# Torsion-free linear connection

## Contents

## Definition

### Symbol-free definition

A linear connection on a differential manifold is said to be **torsion-free** or **symmetric** if it satisfies the following equivalent conditions:

- Its torsion is zero
- Whenever two vector fields are such that their Lie bracket is zero, then the covariant derivative of either with respect to the other are equal.

### Definition with symbols

A linear connection on a differential manifold is said to be **torsion-free** or **symmetric** if it satisfies the following equivalent conditions:

- The torsion of is a zero map, viz:

- Whenever and are vector fields on an open subset such that on , then:

### Definition in local coordinates

In local coordinates, a linear connection is torsion-free if the Christoffel symbols are symmetric in and .

## Facts

### Set of all torsion-free linear connections

`Further information: Affine space of torsion-free linear connections`
Recall that the set of all linear connections is an affine space, viz a translate of a linear subspace (the linear subspace being the maps that are tensorial in both variables).

The set of torsion-free linear connections is an affine subspace of this, in the sense that any affine combination of torsion-free linear connections is again a torsion-free linear connection.

The corresponding linear subspace for torsion-free linear connections are the symmetric 2-tensors.