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arXiv:math.OC/0609566v120Sep2006 A Gauss-Bonnet-like Formula on Two-Dimensional

Summary: arXiv:math.OC/0609566v120Sep2006
A Gauss-Bonnet-like Formula on Two-Dimensional
Almost-Riemannian Manifolds1
Andrei A. Agrachev, Ugo Boscain
SISSA-ISAS, Via Beirut 2-4, 34014 Trieste, Italy
agrachev(-at-)sissa.it, boscain(-at-)sissa.it
Mario Sigalotti
Institut ´Elie Cartan, UMR 7502 Nancy-Universit´e/CNRS/INRIA, POB 239, 54506 Vandoeuvre-l`es-Nancy, France
21st September 2006
Abstract -- We consider a generalization of Riemannian geometry that naturally
arises in the framework of control theory. 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 classical 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. They are special cases of rank-varying sub-Riemannian structures, which
are naturally defined in terms of submodules of the space of smooth vector fields on


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


Collections: Engineering; Mathematics