On proving Minkowski space is a manifold

Question

Since the Minkowski space is just $\mathbb{R}^4$ with a different inner product does that mean it's a manifold ? And if it is are there any easy proofs or a hint, so I can work it myself?

Answer 1

This is an attempt and most likely it's not correct because I am confused about this things in general. That been said, let $M $ to be the Minkowski space and take $x \in M$, $x=(x_1,x_2,x_3,x_4) \in \mathbb{R}^4$ and consider the Identity function $\psi :M\rightarrow \mathbb{R}^4$, $\psi(x)=x$

$\psi $ is an homeomorphism

the Jacobian matrix has rank 4 and $\psi^{-1}(0)=M \cap \mathbb{R}^4$ therefore, $M$ is an 4-d manifold

Did I have any luck with that? Can someone help me and correct me?