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.

Facts

Tensoriality

Further information: Torsion is tensorial

The torsion map is a $(1,2)$ tensor. It is tensorial in both $X$ and $Y$. This means that the value of the torsion of a connection for two vector fields at a point depends only on the values of the vector fields at that point. In other words, $\tau (\mathrm{\nabla })(X,Y)$ at $p$ depends on $\mathrm{\nabla },X(p),Y(p)$ only and does not depend on how $X$ and $Y$ behave elsewhere on the manifold.

Antisymmetry

Further information: Torsion is antisymmetric

We have that the torsion tensor is antisymmetric, i.e., we have:

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

Equivalently, we have that:

$\tau (\mathrm{\nabla })(X,X)=0$