Torsion of a linear connection

This article defines a tensor (viz a section on a tensor bundle over the manifold) of type (1,2)

Definition

Given data

• A differential manifold $M$
• A linear connection $\mathrm{\nabla }$ on $M$ (viz a connection on the tangent bundle $TM$).

Definition part

The torsion of $\mathrm{\nabla }$, denoted as $\tau (\mathrm{\nabla })$, is defined as a map that takes as input 2 vector fields and outputs a third vector field, as follows:

$\tau (\mathrm{\nabla })(X,Y)={\mathrm{\nabla }}_{X}Y-{\mathrm{\nabla }}_{Y}X-[X,Y]$

A linear connection whose torsion is zero is termed a torsion-free linear connection.

Tensoriality

The torsion map is a $(1,2)$ tensor. It is tensorial in both $X$ and $Y$. This can actually be easily checked by hand.