In 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 (or rather the rotation of the Frenet–Serret frame about the tangent vector.) In the geometry of surfaces, the geodesic torsion describes how a surface twists about a curve on the surface. The companion notion of curvature measures how moving frames "roll" along a curve "without twisting."
More generally, on a differentiable manifold equipped with an affine connection (that is, a connection in the tangent bundle), torsion and curvature form the two fundamental invariants of the connection. In this context, torsion gives an intrinsic characterization of how tangent spaces twist about a curve when they are parallel transported; whereas curvature describes how the tangent spaces roll along the curve. Torsion may be described concretely as a tensor, or as a vector-valued two-form on the manifold. If ∇ is an affine connection on a differential manifold, then the torsion tensor is defined, in terms of vector fields X and Y, by
where is the Lie bracket of vector fields.
Torsion is particularly useful in the study of the geometry of geodesics. Given a system of parametrized geodesics, one can specify a class of affine connections having those geodesics, but differing by their torsions. There is a unique connection which absorbs the torsion, generalizing the Levi-Civita connection to other, possibly non-metric situations (such as Finsler geometry). Absorption of torsion also plays a fundamental role in the study of G-structures and Cartan's equivalence method. Torsion is also useful in the study of unparametrized families of geodesics, via the associated projective connection. In relativity theory, such ideas have been implemented in the form of Einstein–Cartan theory.
Other articles related to "torsion tensor, torsion, tensor":
... One application of the torsion of a connection involves the geodesic spray of the connection roughly the family of all affinely parametrized geodesics ... Torsion is the ambiguity of classifying connections in terms of their geodesic sprays Two connections ∇ and ∇′ which have the same affinely parametrized ... the same geodesic spray) differ only by torsion ...
... that action with respect to the metric tensor ... We use the convention that Lagrangian densities are tensor densities ... metric yields equations reminiscent of general relativity where is the Ricci tensor and is the canonical energy-momentum tensor ...