Poincaré's lemma on the Heisenberg group

It is well known that the system ∂xf=a∂xf=a, ∂yf=b∂yf=b on R2R2 has a solution if and only if the closure condition ∂xb=∂ya∂xb=∂ya holds. In this case the solution f   is the work done by the force U=(a,b)U=(a,b) from the origin to the point (x,y)(x,y).This paper deals with a similar problem, where the vector fields ∂x∂x, ∂y∂y are replaced by the Heisenberg vector fields X1X1, X2X2. In this case the sub-Riemannian system X1f=aX1f=a, X2f=bX2f=b has a solution f   if and only if the following integrability conditions hold: X12b=(X1X2+[X1,X2])a, X22a=(X2X1+[X2,X1])b. The question addressed in this paper is whether we can provide a Poincaré-type lemma for the Heisenberg distribution. The positive answer is given by Theorem 2, which provides a result similar to the Poincaré lemma in the integral form. The solution f   in this case is the work done by the force vector field aX1+bX2aX1+bX2 along any horizontal curve from the origin to the current point.