  We study the classical geodesic motions of nonzero rest mass test particles
and photons in (3+1+n)- dimensional warped product spaces. An important feature
of these spaces is that they allow a natural decoupling between the motions in
the (3+1)-dimensional spacetime and those in the extra n dimensions. Using this
decoupling and employing phase space analysis we investigate the conditions for
confinement of particles and photons to the (3+1)- spacetime submanifold. In
addition to providing information regarding the motion of photons, we also show
that these motions are not constrained by the value of the extrinsic curvature.
We obtain the general conditions for the confinement of geodesics in the case
of pseudo-Riemannian manifolds as well as establishing the conditions for the
stability of such confinement. These results also generalise a recent result of
the authors concerning the embeddings of hypersurfaces with codimension one.
