Cohomology of manifolds with structure group \(U(n)\times O(s)\) (Q6038636)

The existence of a (1,1) tensor field \(f\) on a \((2n+s)\)-dimensional manifold \(M\) that satisfies \(f^3 + f =0\) is equivalent to the structure group of \(M\) reducing to \(\mathrm{U}(n)\times \mathrm{O}(s)\). This paper studies the cohomology of special classes of such manifolds, such as \(\mathcal{K}\)-manifolds, \(\mathcal{S}\)-manifolds, and \(\mathcal{C}\)-manifolds which are themselves generalizations of Kähler or Sasakian manifold. In particular a new spectral sequence is defined for \(\mathcal{K}\)-manifolds, and it is proved that the basic cohomology of \(\mathcal{S}\)-manifolds and \(\mathcal{C}\)-manifolds is a topological invariant and that their Hodge numbers are invariant under deformation.