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Opérateurs transversalement elliptiques sur un feuilletage riemannien et applications. (Transversally elliptic operators for Riemannian foliations and appplications). (French) Zbl 0697.57014

The author develops the theory of transversely elliptic operators for a Riemannian foliation \({\mathcal F}\) of a manifold \({\mathcal M}\). Basic differential operators act on the presheaves of basic sections of so- called \({\mathcal F}\)-vector bundles. A basic differential operator D is transversely elliptic if its symbol \(\sigma(D)(x,\xi)\) is an isomorphism for all x of \({\mathcal M}\) and \(\xi\neq \emptyset\). Transversely elliptic operators appear to be Fredholm and admit a Hodge decomposition. They have some cohomology properties which allow, among others, to prove the following, very interesting result of Calabi-Yao type: If \({\mathcal F}\) is transversely Kählerian and a class c in the basic cohomology \(H^ 2({\mathcal M}/{\mathcal F})\) of \({\mathcal F}\) contains at least one basic Kähler form \(\gamma\) for which \((1/2\pi)\gamma\) represents the transverse Chern class of \({\mathcal F}\), then c contains a basic Kähler form \(\omega\) for which \(\gamma\) is the Ricci form of the transverse Kähler metric corresponding to \(\omega\).
Reviewer: P.Walczak

MSC:

57R30 Foliations in differential topology; geometric theory
58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.)
53C12 Foliations (differential geometric aspects)

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