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 2-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 [X,Y] 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). The difference between a connection with torsion, and a corresponding connection without torsion is a tensor, called the contorsion tensor. 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.
The torsion tensor
Let M be a manifold with an affine connection on the tangent bundle (aka covariant derivative) ∇. The torsion tensor (sometimes called the Cartan (torsion) tensor) of ∇ is the vector-valued 2-form defined on vector fields X and Y by
where is the Lie bracket of two vector fields. By the Leibniz rule, T(fX, Y) = T(X, fY) = fT(X, Y) for any smooth function f. So T is tensorial, despite being defined in terms of the connection which is a first order differential operator: it gives a 2-form on tangent vectors, while the covariant derivative is only defined for vector fields.
Components of the torsion tensor
The components of the torsion tensor in terms of a local basis of sections of the tangent bundle can be derived by setting , and by introducing the commutator coefficients . The components of the torsion are then
Here are the connection coefficients defining the connection. If the basis is holonomic then the Lie brackets vanish, . So . In particular (see below), while the geodesic equations determine the symmetric part of the connection, the torsion tensor determines the antisymmetric part.
The torsion form
The torsion form, an alternative characterization of torsion, applies to the frame bundle FM of the manifold M. This principal bundle is equipped with a connection form ω, a gl(n)-valued one-form which maps vertical vectors to the generators of the right action in gl(n) and equivariantly intertwines the right action of GL(n) on the tangent bundle of FM with the adjoint representation on gl(n). The frame bundle also carries a canonical one-form θ, with values in Rn, defined at a frame (regarded as a linear function ) by
where is the projection mapping for the principal bundle and is its push-forward. The torsion form is then
Equivalently, Θ = Dθ, where D is the exterior covariant derivative determined by the connection.
The torsion form is a (horizontal) tensorial form with values in Rn, meaning that under the right action of it transforms equivariantly:
where g acts on the right-hand side through its adjoint representation on Rn.
Torsion form in a frame
The torsion form may be expressed in terms of a connection form on the base manifold M, written in a particular frame of the tangent bundle . The connection form expresses the exterior covariant derivative of these basic sections:
The solder form for the tangent bundle (relative to this frame) is the dual basis of the ei, so that (the Kronecker delta). Then the torsion 2-form has components
In the rightmost expression,