Components of the torsion tensor

The components of the torsion tensor ${\displaystyle T^{c}{}_{ab}}$ in terms of a local basis (e₁, ..., e_n) of sections of the tangent bundle can be derived by setting X = e_i, Y = e_j and by introducing the commutator coefficients γ^k_ije_k := [e_i, e_j]. The components of the torsion are then

${\displaystyle T^{k}{}_{ij}:=\Gamma ^{k}{}_{ij}-\Gamma ^{k}{}_{ji}-\gamma ^{k}{}_{ij},\quad i,j,k=1,2,\ldots ,n.}$

Here ${\displaystyle {\Gamma ^{k}}_{ij}}$ are the connection coefficients defining the connection. If the basis is holonomic then the Lie brackets vanish, ${\displaystyle \gamma ^{k}{}_{ij}=0}$. So ${\displaystyle T^{k}{}_{ij}=2\Gamma ^{k}{}_{[ij]}}$. 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 u ∈ F_xM (regarded as a linear function u : Rⁿ → T_xM) by

${\displaystyle \theta (X)=u^{-1}(\pi _{*}(X))}$

where π  : FM → M is the projection mapping for the principal bundle and π∗ is its push-forward. The torsion form is then

${\displaystyle \Theta =d\theta +\omega \wedge \theta .}$

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 g ∈ GL(n) it transforms equivariantly:

${\displaystyle R_{g}^{*}\Theta =g^{-1}\cdot \Theta }$

where g acts on the right-hand side through its adjoint representation on Rⁿ.

Irreducible decomposition

The torsion tensor can be decomposed into two irreducible parts: a trace-free part and another part which contains the trace terms. Using the index notation, the trace of T is given by

${\displaystyle a_{i}=T^{k}{}_{ik},}$

and the trace-free part is

${\displaystyle B^{i}{}_{jk}=T^{i}{}_{jk}+{\frac {1}{n-1}}\delta ^{i}{}_{j}a_{k}-{\frac {1}{n-1}}\delta ^{i}{}_{k}a_{j},}$

where δⁱ_j is the Kronecker delta.

Intrinsically, one has

${\displaystyle T\in \operatorname {Hom} \left({\textstyle \bigwedge }^{2}{\rm {T}}M,{\rm {T}}M\right).}$

The trace of T, tr T, is an element of T^∗M defined as follows. For each vector fixed X ∈ TM, T defines an element T(X) of Hom(TM, TM) via

${\displaystyle T(X):Y\mapsto T(X\wedge Y).}$

Then (tr T)(X) is defined as the trace of this endomorphism. That is,

${\displaystyle (\operatorname {tr} \,T)(X){\stackrel {\text{def}}{=}}\operatorname {tr} (T(X)).}$

The trace-free part of T is then

${\displaystyle T_{0}=T-{\frac {1}{n-1}}\iota (\operatorname {tr} \,T),}$

where ι denotes the interior product.

The curvature form and Bianchi identities

The curvature form is the gl(n)-valued 2-form

${\displaystyle \Omega =D\omega =d\omega +\omega \wedge \omega }$

where, again, D denotes the exterior covariant derivative. In terms of the curvature form and torsion form, the corresponding Bianchi identities are^[2]

1. ${\displaystyle D\Theta =\Omega \wedge \theta }$
2. ${\displaystyle D\Omega =0.}$

Moreover, one can recover the curvature and torsion tensors from the curvature and torsion forms as follows. At a point u of F_xM, one has^[3]

${\displaystyle {\begin{aligned}R(X,Y)Z&=u\left(2\Omega \left(\pi ^{-1}(X),\pi ^{-1}(Y)\right)\right)\left(u^{-1}(Z)\right),\\T(X,Y)&=u\left(2\Theta \left(\pi ^{-1}(X),\pi ^{-1}(Y)\right)\right),\end{aligned}}}$

where again u : Rⁿ → T_xM is the function specifying the frame in the fibre, and the choice of lift of the vectors via π⁻¹ is irrelevant since the curvature and torsion forms are horizontal (they vanish on the ambiguous vertical vectors).

Twisting of reference frames

In the classical differential geometry of curves, the Frenet-Serret formulas describe how a particular moving frame (the Frenet-Serret frame) twists along a curve. In physical terms, the torsion corresponds to the angular momentum of an idealized top pointing along the tangent of the curve.

The case of a manifold with a (metric) connection admits an analogous interpretation. Suppose that an observer is moving along a geodesic for the connection. Such an observer is ordinarily thought of as inertial since they experience no acceleration. Suppose that in addition the observer carries with themselves a system of rigid straight measuring rods (a coordinate system). Each rod is a straight segment; a geodesic. Assume that each rod is parallel transported along the trajectory. The fact that these rods are physically carried along the trajectory means that they are Lie-dragged, or propagated so that the Lie derivative of each rod along the tangent vanishes. They may, however, experience torque (or torsional forces) analogous to the torque felt by the top in the Frenet-Serret frame. This force is measured by the torsion.

More precisely, suppose that the observer moves along a geodesic path γ(t) and carries a measuring rod along it. The rod sweeps out a surface as the observer travels along the path. There are natural coordinates (t, x) along this surface, where t is the parameter time taken by the observer, and x is the position along the measuring rod. The condition that the tangent of the rod should be parallel translated along the curve is

${\displaystyle \left.\nabla _{\frac {\partial }{\partial t}}{\frac {\partial }{\partial x}}\right|_{x=0}=0.}$

Consequently, the torsion is given by

${\displaystyle \left.T\left({\frac {\partial }{\partial x}},{\frac {\partial }{\partial t}}\right)\right|_{x=0}=\left.\nabla _{\frac {\partial }{\partial x}}{\frac {\partial }{\partial t}}\right|_{x=0}.}$

If this is not zero, then the marked points on the rod (the x = constant curves) will trace out helices instead of geodesics. They will tend to rotate around the observer. Note that for this argument it was not essential that ${\displaystyle \gamma (t)}$ is a geodesic. Any curve would work.

This interpretation of torsion plays a role in the theory of teleparallelism, also known as Einstein–Cartan theory, an alternative formulation of relativity theory.

The torsion of a filament

In materials science, and especially elasticity theory, ideas of torsion also play an important role. One problem models the growth of vines, focusing on the question of how vines manage to twist around objects.^[4] The vine itself is modeled as a pair of elastic filaments twisted around one another. In its energy-minimizing state, the vine naturally grows in the shape of a helix. But the vine may also be stretched out to maximize its extent (or length). In this case, the torsion of the vine is related to the torsion of the pair of filaments (or equivalently the surface torsion of the ribbon connecting the filaments), and it reflects the difference between the length-maximizing (geodesic) configuration of the vine and its energy-minimizing configuration.

Torsion and vorticity

In fluid dynamics, torsion is naturally associated to vortex lines.