In this question manifolds always mean differentiable manifolds and vector fields always mean differentiable vector fields. The following is a stanndard result:
Proposition 1. $(\nabla_X Y_1)(p)=(\nabla_X Y_2)(p)$ for all vector fields $X,Y_1,Y_2$ satisfying $Y_1|_U=Y_2|_U$ for some open neighbourhood $U$ containing $p$ and that $(\nabla_{X_1}Y)(p)=(\nabla_{X_2}Y)(p)$ for vector fields with $X_1(p)=X_2(p)$.
Furthermore the result about bump functions is well known:
Proposition 2. Let $M$ be a manifold, $p \in M$ and $U$ be an open neighbourhood of $p$. Then there exist open neighbourhoods of $p$, $U_1,U_2$ such that $\overline{U_1} \subseteq U_2$, $\overline{U_2} \subseteq U$ as well as a differentiable function $f:M \to [0,1]$ such that $f|_{U_1}=1, f|_{M \setminus U_2}=0$.
The definition of the Christoffel symbols is as follows. For a $p \in M$ and a chart $(U,\varphi)$ around $p$ one knows that the $\frac{\partial}{\partial x_1}|_p,…\frac{\partial}{\partial x_n}|_p$ form a basis of $T_pM$. Hence: $$\nabla_{\frac{\partial}{\partial x_i}}\frac{\partial}{\partial x_j}=\sum_k \Gamma_{ij}^k \frac{\partial}{\partial x_k}$$ for differentiable $\Gamma_{ij}^k:U \to \mathbb{R}.$ What is weird at first is that we calculate the connection of local vector fields instead of global vector fields. This is where Proposition 1 is being used I think:
If $X,Y$ is a local vector field, i.e. $Y: U \to TU,$ and $X:U \to TU$, we can define $$\overline{X}(q):=\begin{cases}
f(q)X(q), q \in U,\\
0, \text{otherwise}
\end{cases},$$ where $f$ is a bump function. Then one can define
$$(\nabla_XY)(q):=
(\nabla_{\overline{X}}\overline{Y})(q), q \in U.$$
This means that one actually computes $\nabla_{\overline{\frac{\partial}{\partial x_i}}}\overline{\frac{\partial}{\partial x_j}}=\sum_k \Gamma_{ij}^k \frac{\partial}{\partial x_k}$.
Another common thing to use is the local representation of a vector field in calculations: In a chart $(U,\varphi)$ one has $X|_U=\sum_{i} X_i \frac{\partial}{\partial x_i}$ and thus $$\nabla_X Y=\sum_i X_i \nabla_{\frac{\partial}{\partial x_i}} Y.$$ Why is that? We know that this property holds for local vector fields, but $\sum_i X_i \frac{\partial}{\partial x_i}$ is again local, meaning one would have to multiply with a bump function $f$. Thus $$\overline{X}=\begin{cases}
\sum_i f(q)X_i(q) \frac{\partial}{\partial x_i}|_q, \ q \in U,\\
0, \text{otherwise}.
\end{cases}$$
However, I don't think this quite works and that one has to extend the $X_i$ by bump functions globally as well, right? If $\overline{X_i}$ are global extensions of the $X_i$ in this way, one can define $$\widetilde{X}(q):=\begin{cases}
f(q) \frac{\partial}{\partial x_i}|_q, \ q \in U\\
0, \text{otherwise}\end{cases}$$ and obtains $X|_U=\sum_i \overline{X_i}|_U\widetilde{X}|_U$ and thus, finally, $$\nabla_X Y=\nabla_{\sum_i \overline{X_i} \widetilde{X}} Y=\sum_i \overline{X_i}\nabla_{\widetilde{X}}Y=\sum_i X_i \nabla_{\frac{\partial}{\partial x_i}} Y.$$
Is this correct? I hope this correctly justifies why one can simply "apply the rules" to local vector fields as well, without bothering whether they are locally or globally defined.
Best Answer
Yes, it's all to do with propositions (1) and (2) which allow us to once and for all extend the meaning of $\nabla$, and then suppress that in the notation, and never speak of it (or bump functions) ever again.
Typically one starts off by defining $\nabla$ as a mapping defined on global vector fields $(X,Y)\mapsto \nabla_XY$. However, because of properties (1) and (2), we can now extend the meaning of $\nabla$ so that it it's arguments aren't local vector fields. Maybe for now I shall use the different symbol $\widetilde{\nabla}$ to denote this extension. Explicitly, given any $p\in M$, any $v\in T_pM$ and any vector field $Y$ defined in some open neighborhood $U$ of $p$, we can define the symbol $\widetilde{\nabla}_vY\in T_pM$ as follows:
By using a chart centered around $p$, it is easy to construct a smooth vector field defined in some open neighborhood of $p$, such that the value of the field at $p$ is the vector $v$. Now, by using a bump function, we can get some global smooth vector field $\widetilde{X}:M\to TM$ such that $\widetilde{X}(p)=v$.
We again use bump functions (take $f,U_1,U_2,p$ as in proposition 2 and $U$ as above, the domain of the vector field $Y$). Now, consider the extension $\widetilde{Y}:M\to TM$ defined as $\widetilde{Y}(q)=f(q)Y(q)$ if $q\in U$ and $0$ otherwise. So, $Y$ and $\widetilde{Y}$ agree in some open neighborhood of $p$ (namely $U_1$).
Define $\widetilde{\nabla}_{v}Y:= \left(\nabla_{\widetilde{X}}\widetilde{Y}\right)(p)\in T_pM$.
Note that proposition (1) tells us that $\widetilde{\nabla}_vY$ is well-defined and doesn't depend on the extensions $\widetilde{X}, \widetilde{Y}$. Now, very clearly, $\widetilde{\nabla}$ is an extension of $\nabla$ in the sense that for any global vector fields $X,Y$ on $M$ and any $p\in M$ and any open neighborhood $U$ of $p$, we have $\widetilde{\nabla}_{X(p)}\left(Y|_U\right)=(\nabla_XY)(p)$.
Next, if $X,Y$ are locally defined vector fields, say on open sets $U,V$ and that $U\cap V\neq\emptyset$, then we define the symbol $\widetilde{\nabla}_XY:U\cap V\to TM$ as $p\mapsto \widetilde{\nabla}_{X(p)}Y$. It is easy to verify that this is a smooth vector field on $U\cap V$. At this stage you can easily verify that $\widetilde{\nabla}$ satisfies all the rules you expect.
As a final step, we avoid the extra symbolism with the tilde and refer to it as $\nabla$. From a very strict sense, it is an abuse of notation, but it's a very convenient and acceptable one because there's no ambiguity in the definition of $\widetilde{\nabla}$; it is uniquely determined by $\nabla$ (and is even an extension of it in the sense described above). This allows us to manipulate local expressions with ease. So the definition of the Christoffel symbols (with respect to a chart $(U,\varphi)$) is properly written as for each $p\in U$, \begin{align} \Gamma^k_{ij}(p)&= dx^k(p)\left(\widetilde{\nabla}_{\frac{\partial}{\partial x^i}(p)}\frac{\partial}{\partial x^j}\right), \end{align} or if we adhere to the convention of dropping the tilde, then $\Gamma^k_{ij}=dx^k\left(\nabla_{\frac{\partial}{\partial x^i}}\frac{\partial}{\partial x^j}\right)$.
As a final remark, note that proposition (1) can be improved further. We actually don't need the full information that the field $Y$ be defined in an open neighborhood of $p$. All we really need is for $Y$ to be defined along some curve $\gamma$ passing through $p$.
Explicitly, suppose $\gamma:I\to M$ is a smooth curve ($I$ an open interval containing the origin) and $Y$ a smooth vector field defined in an open neighborhood of $p:=\gamma(0)$, and let $v=\gamma'(0)\in T_pM$. Now, fix a chart centered at $p$, and denote the coordinate functions as $x^1,\dots, x^n$. Write $v=\sum_{i=1}^nv^i\frac{\partial}{\partial x^i}(p)$, and $Y=\sum_{i=1}^nY^i\frac{\partial}{\partial x^i}$ on the chart domain. Then, by the usual rules, \begin{align} \nabla_vY&=\left(v^i\frac{\partial Y^k}{\partial x^i}(p)+\Gamma^k_{ij}(p)v^iY^j(p)\right)\cdot\frac{\partial}{\partial x^k}(p)\\ &=\left((Y^k\circ \gamma)'(0)+ \Gamma^k_{ij}(p)v^iY^j(p)\right)\cdot\frac{\partial}{\partial x^k}(p), \end{align} hence $\nabla_vY$ depends only on $Y\circ \gamma$, i.e the value of the vector field along the curve (and moreover, it doesn't depend on the actual curve $\gamma$, only on the equivalence class of curves having the same tangent vector).
As a result of this, we can once again extend the meaning of $\nabla$ (or $\widetilde{\nabla}$) so that the expression $\nabla_v\psi$ makes sense, where $v\in T_pM$ is a tangent vector and $\psi:I\to TM$ is a smooth mapping such that for some smooth curve $\gamma:I\to M$ with $\gamma(0)=p$ and $v=\gamma'(0)$, we have that for all $t\in I$, $\psi(t)\in T_{\gamma(t)}M$. In other words, $\psi$ is a vector field along some base curve in $M$ which passes through the point $p$ with tangent vector $v$ (and again this result doesn't depend on $\gamma$, only on its equivalence class). Some texts denote this symbol as $\frac{D\psi}{dt}$ (though more properly it should be something like $\frac{D_{[\gamma]}\psi}{dt}\bigg|_0$).
As a result of this, you can now make sense of the typical expression involving say the geodesic equation $\nabla_{\gamma'}\gamma'=0$, which in a coordinate chart $(U,x^1,\dots, x^n)$ reads $\ddot{\gamma}^i+(\Gamma^i_{jk}\circ \gamma)\dot{\gamma}^j\dot{\gamma}^k=0$, (where $\gamma^i=x^i\circ \gamma$), or in more classical notation, $\ddot{x}^i+\Gamma^i_{jk}\dot{x}^j\dot{x}^k=0$.