@article {2012,
title = {On the Hausdorff volume in sub-Riemannian geometry},
journal = {Calculus of Variations and Partial Differential Equations. Volume 43, Issue 3-4, March 2012, Pages 355-388},
number = {arXiv:1005.0540;},
year = {2012},
publisher = {SISSA},
abstract = {For a regular sub-Riemannian manifold we study the Radon-Nikodym derivative\r\nof the spherical Hausdorff measure with respect to a smooth volume. We prove\r\nthat this is the volume of the unit ball in the nilpotent approximation and it\r\nis always a continuous function. We then prove that up to dimension 4 it is\r\nsmooth, while starting from dimension 5, in corank 1 case, it is C^3 (and C^4\r\non every smooth curve) but in general not C^5. These results answer to a\r\nquestion addressed by Montgomery about the relation between two intrinsic\r\nvolumes that can be defined in a sub-Riemannian manifold, namely the Popp and\r\nthe Hausdorff volume. If the nilpotent approximation depends on the point (that\r\nmay happen starting from dimension 5), then they are not proportional, in\r\ngeneral.},
doi = {10.1007/s00526-011-0414-y},
url = {http://hdl.handle.net/1963/6454},
author = {Andrei A. Agrachev and Davide Barilari and Ugo Boscain}
}
@article {2007,
title = {High-order angles in almost-Riemannian geometry},
number = {SISSA;59/2007/M},
year = {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-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.},
url = {http://hdl.handle.net/1963/1995},
author = {Ugo Boscain and Mario Sigalotti}
}