# Self-intersecting universe

General relativity says space-time is a $4$-dimensional manifold which may have non-zero global curvature. Now if we take a random curve or surface or $n$-fold, it may fail to be a manifold because it self-intersects (and the neigborhoods of a self-intersection point are not homeomorphic to $\mathbb R^n$). Are there any reasons this does not happen with space-time? What would be the physical meaning of the self-intersecting locus of the universe?

• Ok so by Withney theorem for any $4$-dimensional manifold there is an embedding in $\mathbb R^8$. So if trying to find a $4$-fold in $\mathbb R^8$ with locally prescribed curvature we end up self-intersecting, that means there is truly no maniflod satisfying our local conditions, right? However, there might be an "intrinsically self-intersecting" $4$-fold. – Régis Jun 24 '18 at 17:49