Torsion of a linear connection

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

Definition

Given data

Definition part

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

\tau(\nabla)(X,Y) = \nabla_X Y- \nabla_Y X - [X,Y]

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

Tensoriality

Further information: Torsion is tensorial The torsion map is a (1,2) tensor. It is tensorial in both X and Y.