  We prove two new results about the Cauchy problem for nonlinear Schroedinger
equations on four-dimensional compact manifolds. The first one concerns global
wellposedness for Hartree-type nonlinearities and includes approximations of
cubic NLS on the sphere. The second one provides local wellposedness for
quadratic nonlinearities in the case of zonal data on the sphere. Both results
are based on new multilinear Strichartz-type estimates for the Schroedinger
group.
