  In this note, we construct a family of non-singular time-dependent solutions
of a six-dimensional gravitational theory that are warped products of a four
dimensional bouncing cosmological solution and a two dimensional internal
manifold. The warp factor is time-dependent and breaks translation invariance
along one of the internal directions. When the warp factor is periodic in time,
the non-compact part of the geometry bounces periodically. The six dimensional
geometry is supported by matter that does not violate the null energy
condition. We show that this 6D geometry does not admit a closed trapped
surface and hence the Hawking-Penrose singularity theorems do not apply to
these solutions. We also present examples of singular solutions where the
topology of the internal manifold changes dynamically.
