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 video and in particular this part to see how geodesics can be obtained from them.

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} .