# Formula Christoffel Symbols (without Torsion)

## Christoffel symbols

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

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

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