Indifferential geometry, thetorsion tensor is atensor that is associated to anyaffine connection. The torsion tensor is abilinear map of two input vectors${\displaystyle X,Y}$, that produces an output vector${\displaystyle T(X,Y)}$ representing the displacement within a tangent space when the tangent space is developed (or "rolled") along an infinitesimal parallelogram whose sides are${\displaystyle X,Y}$. It isskew symmetric in its inputs, because developing over the parallelogram in the opposite sense produces the opposite displacement, similarly to how ascrew moves in opposite ways when it is twisted in two directions.

Torsion is particularly useful in the study of the geometry ofgeodesics. 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 whichabsorbs the torsion, generalizing theLevi-Civita connection to other, possibly non-metric situations (such asFinsler geometry). The difference between a connection with torsion, and a corresponding connection without torsion is a tensor, called thecontorsion tensor. Absorption of torsion also plays a fundamental role in the study ofG-structures andCartan's equivalence method. Torsion is also useful in the study of unparametrized families of geodesics, via the associatedprojective connection. Inrelativity theory, such ideas have been implemented in the form ofEinstein–Cartan theory.

LetM be a manifold with anaffine connection on thetangent bundle (akacovariant derivative) ∇. Thetorsion tensor (sometimes called theCartan (torsion)tensor) of ∇ is thevector-valued 2-form defined onvector fieldsX andY by^[1]

${\displaystyle T(X,Y):={\mathrm{\nabla }}_{X}Y-{\mathrm{\nabla }}_{Y}X-[X,Y]}$

where[X,Y] is theLie bracket of two vector fields. By theLeibniz rule,T(fX,Y) =T(X,fY) =fT(X,Y) for anysmooth functionf. SoT istensorial, despite being defined in terms of theconnection 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.

The components of the torsion tensor${\displaystyle T^c{}_{ab}}$ in terms of a localbasis(e₁, ...,e_n) ofsections of the tangent bundle can be derived by settingX =e_i,Y =e_j and by introducing the commutator coefficientsγ^k_ije_k := [e_i,e_j]. The components of the torsion are then^[2]

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

Here${\displaystyle {{\mathrm{\Gamma }}^k}_{ij}}$ are theconnection coefficients defining the connection. If the basis isholonomic then the Lie brackets vanish,${\displaystyle {\gamma }^k{}_{ij}=0}$. So${\displaystyle T^k{}_{ij}=2{\mathrm{\Gamma }}^k{}_{[ij]}}$. In particular (see below), while thegeodesic equations determine the symmetric part of the connection, the torsion tensor determines the antisymmetric part.

Thetorsion form, an alternative characterization of torsion, applies to theframe bundle FM of the manifoldM. Thisprincipal bundle is equipped with aconnection formω, agl(n)-valued one-form which maps vertical vectors to the generators of the right action ingl(n) and equivariantly intertwines the right action of GL(n) on the tangent bundle of FM with theadjoint representation ongl(n). The frame bundle also carries acanonical one-form θ, with values inRⁿ, defined at a frameu ∈ F_xM (regarded as a linear functionu :Rⁿ → T_xM) by^[3]

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

whereπ  : FM →M is the projection mapping for the principal bundle andπ∗ is its push-forward. The torsion form is then^[4]

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

Equivalently, Θ =Dθ, whereD is theexterior covariant derivative determined by the connection.

The torsion form is a (horizontal)tensorial form with values inRⁿ, meaning that under the right action ofg ∈ GL(n) it transformsequivariantly:

${\displaystyle R_g^{\ast }\mathrm{\Theta }=g^{-1}\cdot \mathrm{\Theta }}$

where${\displaystyle g^{-1}}$ acts on the right-hand side by its canonical action onRⁿ.

The torsion form may be expressed in terms of aconnection form on the base manifoldM, written in a particular frame of the tangent bundle(e₁, ...,e_n). The connection form expresses the exterior covariant derivative of these basic sections:^[5]

${\displaystyle D{\mathbf{e}}_i={\mathbf{e}}_j{{\omega }^j}_i.}$

Thesolder form for the tangent bundle (relative to this frame) is thedual basisθⁱ ∈ T^∗M of thee_i, so thatθⁱ(e_j) =δⁱ_j (theKronecker delta). Then the torsion 2-form has components

${\displaystyle {\mathrm{\Theta }}^k=d{\theta }^k+{{\omega }^k}_j\wedge {\theta }^j={T^k}_{ij}{\theta }^i\wedge {\theta }^j.}$

In the rightmost expression,

${\displaystyle {T^k}_{ij}={\theta }^k\left({\mathrm{\nabla }}_{{\mathbf{e}}_i}{\mathbf{e}}_j-{\mathrm{\nabla }}_{{\mathbf{e}}_j}{\mathbf{e}}_i-\left[{\mathbf{e}}_i,{\mathbf{e}}_j\right]\right)}$

are the frame-components of the torsion tensor, as given in the previous definition.

It can be easily shown that Θⁱ transforms tensorially in the sense that if a different frame

${\displaystyle {\tilde{\mathbf{e}}}_i={\mathbf{e}}_j{g^j}_i}$

for some invertible matrix-valued function (g^j_i), then

${\displaystyle {\tilde{\mathrm{\Theta }}}^i={{\left(g^{-1}\right)}^i}_j{\mathrm{\Theta }}^j.}$

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-formθ onM corresponding to the identity endomorphism of the tangent bundle under the duality isomorphismEnd(TM) ≈ TM ⊗ T^∗M. Then the torsion 2-form is a section

${\displaystyle \mathrm{\Theta }\in \text{Hom}\left({\bigwedge }^2\mathrm{T}M,\mathrm{T}M\right)}$

given by

${\displaystyle \mathrm{\Theta }=D\theta ,}$

whereD is theexterior covariant derivative. (Seeconnection form for further details.)

The torsion tensor can be decomposed into twoirreducible parts: atrace-free part and another part which contains the trace terms. Using theindex notation, the trace ofT is given by

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

and the trace-free part is

${\displaystyle B^i{}_{jk}=T^i{}_{jk}+\frac{1}{n-1}{\delta }^i{}_j{a}_k-\frac{1}{n-1}{\delta }^i{}_k{a}_j,}$

whereδⁱ_j is theKronecker delta.

Intrinsically, one has

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

The trace ofT, trT, is an element of T^∗M defined as follows. For each vector fixedX ∈ TM,T defines an elementT(X) ofHom(TM, TM) via

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

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

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

The trace-free part ofT is then

${\displaystyle T_0=T-\frac{1}{n-1}\iota (\mathrm{tr}T),}$

whereι denotes theinterior product.

Thecurvature tensor of ∇ is a mappingTM × TM → End(TM) defined on vector fieldsX,Y, andZ by

${\displaystyle R(X,Y)Z={\mathrm{\nabla }}_X{\mathrm{\nabla }}_{Y}Z-{\mathrm{\nabla }}_Y{\mathrm{\nabla }}_{X}Z-{\mathrm{\nabla }}_{[X,Y]}Z.}$

For vectors at a point, this definition is independent of how the vectors are extended to vector fields away from the point (thus it defines a tensor, much like the torsion).

TheBianchi identities relate the curvature and torsion as follows.^[6] Let${\displaystyle \mathfrak{S}}$ denote thecyclic sum overX,Y, andZ. For instance,

${\displaystyle \mathfrak{S}\left(R\left(X,Y\right)Z\right):=R(X,Y)Z+R(Y,Z)X+R(Z,X)Y.}$

Then the following identities hold

1. Bianchi's first identity:

   ${\displaystyle \mathfrak{S}\left(R\left(X,Y\right)Z\right)=\mathfrak{S}\left(T\left(T(X,Y),Z\right)+\left({\mathrm{\nabla }}_{X}T\right)\left(Y,Z\right)\right)}$

2. Bianchi's second identity:

   ${\displaystyle \mathfrak{S}\left(\left({\mathrm{\nabla }}_{X}R\right)\left(Y,Z\right)+R\left(T\left(X,Y\right),Z\right)\right)=0}$

Thecurvature form is thegl(n)-valued 2-form

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

where, again,D denotes the exterior covariant derivative. In terms of the curvature form and torsion form, the corresponding Bianchi identities are^[7]

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

Moreover, one can recover the curvature and torsion tensors from the curvature and torsion forms as follows. At a pointu of F_xM, one has^[8]

where againu :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).

The torsion is a manner of characterizing the amount of slipping or twisting that a plane does when rolling along a surface or higher dimensionalaffine manifold.^[9]

For example, consider rolling a plane along a small circle drawn on a sphere. If the plane does not slip or twist, then when the plane is rolled all the way along the circle, it will also trace a circle in the plane. It turns out that the plane will have rotated (despite there being no twist whilst rolling it), an effect due to thecurvature of the sphere. But the curve traced out will still be a circle, and so in particular a closed curve that begins and ends at the same point. On the other hand, if the plane were rolled along the sphere, but it was allowed it to slip or twist in the process, then the path the circle traces on the plane could be a much more general curve that need not even be closed. 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 vectorsv andw, in a space and rolling thetangent 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${\displaystyle T(v,w)}$. Thus the torsion tensor is a tensor: a (bilinear) function of two input vectorsv andw that produces an output vector${\displaystyle T(v,w)}$. It isskew symmetric in the argumentsv andw, a reflection of the fact that traversing the circuit in the opposite sense undoes the original displacement, in much the same way that twisting ascrew in opposite directions displaces the screw in opposite ways. The torsion tensor thus is related to, although distinct from, thetorsion of a curve, as it appears in theFrenet–Serret formulas: the torsion of a connection measures a dislocation of a developed curve out of its plane, while the torsion of a curve is also a dislocation out of itsosculating plane. In the geometry of surfaces, thegeodesic torsion describes how a surface twists about a curve on the surface. The companion notion ofcurvature measures how moving frames roll along a curve without slipping or twisting.

Consider the (flat)Euclidean space${\displaystyle M={\mathbb{R}}^3}$. On it, we put a connection that is flat, but with non-zero torsion, defined on the standard Euclidean frame${\displaystyle e_1,e_2,e_3}$ by the (Euclidean)cross product:$${\displaystyle {\mathrm{\nabla }}_{e_i}{e}_j=e_i\times e_j.}$$Consider now the parallel transport of the vector${\displaystyle e_2}$ along the${\displaystyle e_1}$ axis, starting at the origin. The parallel vector field${\displaystyle X(x)=a(x)e_2+b(x)e_3}$ thus satisfies${\displaystyle X(0)=e_2}$, and the differential equationThus${\displaystyle \dot{a}=b,\dot{b}=-a}$, and the solution is${\displaystyle X=\cos x{e}_2-\sin x{e}_3}$.

Now the tip of the vector${\displaystyle X}$, as it is transported along the${\displaystyle e_1}$ axis traces out the helix$${\displaystyle x{e}_1+\cos x{e}_2-\sin x{e}_3.}$$Thus we see that, in the presence of torsion, parallel transport tends to twist a frame around the direction of motion, analogously to the role played by torsion in the classicaldifferential geometry of curves.

One interpretation of the torsion involves the development of a curve.^[10] Suppose that a piecewise smooth closed loop${\displaystyle \gamma :[0,1]\to M}$ is given, based at the point${\displaystyle p\in M}$, where${\displaystyle \gamma (0)=\gamma (1)=p}$. We assume that${\displaystyle \gamma }$ is homotopic to zero. The curve can be developed into the tangent space at${\displaystyle p}$ in the following manner. Let${\displaystyle {\theta }^i}$ be a parallel coframe along${\displaystyle \gamma }$, and let${\displaystyle x^i}$ be the coordinates on${\displaystyle T_{p}M}$ induced by${\displaystyle {\theta }^i(p)}$. A development of${\displaystyle \gamma }$ is a curve${\displaystyle \tilde{\gamma }}$ in${\displaystyle T_{p}M}$ whose coordinates${\displaystyle x^i=x^i(t)}$ sastify the differential equation$${\displaystyle d{x}^i={\gamma }^{\ast }{\theta }^i.}$$If the torsion is zero, then the developed curve${\displaystyle \tilde{\gamma }}$ is also a closed loop (so that${\displaystyle \tilde{\gamma }(0)=\tilde{\gamma }(1)}$). On the other hand, if the torsion is non-zero, then the developed curve may not be closed, so that${\displaystyle \tilde{\gamma }(0)\ne \tilde{\gamma }(1)}$. Thus the development of a loop in the presence of torsion can become dislocated, analogously to ascrew dislocation.^[11]

The foregoing considerations can be made more quantitative by considering a small parallelogram, originating at the point${\displaystyle p\in M}$, with sides${\displaystyle v,w\in T_{p}M}$. Then the tangent bivector to the parallelogram is${\displaystyle v\wedge w\in {\mathrm{\Lambda }}^2{T}_{p}M}$. 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${\displaystyle \mathrm{\Theta }(v,w)}$, where${\displaystyle \mathrm{\Theta }}$ is the torsion tensor, up to higher order terms in${\displaystyle v,w}$. This displacement is directly analogous to theBurgers vector of crystallography.^[12][13]

More generally, one can also transport amoving frame along the curve${\displaystyle \tilde{\gamma }}$. Thelinear transformation that the frame undergoes between${\displaystyle t=0,t=1}$ is then determined by the curvature of the connection. Together, the linear transformation of the frame and the translation of the starting point from${\displaystyle \tilde{\gamma }(0)}$ to${\displaystyle \tilde{\gamma }(1)}$ comprise theholonomy of the connection.

Inmaterials science, and especiallyelasticity 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.^[14] 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 ahelix. 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.

Influid dynamics, torsion is naturally associated tovortex lines.

Suppose that a connection${\displaystyle D}$ is given in three dimensions, with curvature 2-form${\displaystyle {\mathrm{\Omega }}_a^b}$ and torsion 2-form${\displaystyle {\mathrm{\Theta }}^a=D{\theta }^a}$. Let${\displaystyle {\eta }_{abc}}$ be the skew-symmetricLevi-Civita tensor, and$${\displaystyle t_a=\frac{1}{2}{\eta }_{abc}\wedge {\mathrm{\Omega }}^{bc},}$$$${\displaystyle s_{ab}=-{\eta }_{abc}\wedge {\mathrm{\Theta }}^c.}$$Then the Bianchi identities (?)The Bianchi identities are$${\displaystyle D{\mathrm{\Omega }}_b^a=0,D{\mathrm{\Theta }}^a={\mathrm{\Omega }}_b^a\wedge {\theta }^b.}$$imply that${\displaystyle D{t}_a=0}$ and$${\displaystyle D{s}_{ab}={\theta }_a\wedge t_b-{\theta }_b\wedge t_a.}$$These are the equations satisfied by an equilibrium continuous medium with moment density${\displaystyle s_{ab}}$.^[15]

Suppose thatγ(t) is a curve onM. Thenγ is anaffinely parametrized geodesic provided that

${\displaystyle {\mathrm{\nabla }}_{\dot{\gamma }(t)}\dot{\gamma }(t)=0}$

for all timet in the domain ofγ. (Here the dot denotes differentiation with respect tot, which associates with γ the tangent vector pointing along it.) Each geodesic is uniquely determined by its initial tangent vector at timet = 0,${\displaystyle \dot{\gamma }(0)}$.

One application of the torsion of a connection involves thegeodesic 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 geodesics (i.e., the same geodesic spray) differ only by torsion.^[16]

More precisely, ifX andY are a pair of tangent vectors atp ∈M, then let

${\displaystyle \mathrm{\Delta }(X,Y)={\mathrm{\nabla }}_X\tilde{Y}-{\mathrm{\nabla }}_X^{\prime }\tilde{Y}}$

be the difference of the two connections, calculated in terms of arbitrary extensions ofX andY away fromp. By theLeibniz product rule, one sees that Δ does not actually depend on howX andY′ are extended (so it defines a tensor onM). LetS andA be the symmetric and alternating parts of Δ:

${\displaystyle S(X,Y)=\frac{1}{2}\left(\mathrm{\Delta }(X,Y)+\mathrm{\Delta }(Y,X)\right)}$

${\displaystyle A(X,Y)=\frac{1}{2}\left(\mathrm{\Delta }(X,Y)-\mathrm{\Delta }(Y,X)\right)}$

Then

• ${\displaystyle A(X,Y)=\frac{1}{2}\left(T(X,Y)-T^{\prime }(X,Y)\right)}$ is the difference of the torsion tensors.
• ∇ and ∇′ define the same families of affinely parametrized geodesics if and only ifS(X,Y) = 0.

In other words, the symmetric part of the difference of two connections determines whether they have the same parametrized geodesics, whereas the skew part of the difference is determined by the relative torsions of the two connections. Another consequence is:

• Given any affine connection ∇, there is a unique torsion-free connection ∇′ with the same family of affinely parametrized geodesics. The difference between these two connections is in fact a tensor, thecontorsion tensor.

This is a generalization of thefundamental theorem of Riemannian geometry to general affine (possibly non-metric) connections. Picking out the unique torsion-free connection subordinate to a family of parametrized geodesics is known asabsorption of torsion, and it is one of the stages ofCartan's equivalence method.

• Contorsion tensor
• Curtright field
• Curvature tensor
• Levi-Civita connection
• Torsion coefficient
• Torsion of curves

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• Differential geometry
• Connection (mathematics)
• Curvature (mathematics)
• Tensors

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