Poincaré's lemma on the Heisenberg group

It is well known that the system ? x f = a , ? y f = b on R 2 has a solution if and only if the closure condition ? x b = ? y a holds. In this case the solution f is the work done by the force U = ( a , b ) from the origin to the point ( x , y ) .This paper deals with a similar problem, where the vector fields ? x , ? y are replaced by the Heisenberg vector fields X 1 , X 2 . In this case the sub-Riemannian system X 1 f = a , X 2 f = b has a solution f if and only if the following integrability conditions hold: X 1 2 b = ( X 1 X 2 + X 1 , X 2 ) a , X 2 2 a = ( X 2 X 1 + X 2 , X 1 ) b . The question addressed in this paper is whether we can provide a Poincare-type lemma for the Heisenberg distribution. The positive answer is given by Theorem 2, which provides a result similar to the Poincare lemma in the integral form. The solution f in this case is the work done by the force vector field a X 1 + b X 2 along any horizontal curve from the origin to the current point.