Torsion-free linear connection

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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.