Abstract
The author proves the Poincaré lemma on some (n + 1)-dimensional corank 1 sub-Riemannian structures, formulating the \(\frac{{\left( {n - 1} \right)n\left( {{n^2} + 3n - 2} \right)}}{8}\) 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.
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Acknowledgements
The author thanks Professor Philippe G. Ciarlet for his invitation to the City University of Hong Kong where the present work has been initiated. He is also grateful to Professors Ovidiu Calin and Der-Chen Chang for their suggestions and remarks.
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Dedicated to Philippe G. Ciarlet on the occasion of his 80th birthday
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Kristály, A. Poincaré’s Lemma on Some Non-Euclidean Structures. Chin. Ann. Math. Ser. B 39, 297–314 (2018). https://doi.org/10.1007/s11401-018-1065-5
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DOI: https://doi.org/10.1007/s11401-018-1065-5