Torsion-free linear connection

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 $\nabla$ on a differential manifold $M$ is said to be torsion-free or symmetric if it satisfies the following equivalent conditions:

• The torsion of $\nabla$ is a zero map, viz:

$\tau(\nabla) = (X,Y) \mapsto \nabla_XY - \nabla_YX - [X,Y] = 0$

• Whenever $X$ and $Y$ are vector fields on an open subset $U \subset M$ such that $[X,Y]= 0$ on $U$, then:

$\nabla_X Y = \nabla_YX$

Definition in local coordinates

In local coordinates, a linear connection is torsion-free if the Christoffel symbols $\Gamma_{ij}^k$ are symmetric in $i$ and $j$.

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.