  Using methods originating in the theory of intersection spaces, specifically
a de Rham type description of the real cohomology of these spaces by a complex
of global differential forms, we show that the Leray-Serre spectral sequence
with real coefficients of a flat fiber bundle of smooth manifolds collapses if
the fiber is Riemannian and the structure group acts isometrically. The proof
is largely topological and does not need a metric on the base or total space.
We use this result to show further that if the fundamental group of a smooth
aspherical manifold acts isometrically on a Riemannian manifold, then the
equivariant real cohomology of the Riemannian manifold can be computed as a
direct sum over the cohomology of the group with coefficients in the (generally
twisted) cohomology modules of the manifold. Our results have consequences for
the Euler class of flat sphere bundles. Several examples are discussed in
detail, for instance an action of a free abelian group on a flag manifold.
