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Zbl 0697.57014
El Kacimi-Alaoui, Aziz
Opérateurs transversalement elliptiques sur un feuilletage riemannien et applications. (Transversally elliptic operators for Riemannian foliations and appplications). (French)
[J] Compos. Math. 73, No.1, 57-106 (1990). ISSN 0010-437X; ISSN 1570-5846/e

The author develops the theory of transversely elliptic operators for a Riemannian foliation ${\cal F}$ of a manifold ${\cal M}$. Basic differential operators act on the presheaves of basic sections of so- called ${\cal F}$-vector bundles. A basic differential operator D is transversely elliptic if its symbol $\sigma(D)(x,\xi)$ is an isomorphism for all x of ${\cal M}$ and $\xi\ne \emptyset$. 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 ${\cal F}$ is transversely Kählerian and a class c in the basic cohomology $H\sp 2({\cal M}/{\cal F})$ of ${\cal F}$ contains at least one basic Kähler form $\gamma$ for which $(1/2\pi)\gamma$ represents the transverse Chern class of ${\cal F}$, then c contains a basic Kähler form $\omega$ for which $\gamma$ is the Ricci form of the transverse Kähler metric corresponding to $\omega$.
[P.Walczak]

MSC 2000:: *57R30 Foliations; geometric theory
58J60 Relations with special manifold structures
53C12 Foliations (differential geometry)

Keywords: transversely elliptic operators for a Riemannian foliation; basic sections; basic differential operator; Hodge decomposition; transversely Kählerian; basic Kähler form; transverse Chern class; Ricci form of the transverse Kähler metric

Cited in: Zbl 1169.14015 Zbl 1046.53029

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