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 $k = 1$ :

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

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

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

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

For $k = 2$ :

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

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

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

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

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., $Γ_{i j}^{k} = Γ_{j i}^{k}$ .

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.

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Author of the notes: Antonio J. Pan-Collantes

antonio.pan@uca.es

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