  Many classical results in relativity theory concerning spherically symmetric
space-times have easy generalizations to warped product space-times, with a
two-dimensional Lorentzian base and arbitrary dimensional Riemannian fibers. We
first give a systematic presentation of the main geometric constructions, with
emphasis on the Kodama vector field and the Hawking energy; the construction is
signature independent. This leads to proofs of general Birkhoff-type theorems
for warped product manifolds; our theorems in particular apply to situations
where the warped product manifold is not necessarily Einstein, and thus can be
applied to solutions with matter content in general relativity. Next we
specialize to the Lorentzian case and study the propagation of null expansions
under the assumption of the dominant energy condition. We prove several
non-existence results relating to the Yamabe class of the fibers, in the spirit
of the black-hole topology theorem of Hawking-Galloway-Schoen. Finally we
discuss the effect of the warped product ansatz on matter models. In particular
we construct several cosmological solutions to the Einstein-Euler equations
whose spatial geometry is generally not isotropic.
