Cohomology of manifolds with structure group U(n) times O(s)

Abstract

We introduce a new spectral sequence for the study of {mathcal {K}}-manifolds which arises by restricting the spectral sequence of a Riemannian foliation to forms invariant under the flows of {xi _1,ldots ,xi _s}. We use this sequence to generalize a number of theorems from K-contact geometry to {mathcal {K}}-manifolds. Most importantly we compute the cohomology ring and harmonic forms of {mathcal {S}}-manifolds in terms of primitive basic cohomology and primitive basic harmonic forms (respectively). As an immediate consequence of this we get that the basic cohomology of {mathcal {S}}-manifolds are a topological invariant. We also show that the basic Hodge numbers of {mathcal {S}}-manifolds are invariant under deformations. Finally, we provide similar results for {mathcal {C}}-manifolds.