# (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?

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).