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

Abstract

AbstractWe introduce a new spectral sequence for the study of $${\mathcal {K}}$$ K -manifolds which arises by restricting the spectral sequence of a Riemannian foliation to forms invariant under the flows of $$\{\xi _1,\ldots ,\xi _s\}$$ { ξ 1 , … , ξ s } . We use this sequence to generalize a number of theorems from K-contact geometry to $${\mathcal {K}}$$ K -manifolds. Most importantly we compute the cohomology ring and harmonic forms of $${\mathcal {S}}$$ 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 $${\mathcal {S}}$$ S -manifolds are a topological invariant. We also show that the basic Hodge numbers of $${\mathcal {S}}$$ S -manifolds are invariant under deformations. Finally, we provide similar results for $${\mathcal {C}}$$ C -manifolds.