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| Poincar\'e's Lemma on Some Non-Euclidean Structures |
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Citation： |
Alexandru KRIST'ALY.Poincar\'e's Lemma on Some Non-Euclidean Structures[J].Chinese Annals of Mathematics B,2018,39(2):297~314 |
| Page view： 1336
Net amount： 888 |
Authors： |
Alexandru KRIST'ALY; |
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| Abstract： |
The author proves the Poincar\'e lemma on some $(n+1)$-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. The proof in this paper 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. |
Keywords： |
Poincar'e lemma, Ces`aro-Volterra path integral, Sub-Riemannianmanifolds |
Classification： |
53C17 |
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