Gauss lemma tangent space identification

Question

In Gauss' lemma (in riemannian geometry), many books say that we can identify $T_v(T_xM)$ with $T_xM$ where $(x,v) \in TM$. How can I see how to identify these two spaces?

Answer 1

In general, if $V$ is a finite-dimensional real vector space and $a \in V$, we have a canonical isomorphism $\varphi : V \to T_aV$ given by $v \mapsto D_v|_a$ where $$D_v|_af = \frac{d}{dt}f(a + tv)\bigg|_{t=0}.$$ Geometrically, each vector $v$ defines a directional derivative at $a$ (in the direction of $v$), and these are the only derivations of $C^{\infty}(V)$ at $a$. The fact that the map $\varphi$ is an isomorphism is proved in Lee's Introduction to Smooth Manifolds (second edition), Proposition $3.13$.