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A "Gauss{Bonnet Formula" for Contact Sub-Riemannian Manifolds

Summary: A "Gauss{Bonnet Formula" for Contact
Sub-Riemannian Manifolds
A. A. Agrachev 
We study 3-dimensional manifolds endowed with oriented contact
sub-Riemannian structures. The Euler characteristic class of the con-
tact structure is presented as the rotation class of a volume preserving
vector eld constructed in terms of fundamental di erential invariants
of the sub-Riemannian metric.
1. Let M be a smooth 3-dimensional manifold. A contact sub-Riemannian
structure is a pair ; hji; where  = f q g q2M ;  q  T q M; is a contact
structure on M and hji = fhji q g q2M is a smooth with respect to q family
of Euclidean inner products
(v 1 ; v 2 ) 7! hv 1 jv 2 i q ; v 1 ; v 2 2  q ;
de ned on  q : A Lipschitzian curve  : [0; 1] ! M is called admissible for
 if d(t)
dt 2  (t) for almost all t 2 [0; 1]: The length of an admissible curve
 is the integral


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


Collections: Engineering; Mathematics