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De Rham-Hodge theory for Riemannian foliations. (English) Zbl 0637.53043
Let M be a compact orientable manifold, F be a transversely oriented Riemannian foliation of M and \(g_ M\) be a bundle-like metric on M. \((\Omega^._ B,d_ B)\) denotes the complex of basic forms. The usual global scalar product on forms restricts on \(\Omega^._ B\) to a scalar product. The adjoint \(\delta_ B\) of \(d_ B\) and the Laplacian \(\Delta_ B=\delta_ Bd_ B+d_ B\delta_ B\) are therefore defined. Let \(H^ r_ B\) denote the kernel of \(\Delta_ B\) on \(\Omega^ r_ B\). The authors prove that there is a decomposition into mutually orthogonal subspaces \[ \Omega^ r_ B=im d_ B\oplus im \delta_ B\oplus H^ r_ B \] with finite-dimensional \(H^ r_ B\).
Reviewer: A.Piatkowski

MSC:
53C12 Foliations (differential geometric aspects)
58A14 Hodge theory in global analysis
57R30 Foliations in differential topology; geometric theory
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