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

## TL;DR

This paper develops new singularity theorems for warped-product spacetimes to analyze the stability of extra dimensions, providing conditions on warping functions and fiber geometry that lead to geodesic incompleteness.

## Contribution

It introduces novel singularity theorems for warped products and links warping functions to the stability of extra dimensions against perturbations.

## Key findings

- Derived explicit conditions for geodesic incompleteness in warped spacetimes
- Linked warping function restrictions to fiber geometry
- Solved parallel transportation conditions in warped products

## Abstract

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