Christoffel symbols

Let $M$ be a manifold. To specify locally a covariant derivative operator $\nabla$ it suffices to fix a local chart ${x_{i}}$ and provide the functions $Γ_{i j}^{k}$ such that

\nabla_{\partial_{x_{i}}} \partial_{x_{j}} = Γ_{i j}^{k} \partial_{x_{k}}

which are called the Christoffel symbols (of second kind) when the covariant derivative is coming from the Levi-Civita connection of a Riemannian metric! They correspond to the vector bundle connection#Connection form.

The formula to compute the Christoffel symbols from the Riemannian metric can be deduced from the defining conditions 1. and 2. of the Levi-Civita connection (see, for example, wikipedia or this video) giving rise to

Γ_{i j}^{k} = \frac{1}{2} g^{k l} (\partial_{i} g_{j l} + \partial_{j} g_{i l} - \partial_{l} g_{i j})

In 2 dimensions:
For $k = 1$ :

$Γ_{11}^{1} = \sum_{l = 1}^{2} \frac{1}{2} g^{1 l} (\partial_{1} g_{1 l} + \partial_{1} g_{1 l} - \partial_{l} g_{11}) = \frac{1}{2} g^{11} \partial_{1} g_{11} + \frac{1}{2} g^{12} (2 \partial_{1} g_{21} - \partial_{2} g_{11})$
$Γ_{12}^{1} = \sum_{l = 1}^{2} \frac{1}{2} g^{1 l} (\partial_{1} g_{2 l} + \partial_{2} g_{1 l} - \partial_{l} g_{12}) = \frac{1}{2} g^{11} \partial_{2} g_{11} + \frac{1}{2} g^{12} \partial_{1} g_{22}$
$Γ_{21}^{1} = \sum_{l = 1}^{2} \frac{1}{2} g^{1 l} (\partial_{2} g_{1 l} + \partial_{1} g_{2 l} - \partial_{l} g_{21}) = \frac{1}{2} g^{11} \partial_{2} g_{11} + \frac{1}{2} g^{12} \partial_{1} g_{22}$
$Γ_{22}^{1} = \sum_{l = 1}^{2} \frac{1}{2} g^{1 l} (\partial_{2} g_{2 l} + \partial_{2} g_{2 l} - \partial_{l} g_{22}) = \frac{1}{2} g^{11} (2 \partial_{2} g_{21} - \partial_{1} g_{22}) + \frac{1}{2} g^{12} \partial_{2} g_{22}$

For $k = 2$ :

$Γ_{11}^{2} = \frac{1}{2} g^{21} \partial_{1} g_{11} + \frac{1}{2} g^{22} (2 \partial_{1} g_{21} - \partial_{2} g_{11})$
$Γ_{12}^{2} = \frac{1}{2} g^{21} \partial_{2} g_{11} + \frac{1}{2} g^{22} \partial_{1} g_{22}$
$Γ_{21}^{2} = \frac{1}{2} g^{21} \partial_{2} g_{11} + \frac{1}{2} g^{22} \partial_{1} g_{22}$
$Γ_{22}^{2} = \frac{1}{2} g^{21} (2 \partial_{2} g_{21} - \partial_{1} g_{22}) + \frac{1}{2} g^{22} \partial_{2} g_{22}$

Since the metric tensor $g_{i j}$ is symmetric (i.e., $g_{i j} = g_{j i}$ ), it follows that the Christoffel symbols are symmetric in their lower indices, i.e., $Γ_{i j}^{k} = Γ_{j i}^{k}$ .

Intuition

See this vide for the meaning of Christoffel symbols. See this video and in particular this part to see how geodesics can be obtained from them.
relation of Lie derivative, covariant derivative and torsion

In practice

Given the vector field $V = V^{i} \partial_{x_{i}}$ , the covariant derivative $\nabla_{\partial_{x_{j}}} V$ is computed using the following formula:

(\nabla_{\partial_{x_{j}}} V)^{k} = \partial_{j} V^{k} + Γ_{i j}^{k} V^{i}

See also linear connection#Extension to tensor fields for using the covariant derivatives on tensors with components.

Change of chart

\begin{aligned} Γ_{(y) j k}^{i} & := d y^{i} : (\nabla_{\frac{\partial}{\partial y^{j}}} \frac{\partial}{\partial y^{k}}) \\ = \frac{\partial y^{i}}{\partial x^{q}} d x^{q} : (\nabla_{\frac{\partial x^{p}}{\partial y^{j}} \frac{\partial}{\partial x^{p}}} \frac{\partial x^{s}}{\partial y^{k}} \frac{\partial}{\partial x^{s}}) \\ = \frac{\partial y^{i}}{\partial x^{q}} d x^{q} : (\frac{\partial x^{p}}{\partial y^{j}} [\frac{\partial}{\partial x^{p}} (\frac{\partial x^{s}}{\partial y^{k}}) \frac{\partial}{\partial x^{s}} + \frac{\partial x^{s}}{\partial y^{k}} (\nabla_{\frac{\partial}{\partial x^{p}}} \frac{\partial}{\partial x^{s}})]) \\ = \frac{\partial y^{i}}{\partial x^{q}} d x^{q} : (\frac{\partial x^{p}}{\partial y^{j}} [\frac{\partial}{\partial x^{p}} (\frac{\partial x^{s}}{\partial y^{k}}) \frac{\partial}{\partial x^{s}} + \frac{\partial x^{s}}{\partial y^{k}} Γ_{(x) s p}^{m} \frac{\partial}{\partial x^{m}}]) \\ = \frac{\partial y^{i}}{\partial x^{q}} \frac{\partial x^{p}}{\partial y^{j}} (\frac{\partial}{\partial x^{p}} (\frac{\partial x^{s}}{\partial y^{k}}) δ_{s}^{q} + \frac{\partial x^{s}}{\partial y^{k}} Γ_{(x) s p}^{m} δ_{m}^{q}) \\ = \frac{\partial y^{i}}{\partial x^{q}} \frac{\partial x^{p}}{\partial y^{j}} \frac{\partial}{\partial x^{p}} (\frac{\partial x^{q}}{\partial y^{k}}) + \frac{\partial y^{i}}{\partial x^{q}} \frac{\partial x^{p}}{\partial y^{j}} \frac{\partial x^{s}}{\partial y^{k}} Γ_{(x) s p}^{q} \\ = \frac{\partial y^{i}}{\partial x^{q}} \frac{\partial}{\partial y^{j}} (\frac{\partial x^{q}}{\partial y^{k}}) + \frac{\partial y^{i}}{\partial x^{q}} \frac{\partial x^{p}}{\partial y^{j}} \frac{\partial x^{s}}{\partial y^{k}} Γ_{(x) s p}^{q} \\ = \frac{\partial y^{i}}{\partial x^{q}} \frac{\partial^{2} x^{q}}{\partial y^{j} \partial y^{k}} + \frac{\partial y^{i}}{\partial x^{q}} \frac{\partial x^{s}}{\partial y^{j}} \frac{\partial x^{p}}{\partial y^{k}} Γ_{(x) s p}^{q}, \end{aligned}

See Schuller.

Generalization?

I think that for other local frame ${e_{i}}$ of $T M$ , the functions $Γ_{i j}^{k}$ such that

\nabla_{e_{i}} e_{j} = Γ_{i j}^{k} e_{k},

playing the same role of Christoffel symbols.

This idea works also for a vector bundle connection on a vector bundle $E$ , not only the particular case of $T M$ . This case is explained here: it is called the connection form, not to be confused with the connection 1-form of a connection on a general bundle, although there is a relation explained here.

We can say that in terms of the connection form $Θ_{i}^{k}$ , we have

\partial_{x_{j}} ⌟ Θ_{i}^{k} = Γ_{i j}^{k} .