(pseudo-)Riemannian manifolds and global coordinates

Question

I have a question about (pseudo-)Riemannian manifolds. In general relativity one often assumes that there exists a global set of coordinates describing the whole manifold. My question is: Does there always exist a global set of coordinates describing the whole manifold or are there manifolds that only permit local coordinates in a neighbourhood of some point $p \in M$ of the manifold?

Answer 1

What exactly do you mean by "global set of coordinates"? (i.e. how "global" is your global?)

If you mean it to be on entire manifold then no. Even just the $\mathbb{S}^n$ (in Riemannian case) or (anti) de Sitter $dS^n$ or $AdS^n$ doesn't allow this (e.g. there is no "global" coordinates on $\mathbb{S}^1$ since going round it once must give a different value).

On the other hand, on a large open set this is true, provided you have completeness and smoothness. By large I mean, for example, almost everywhere w.r.t. volume measure. As a consequence, for calculation of integrals you can just work inside one coordinate chart.