Is Minkowski space locally Euclidean?

Question

The Minkowski spacetime $\mathbb{R}^{1,3}$ is said to be a manifold (isomorphic to $SO^{1,3}$. But according to the definition of a manifold it should be locally euclidean. However, this seems to be wrong, in general relativity your pseudo riemmanian manifold is locally minkowskian, if the above statement was true, it would also be possible to make it locally euclidean.

I think I am missing a major point in connecting, "A manifold is a locally Euclidean topological space" and "Minkowski space is a manifold".

Answer 1

The definition of a manifold as such does not take metric structures into account. I just requires local diffeomorphism (homeomorphic in case of topological manifolds) with some $\mathbb{R}^n$, which sometimes (sloppily) is rephrased as locally diffeomorphic to a Euclidean space. This is still correct, cause a diffeomorphism does not have to preserve scalar products. Minkowski space is a manifold with additional structure (the Lorentz metric).

Answer 2

Minkowski space $ \scr{M} = (\mathbb{R}^4,g$) is a manifold $\mathbb{R}^4$ equipped with the flat Lorentz metric $$g = - dt \otimes dt + dx \otimes dx + dy \otimes dy + dz \otimes dz$$ in standard coordinates ($t,x,y,z$) in $\mathbb{R}^4$. Its obviously locally euclidean by the identity map $Id : \mathbb{R}^4 \rightarrow \mathbb{R}^4$ which is a chart map for the global chart $(\mathbb{R}^4,Id)$. As @Thomas said, the definition of smooth manifold does not required additional structure (such as metric structure) other than : Topological structure, that is the topological space must be second countable and Hausdorff, and locally euclidean. The topology for $\scr{M}$ can be chosen other than standard topology in $\mathbb{R}^4$. I dont' really know for sure the recent development for this, but the well-known possible choices are Path (or Zeeman) Topology, which is the finest topology such that for any Euclidean continous timelike curve $\alpha : I \rightarrow \scr{M}$, the image $\alpha(I)$ inherit ordinary euclidean topology on $\scr{M}$ as a subspace topology. So that this topology gives a notion of "nearness" of events on the timelike worldline $\alpha(I)$. The other newer topology constructed by Hawking, King and McCarthy which is includes more structure than Zeeman topology.