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De Rham decomposition theorems for foliated manifolds

Robert A. Blumenthal, James J. Hebda (1983)

Annales de l'institut Fourier

We prove that if M is a complete simply connected Riemannian manifold and F is a totally geodesic foliation of M with integrable normal bundle, then M is topologically a product and the two foliations are the product foliations. We also prove a decomposition theorem for Riemannian foliations and a structure theorem for Riemannian foliations with recurrent curvature.

Décomposition de Hodge basique pour un feuilletage riemannien

Aziz El Kacimi-Alaoui, Gilbert Hector (1986)

Annales de l'institut Fourier

Soit un feuilletage de codimension n sur une variété compacte M . On montre que le complexe des formes basiques Ω * ( M / ) admet une décomposition de Hodge. Il en résulte que la cohomologie basique H * ( M / ) de ( M , ) est de dimension finie et vérifie la dualité de Poincaré si et seulemnt si H n ( M / ) 0 .

Deformations of Lie brackets: cohomological aspects

Marius Crainic, Ieke Moerdijk (2008)

Journal of the European Mathematical Society

We introduce a new cohomology for Lie algebroids, and prove that it provides a differential graded Lie algebra which “controls” deformations of the structure bracket of the algebroid.

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