Isometry between Minkowski space and Tangent space in an article by Stefan Waldmann [closed]

Question

In the notes Geometric Wave Equations by Stefan Waldmann at page 70 they have

Having a fixed Lorentz metric $g$ on a spacetime manifold $M$ we can now transfer the notions of special relativity, see e.g. 50 , to $(M, g)$. In fact, each tangent space $\left(T_{p} M, g_{p}\right)$ is isometrically isomorphic to Minkowski spacetime $\left(\mathbb{R}^{n}, \eta\right)$ with $\eta=\operatorname{diag}(+1,-1, \ldots,-1)$, by choosing a Lorentz frame: there exist tangent vectors $e_{i} \in T_{p} M$ with $i=1, \ldots, n$ such that $$ g_{p}\left(e_{i}, e_{j}\right)=\eta_{i j}=\pm \delta_{i j} . $$

We say that two manifolds $M$ and $N$ are isometric if for all $v \in T_pM$ and a map $\phi:M\rightarrow N$ such that

$g(v,v)=g'(\phi_*v,\phi_*v)$ where $g$ is a metric in $M$ , $g'$ is a metric in $N$ and $\phi_*$ denotes a pushfoward.

Now the definition of isometry refers to two manifolds, but in the notes they are claiming an isometry between a manifold and a tangent space.

How can a tangent space be isometric to a manifold?

Answer 1

I think you are confused by two different uses for the word "isometry".

There is first the notion of a linear isometry between vector spaces equipped with a non-degenerate bilinear form. This is the notion that is used on page 70 of the paper you refer to. It simply says that, for a Lorentz metric $g$ on a manifold $M$, for every $m\in M$ the tangent space $(T_mM,g)$ is linearly isometric to the Minskowsky space $(\mathbb{R}^n,\eta)$.

There is then the notion of an isometry between two pseudo-riemannian manifolds, that is a smooth map $\phi:M\to N$ such that for every $m\in M$ the tangent map $d\phi_m:T_mM\to T_{\phi(m)}N$ is a linear isometry.