Summary: A "Gauss{Bonnet Formula" for Contact
SubRiemannian Manifolds
A. A. Agrachev
Abstract
We study 3dimensional manifolds endowed with oriented contact
subRiemannian 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 dierential invariants
of the subRiemannian metric.
1. Let M be a smooth 3dimensional manifold. A contact subRiemannian
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 ;
dened 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
1
R
