Torsion
In mathematics, torsion has several meanings, mostly unrelated to each other.
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 <math>r\left( t \right)<math>, the torsion at a given value of <math>t<math> is
<math>\tau = {{\det \left( {r',r'',r'''} \right)} \over {\left\| {r' \times r''} \right\|}} = {{r'\left( {r'' \times r'''} \right)} \over {\left\| {r' \times r''} \right\|}}<math>
Torsion of connection
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
- <math>T(u,v)=\nabla_u v - \nabla_v u -[u,v]<math>
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.
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 all its elements have infinite 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 Hi(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.
See also:
Torsion is also sometimes used for the medical condition commonly known as bloat.
