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On the Hausdorff volume in sub-Riemannian geometry Andrei Agrachev

Summary: On the Hausdorff volume in sub-Riemannian geometry
Andrei Agrachev
SISSA, Trieste, Italy and MIAN, Moscow, Russia - agrachev@sissa.it
Davide Barilari
SISSA, Trieste, Italy - barilari@sissa.it
Ugo Boscain
CNRS, CMAP Ecole Polytechnique, Paris, France - boscain@cmap.polytechnique.fr
January 19, 2011
For a regular sub-Riemannian manifold we study the Radon-Nikodym derivative of the spher-
ical Hausdorff measure with respect to a smooth volume. We prove that this is the volume of
the unit ball in the nilpotent approximation and it is always a continuous function. We then
prove that up to dimension 4 it is smooth, while starting from dimension 5, in corank 1 case,
it is C3
(and C4
on every curve) but in general not C5
. These results answer to a question
addressed by Montgomery about the relation between two intrinsic volumes that can be defined
in a sub-Riemannian manifold, namely the Popp and the Hausdorff volume. If the nilpotent
approximation depends on the point (that may happen starting from dimension 5), then they


Source: Agrachev, Andrei - Functional Analysis Sector, Scuola Internazionale Superiore di Studi Avanzati (SISSA)


Collections: Engineering; Mathematics