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Differential forms on contact manifolds. (Formes différentielles sur les variétés de contact.) (French) Zbl 0973.53524

This paper is the result of the author’s thesis written under the direction of M. Gromov and P. Pansu.
In the first part the author defines the concepts of differential forms, codifferential and Laplacian on contact manifolds. In the second part a special connection, adapted to the contact structure, is defined and, using Weitzenbock methods, some vanishing theorems are obtained. In particular, under a certain “positivity” condition on the curvature of the connection adapted to the contact structure, it is proved that \(H^k(M,\mathbb{R})=0\) for \(k\neq n,n+1\), where \(M\) is a \((2n+1)\)-dimensional pseudo-Hermitian manifold. Moreover, for \(n=1\) a vanishing theorem for \(H^1(M,\mathbb{R})\) is obtained. In the last section Carnot-Carathéodory geodesics are considered and a pseudo-Hermitian version of Myers’ theorem is proved for a pseudo-Hermitian complete 3-dimensional manifold.

MSC:

53D10 Contact manifolds (general theory)
58A10 Differential forms in global analysis
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
58A12 de Rham theory in global analysis
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