# (pseudo-)Riemannian manifolds and global coordinates

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?

## 1 Answer

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.