Torsion-free
Encyclopedia
In mathematics
, the term torsion-free may refer to several unrelated notions:
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the term torsion-free may refer to several unrelated notions:
- In abstract algebra, a group is torsion-free if the only element of finite order is the identity.
- In abstract algebra, a module is torsion-free if it has no nonzero torsion elements.
- In differential geometry, an affine connectionAffine connectionIn the branch of mathematics called differential geometry, an affine connection is a geometrical object on a smooth manifold which connects nearby tangent spaces, and so permits tangent vector fields to be differentiated as if they were functions on the manifold with values in a fixed vector space...
is torsion-free if its torsion tensorTorsion tensorIn differential geometry, the notion of torsion is a manner of characterizing a twist or screw of a moving frame around a curve. The torsion of a curve, as it appears in the Frenet-Serret formulas, for instance, quantifies the twist of a curve about its tangent vector as the curve evolves In the...
vanishes.