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New singularity theorems are derived for generic warped-product spacetimes ofany dimension. The main purpose is to analyze the stability of (compact orlarge) extra dimensions against dynamical perturbations. To that end, the baseof the warped product is assumed to be our visible 4-dimensional world, whilethe extra dimensions define the fibers, hence we consider "extra-dimensionalevolution". Explicit conditions on the warping function that lead to geodesicincompleteness are given. These conditions can be appropriately rewritten,given a warping function, as restrictions on the intrinsic geometry of thefibers ---i.e. the extra dimensional space. To find the results, the conditionsfor parallel transportation in warped products in terms of their projectionsonto the base and the fibers have been solved, a result of independentmathematical interest that have been placed on an Appendix.

Singularity theorems for warped products and the stability of spatial extra dimensions