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I have some questions regarding connections and christoffel symbols. The definition of connections im working with is simpler than the more general definition on a vector/tensor bundle, it is the following:

Let $M$ be a smooth manifold, $f_1,f_2\in\mathcal{C}^\infty$, $X_1,X_2,Y_1,Y_2\in\mathcal{V}(M)$, and $a_1,a_2\in\mathbb{R}$

A connection on $M$ is a map $\nabla:\mathcal{V}(M)\times\mathcal{V}(M)\to\mathcal{V}(M),(X,Y)\mapsto\nabla_XY$ satisfying the following properties:\begin{align*}\nabla_{f_1X_1+f_2X_2}Y&=f_1\nabla_{X_1}Y+f_2\nabla_{X_2}Y\\\nabla_X(a_1Y_1+a_2Y_2)&=a_1\nabla_XY_1+a_2\nabla_XY_2\\\nabla_X(f_1Y)&=f_1\nabla_XY+(Xf_1)Y\end{align*}

A connection can be expressed in local coordinates using connection coefficients/Christoffel symbols:$$\nabla_{E_i}E_j=\Gamma_{ij}^kE_k$$ where $(E_i)$ is a local smooth frame. My questions are

  1. Are connections uniquely determined through their connection coefficients?
  2. How do I interpret these connection coefficients? Are they constants for the entire chart/local smooth frame I'm in? Or do they depend on where I am within that chart as well?
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Re 1.: Locally, when can define a connection $\nabla$ by picking a smooth local frame $E_{1}, E_{2}, \ldots , E_{n}$ on an open set $U\subseteq M$ and defining the connection on the frame.

That is, with a smooth local frame $E_{1}, E_{2}, \ldots , E_{n}$ and connection coefficients/Christoffel symbols $\Gamma_{ij}^{k} \in \mathcal{C}^{\infty}(U)$ defined by $$ \nabla_{E_{i}}E_{j} = \Gamma_{ij}^{k}E_{k},$$ one obtains a local connection if one extends the connection to vector fields $\mathcal{V}(U)$ via the stated properties.

Namely, suppose that $X, Y \in \mathcal{V}(U)$. Then one can express $X$ and $Y$ relative to the local frame as $$ X = p^{i}E_{i} \quad and \quad Y = q^{i}E_{i}, $$ where $p^{i}, q^{i}$,$1 \le i \le n$, are smooth functions on ${U}$ (i.e. , $p^{i}, q^{i}\in \mathcal{C}^{\infty}\left(U\right)$). Using the outlined properties of the connection, one has the following: \begin{align*} \nabla_{X} Y & = \nabla_{p^iE_i}q^jE_{j}\\ &= p^{i}\nabla_{E_i}q^jE_{j} \quad \textrm{(linearity across functions in the first slot)}\\ & = p^i\left(q^{j}\nabla_{E_i}E_{j} + E_i(q^j)E_j\right) \quad \textrm{("product rule" with functions in the second slot)}\\ &= p^i\left(q^{j} \Gamma_{ij}^{k}E_{k}+ E_i(q^j)E_j\right) \quad \textrm{(Using expansion of $\nabla_{E_i}E_j = \Gamma_{ij}^{k}E_{k}$)}\\ &= p^iq^{j} \Gamma_{ij}^{k}E_{k}+ p^{i}E_i(q^k)E_k \quad \textrm{(Changing summation indices to get everything on same basis)}\\ &= \left(p^iq^{j} \Gamma_{ij}^{k}+ p^{i}E_i(q^k)\right)E_k\\ &= \left(p^iq^{j} \Gamma_{ij}^{k}+ X(q^k)\right)E_k \quad \textrm{(noting that $X = p^iE_i$)} \end{align*}

Re 2.: The connection coefficients/Christoffel symbols depend on location in the chart and are generally non-constant smooth functions. One of the keys is how the connection coefficients change when one changes frames (they don't change like tensors). The issue here is that this is not purely a point wise condition: The value of $\nabla_{X}Y$ at a point $p$ depends on both the value of $X$ at the point $p$ and the values of $Y$ along a curve that is tangent to the vector field $X$ at the point $p$ (i.e., some curve with $c(0) = p$ and $c^{\prime}(0) = X(p)$).

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