Homogeneous Spaces, Tits Buildings and Isoparametric Hypersurfaces

We classify 1-connected compact homogeneous spaces which have the same rational cohomology as a product of spheres $/mahtbb{S}^{n_1}/times/mathbb{S}^{n_2}$, with $3/leq n_1/leq n_2$ and $n_2$ odd. As an application, we classify compact generalized quadrangles (buildings of type $C_2)$ which admit a point transitive automorphism group, and isoparametric hypersurfaces which admit a transitive isometry group on one focal manifold.