# Formula Christoffel Symbols (without Torsion)

## Christoffel symbols

`$$ \Gamma^{ \class{red}{c} }_{\;\class{blue}{a}\class{green}{b}} $$`Unit

`$$ $$`

Christoffel symbols are used to extend a partial derivative (on a flat manifold) to a covariant derivative (on a curved manifold). The indices \( \class{red}{c}\), \(\class{blue}{a}\), \( \class{green}{b} \) and \(s\) take the values between 0 and 3 in the four-dimensional case (time + 3d space).

Here \(s\) is a summation index. Einstein summation convention is used here.

## Metric tensor

`$$ g_{\class{green}{b}s} $$`Unit

`$$ $$`

The metric tensor determines the distances and angles in a curved space. In a chosen basis and in the four-dimensional case, the metric tensor is a symmetric 4x4 matrix.

## Inverse metric tensor

`$$ g^{\class{red}{c}s} $$`Unit

`$$ $$`

Contravariant metric tensor is the inverse of the metric tensor. In a basis it is the inverse matrix.