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May 19, 2022 at 16:03 comment added DamienC A manifold is something that locally looks like a finite dimensional R-vector space. As such, any finite dimensional R-vector space is a manifold. I don't understand what you don't understand.
May 18, 2022 at 17:11 comment added amilton moreira $R^n$ can be a manifold without being a vector space
May 18, 2022 at 14:50 comment added DamienC @amiltonmoreira would you refuse to consider $\mathbb{R}^n$ as a vector space because several pages before it had been considered as a manifold? Every finite dimensional vector space equipped with a nondegenerate pairing is an example of a pseudo-riemannian manifold. Example 2.1.22 of loc.cit. essentially says this (it is even written "constant metric", with "constant" emphazised).
May 18, 2022 at 14:36 comment added amilton moreira At page 64 example 2.1.22 he defined Minskowsky spacetime as a manifold not as vector space.
May 18, 2022 at 9:51 history answered DamienC CC BY-SA 4.0