The author proves the Poincaré lemma on some (n + 1)-dimensional corank 1 sub-Riemannian structures, formulating the (n−1)n(n2+3n−2)8 necessarily and sufficiently “curl-vanishing” compatibility conditions. In particular, this result solves partially an open problem formulated by Calin and Chang. The proof in this paper 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.
- Cesàro-Volterra path integral
- Poincaré lemma
- Sub-Riemannian manifolds
ASJC Scopus subject areas
- Applied Mathematics