Torsion

In elementary differential geometry in three dimensions, the torsion of a curve measures how sharply it is twisting. It is analogous to curvature in two dimensions. Given a vector-valued function r(t), the torsion at a given value of t is

((r' × r")· r⁽³⁾ ) ||r' × r"||^-2.

A second meaning of torsion in differential geometry is the torsion tensor, which depends on a connection. It is a (1,2) tensor given by the formula where [u,v] is the Lie bracket of the two vector fields.

Torsion free connections are considered most frequently - the Levi-Civita connection is assumed to have zero torsion, for instance.