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

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 $\{\xi_1,...,\xi_s\}$. 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.