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Categories: Differential geometry | Riemannian geometry

Torsion

In mathematics, the term torsion has several meanings, mostly unrelated to each other.

Contents

1 Differential geometry of curves

2 Torsion tensor of a connection

3 Abstract algebra

4 Topology

5 See also:

Differential geometry of curves

In the elementary differential geometry of curves in three dimensions, the torsion of a curve measures how sharply it is twisting. It is analogous to curvature in two dimensions. Given a function r(t) with values in R³, the torsion at a given value of $t$ is

$\tau = {{\det \left( {r',r'',r'''} \right)} \over {\left\| {r' \times r''} \right\|^2}} = {{\left( {r' \times r''} \right)\cdot r'''} \over {\left\| {r' \times r''} \right\|^2}}.$

Here the primes denote the derivatives of r with respect to t; if the cross product in the denominator is zero, the torsion τ is defined to be zero as well.

The torsion of a curve will be zero if and only if the curve sits inside a fixed plane. It is positive for right-handed spirals and negative for left-handed ones.

Torsion tensor of a connection

A second meaning of torsion in differential geometry is the torsion tensor, which depends on an affine connection $\nabla$ . It is a (1,2) tensor given by the formula

$T(u,v)=\nabla_u v - \nabla_v u -[u,v]$

where $[u, v]$ is the Lie bracket of the two vector fields.

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

See also Connection form.

Abstract algebra

In abstract algebra, the torsion subgroup of an abelian group consists of all elements of finite order. An abelian group is called torsion-free if and only if the identity is the only element that has finite order. (This concept generalises to that of a torsion module.) In the Tor functors of homological algebra, which arise because tensor product does not in general preserve exact sequences, the symbol Tor does stand for this kind of algebraic torsion, historically speaking anyway. These functors were introduced in order to make systematic the universal coefficient theorem of homology theory, in cases where the homology groups H_i(X,Z) of a space X had some torsion.

Topology

Some topological invariants are called torsions: for example the Reidemeister-Schreier torsion of a group acting on a finite complex; and also the analytic torsion defined using Laplacians.

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