineering , operations management , economics , engineering , finance

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 almostRiemannian structures. The main result of the paper is a generalization to almost-Riemannian structures of the Gauss-Bonnet formula for domains with piecewise-C 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. MSC: 49j15, 53c17 PREPRINT SISSA SISSA 59/2007/M

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