Poincaré ’ S Lemma on Some Non - Euclidean Structures

In this paper we prove the Poincaré lemma on some n-dimensional corank 1 sub-Riemannian structures, formulating the (n−1)n(n +3n−2) 8 necessarily and sufficiently ’curl-vanishing’ compatibility conditions. In particular, this result solves partially an open problem formulated by Calin and Chang. Our proof is based on a Poincaré lemma stated on Riemannian manifolds and a suitable Cesàro-Volterra path integral formula established in local coordinates. As a byproduct, a Saint-Venant lemma is also provided on generic Riemannian manifolds. Some examples are presented on the hyperbolic space and Carnot/Heisenberg groups.