Poincar\'e's lemma on some non-Euclidean structures

TL;DR

This paper establishes a Poincaré lemma for certain non-Euclidean sub-Riemannian structures, providing compatibility conditions and solving part of an open problem in geometric analysis.

Contribution

It formulates necessary and sufficient curl-vanishing conditions for the Poincaré lemma on specific sub-Riemannian manifolds, extending classical results to non-Euclidean geometries.

Findings

Proves Poincaré lemma on corank 1 sub-Riemannian structures

Derives compatibility conditions for curl-vanishing

Provides examples on hyperbolic space and Carnot groups

Abstract

In this paper we prove the Poincar\'e lemma on some $n$-dimensional corank 1 sub-Riemannian structures, formulating the $\frac{(n-1)n(n^2+3n-2)}{8}$ 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 Poincar\'e lemma stated on Riemannian manifolds and a suitable Ces\`aro-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.