Local expression of vector field along a curve

Question

In Chapter 2 of Do Carmo's Riemannian Geometry, the following is stated:

Question: In the expression for $V$, why is $X_j=X_j(c(t))$?

Answer 1

The vector field $V$ is defined along $c$. In local coordinates, the restrictions to $c$ of the coordinate vector fields $X_j$ form a basis for the vector fields along $c$, so that each vector field along $V$ is a $C^\infty$-linear combination of the $X_j$ in the local coordinate chart $(\mathbf{x},U)$. When we choose a parametrization for $c$, we have a parametrization for the local coordinate expression of $V$ by pulling back to $\Bbb{R}$: $$ V(t) = \sum_j v^j(t)X_j(c(t)) $$

tl;dr: The vector field $X_j$ is defined in $U\subseteq M$ so its pullback to $\Bbb{R}$ is $X_j\circ c$. It is the smooth manifold equivalent of a frame field on a curve from your undergraduate curves-and-surfaces class.

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There's some bundle chicanery going on here behind the scenes. The following diagram chase is as much for my own edification as to answer the question.

We start with a curve $c: L\to M$ where $c$ is a smooth map embedding the smooth $1$-manifold $L$ into $M$, our smooth $n$-manifold. A vector field "along $c$" is a section of the pullback of $TM$ by $c$, which makes the following diagram commute: $$\require{AMScd} \begin{CD} c^*TM @>{}>> TM\\ @VVV @VVV \\ L @>{c}>> M \end{CD}$$

When we write a vector field in local coordinates, we're really pulling back sections of $TM\to M$ by a local coordinate map: $$\require{AMScd} \begin{CD} \mathbf{x}^*TM @>{}>> TM\\ @VVV @VVV \\ U @>{\mathbf{x}}>> M \end{CD}$$ In particular, the vector fields $X_j$ are "really" canonical sections of $\mathbf{x}^*TM\to U$ forming a smooth pointwise basis for each fiber. If $V$ is a field defined on $M$, then its local coordinate expression is the pulled-back section $\mathbf{x}^*V: U\to \mathbf{x}^*TM$ when expanded in the basis $X_j$.

A smooth parametrization of $c$ is a choice of local coordinates $\Bbb{R}\to L$ composed with the map $c$ and yields a third commutative diagram: $$\require{AMScd} \begin{CD} \phi^*(c^*TM) @>{}>> c^*TM \\ @VVV @VVV \\ \Bbb{R} @>{\phi}>> L \end{CD}$$

Now let's put our three diagrams together to see what happens when we take a vector field $V$ on $c$, that is a section $V: L\to c^*TM$, and write it in local coordinates after restricting to a single component of $c(L)\cap U$.

$$\require{AMScd} \begin{CD} \phi^*(c^*TM) @>{}>> c^*TM @>{}>> TM @<{}<< \mathbf{x}^*TM \\ @VVV @VVV @VVV @VVV \\ \Bbb{R} @>{\phi}>> L @>{c}>> M @<{\mathbf{x}}<< U \end{CD}$$

By choosing a parametrization, we pull back $V$ to a smooth section $\phi^*V: \Bbb{R}\to \phi^*(c^*TM)$ of the parametrized bundle. By restricting to a single component of $c(L)\cap U$, we compose with $\mathbf{x}^{-1}$ to get a local coordinate expression $(\mathbf{x}^{-1}\circ c\circ \phi)(t) = (x_1(t), x_2(t), ..., x_n(t))$.

Finally, we pull back the local coordinate bundle $\mathbf{x}^*TM$ and its canonical basis $X_j$ along the map $\mathbf{x}^{-1}\circ c\circ\phi$ to a vector bundle over $\Bbb{R}$ which is bundle-isomorphic to $c^*TM\to L$ and the image of $V$ under this isomorphism is the local coordinate expression of $V$.

There's probably some category theoretic term encapsulating and abstracting this entire process I just haltingly described, but I'm not a category theorist :)