Difference between revisions of "Torsion of a linear connection"

From Diffgeom
Jump to: navigation, search
m (5 revisions)
(No difference)

Revision as of 20:12, 18 May 2008

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


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.


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