Torsion 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.
Let M be a manifold with an affine connection on the tangent bundle (aka covariant derivative) ∇.
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.
in terms of a local basis (e1, ..., en) of sections of the tangent bundle can be derived by setting X = ei, Y = ej and by introducing the commutator coefficients γkijek := [ei, ej].
The frame bundle also carries a canonical one-form θ, with values in Rn, defined at a frame u ∈ FxM (regarded as a linear function u : Rn → TxM) by[3] where π : FM → M is the projection mapping for the principal bundle and π∗ is its push-forward.
The torsion form is then[4] Equivalently, Θ = Dθ, where D is the exterior covariant derivative determined by the connection.
The connection form expresses the exterior covariant derivative of these basic sections:[5] The solder form for the tangent bundle (relative to this frame) is the dual basis θi ∈ T∗M of the ei, so that θi(ej) = δij (the Kronecker delta).
It can be easily shown that Θi transforms tensorially in the sense that if a different frame for some invertible matrix-valued function (gji), then In other terms, Θ is a tensor of type (1, 2) (carrying one contravariant and two covariant indices).
Alternatively, the solder form can be characterized in a frame-independent fashion as the TM-valued one-form θ on M corresponding to the identity endomorphism of the tangent bundle under the duality isomorphism End(TM) ≈ TM ⊗ T∗M.
Using the index notation, the trace of T is given by and the trace-free part is where δij is the Kronecker delta.
For instance, Then the following identities hold The curvature form is the gl(n)-valued 2-form where, again, D denotes the exterior covariant derivative.
At a point u of FxM, one has[8] where again u : Rn → TxM 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).
The torsion is a manner of characterizing the amount of slipping or twisting that a plane does when rolling along a surface or higher dimensional affine manifold.
It turns out that the plane will have rotated (despite there being no twist whilst rolling it), an effect due to the curvature of the sphere.
The torsion is a way to quantify this additional slipping and twisting while rolling a plane along a curve.
Thus the torsion tensor can be intuitively understood by taking a small parallelogram circuit with sides given by vectors v and w, in a space and rolling the tangent space along each of the four sides of the parallelogram, marking the point of contact as it goes.
When the circuit is completed, the marked curve will have been displaced out of the plane of the parallelogram by a vector, denoted
The companion notion of curvature measures how moving frames roll along a curve without slipping or twisting.
On it, we put a connection that is flat, but with non-zero torsion, defined on the standard Euclidean frame
[11] The foregoing considerations can be made more quantitative by considering a small parallelogram, originating at the point
The development of this parallelogram, using the connection, is no longer closed in general, and the displacement in going around the loop is translation by the vector
In materials science, and especially elasticity theory, ideas of torsion also play an important role.
In its energy-minimizing state, the vine naturally grows in the shape of a helix.
In fluid dynamics, torsion is naturally associated to vortex lines.
These are the equations satisfied by an equilibrium continuous medium with moment density
(Here the dot denotes differentiation with respect to t, which associates with γ the tangent vector pointing along it.)
Each geodesic is uniquely determined by its initial tangent vector at time t = 0,
Torsion is the ambiguity of classifying connections in terms of their geodesic sprays: More precisely, if X and Y are a pair of tangent vectors at p ∈ M, then let be the difference of the two connections, calculated in terms of arbitrary extensions of X and Y away from p. By the Leibniz product rule, one sees that Δ does not actually depend on how X and Y′ are extended (so it defines a tensor on M).
Picking out the unique torsion-free connection subordinate to a family of parametrized geodesics is known as absorption of torsion, and it is one of the stages of Cartan's equivalence method.
