01180nas a2200109   4500008004300000245005200043210005000095520085100145100001700996700002101013856003601034       2007                        en_Ud  00aHigh-order angles in almost-Riemannian geometry0 aHighorder angles in almostRiemannian geometry3 aLet 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-C2 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.1 aBoscain, Ugo1 aSigalotti, Mario  uhttp://hdl.handle.net/1963/1995