I am reading the book by Lee, Introduction to Riemannian manifolds which I think is very nice. However among the things I don't understand is the following :
Theorem 4.24 (Covariant Derivative Along a Curve).
Let $M$ be a smooth mani- fold with or without boundary and let
$\nabla$ be a connection in $TM$ . For each smooth curve $\gamma : I \to M$ , the connection determines a unique operator $D_t:\mathfrak{X}(\gamma)\to\mathfrak{X}(\gamma)$,called the covariant derivative along $\gamma$, satisfying the
following properties:(i) LINEARITY OVER $\mathbb{R}$: $D_t(aV+bW)=aD_tV+bD_tW$ for $a,b\in\mathbb{R}$:
(ii) PRODUCT RULE :
$D_t(f V)=f'+ D_t V$ for $f\in C^\infty(I)$
(iii) If $V \in \mathfrak{X}(M)$ is extendible, then for every
extension $\tilde{V}$ of $V$$D_t V(t)= \nabla_{\gamma'(t)}\tilde{V}$.
There is an analogous operator on the space of smooth tensor fields of
any type along $\gamma$ .
But in property (iii), If it was "$V\in \mathfrak{X}(M)$ and $\gamma'$ are both extendible, then for every extension $\tilde{V}$ and $\tilde{\gamma'}, D_tV(t)=(\nabla_{\tilde{\gamma'}}\tilde{V})(\gamma(t))$", it would make more sense to me, because here $\gamma'(t)$ is just a tangent vector and not a vector field. There is something I am not getting right.
Best Answer
I think you want use Proposition 4.5: $\nabla_X Y |_p$ depends only on the value of $X$ at $p$ (as well as the values of $Y$ in a neighborhood of $p$). Then, for a tangent vector $v \in T_p M$ we interpret the expression $\nabla_v Y$ by arbitrarily extending $v$ to a vector field in a neighborhood of $p$; the proposition implies that the result is independent of extension.