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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 — together with 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

- $T(X,Y)\; =\; nabla\_XY-nabla\_YX\; -\; [X,Y]$

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.

- $T(X,\; Y)\; :=\; nabla\_X\; Y\; -\; nabla\_Y\; X\; -\; [X,Y]$

where [X,Y] 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 non-tensorial covariant derivative: it gives a 2-form on tangent vectors, while the covariant derivative is only defined for vector fields.

- $R(X,Y)Z\; =\; nabla\_Xnabla\_YZ\; -\; nabla\_Ynabla\_XZ\; -\; nabla\_\{[X,Y]\}Z.$

The Bianchi identities relate the curvature and torsion as follows. Let $mathfrak\{S\}$ denote the cyclic sum over X, Y, and Z. For instance,

- $mathfrak\{S\}left(R(X,Y)Zright)\; :=\; R(X,Y)Z\; +\; R(Y,Z)X\; +\; R(Z,X)Y.$

1. Bianchi's first identity:

- $mathfrak\{S\}left(R(X,Y)Zright)\; =\; mathfrak\{S\}left(T(T(X,Y),Z)+(nabla\_XT)(Y,Z)right)$

2. Bianchi's second identity:

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

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

If the basis is holonomic then the Lie brackets vanish, $gamma^k\{\}\_\{ij\}=0$. So $T^k\{\}\_\{ij\}=2Gamma^k\{\}\_\{[ij]\}$. In particular (see below) while the geodesic equations determine the symmetric part of the connection, the torsion tensor determines the antisymmetric part.

- $theta(X)\; =\; u^\{-1\}(dpi(X))$

- $Theta\; =\; dtheta\; +\; omegawedgetheta.$

The torsion form is a (horizontal) tensorial form with values in R^{n}, meaning that under the right action of g ∈ Gl(n) it transforms equivariantly:

- $R\_g^*Theta\; =\; g^\{-1\}cdotTheta$

- $Omega\; =\; Domega\; =\; domega\; +\; omegawedgeomega$

- $DTheta\; =\; Omegawedgetheta$
- $DOmega\; =\; 0.,$

Moreover, one can recover the curvature and torsion tensors from the curvature and torsion forms as follows. At a point u of F_{x}M, one has

- $R(X,Y)Z\; =\; uleft(2Omega(pi^\{-1\}(X),pi^\{-1\}(Y))right)(u^\{-1\}(Z)),$

- $T(X,Y)\; =\; uleft(2Theta(pi^\{-1\}(X),pi^\{-1\}(Y))right),$

- $D\{mathbf\; e\}\_i\; =\; sum\_\{j=1\}^n\; \{mathbf\; e\}\_jomega\_i^j.$

- $Theta^k\; =\; dtheta^k\; +\; sum\_\{j=1\}^nomega^k\_jwedgetheta^j\; =\; sum\_\{i,j\}T\_\{ij\}^k\; theta^iwedgetheta^j.$

In the rightmost expression,

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

It can be easily shown that Θ^{i} transforms tensorially in the sense that if a different frame

- $tilde\{mathbf\; e\}\_i\; =\; sum\_j\; mathbf\; e\_j\; g\_i^j$

- $tilde\{Theta\}^i\; =\; (g^\{-1\})^i\_jTheta^j.$

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. Then the torsion two-form is a section of

- $Thetaintext\{Hom\}(wedge^2\; TM,\; TM)$

- $Theta\; =\; Dtheta,,$

- $a\_i\; =\; T^k\_\{ik\},$

- $B^i\_\{jk\}\; =\; T^i\_\{jk\}\; +\; frac\{1\}\{n-1\}delta^i\_ja\_k-frac\{1\}\{n-1\}delta^i\_ka\_j$

Intrinsically, one has

- $Tin\; operatorname\{Hom\}left(wedge^2\; TM,\; TMright).$

- $T(X)\; :\; Y\; mapsto\; T(Xwedge\; Y).$

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

The trace-free part of T is then

- $T\_0\; =\; T\; -\; frac\{1\}\{n-1\}iota(operatorname\{tr\}\; ,T)$

As a consequence of the Bianchi identity, the one-form tr T is a closed one-form:

- $d(operatorname\{tr\},\; T)\; =\; 0$

where d is the exterior derivative.

- $dot\{C\}\_t\; =\; tau\_t^0dot\{x\}\_t,quad\; C\_0\; =\; 0$

- $tau\_t^0\; :\; T\_\{x\_t\}M\; to\; T\_\{x\_0\}M$

In particular, if x_{t} is a closed loop, then C_{t} may or may not also be closed depending on the torsion of the connection. Thus the torsion is interpreted as a screw dislocation of the development of a curve. In this way, the torsion is associated with a translational component to the holonomy of the connection. The companion notion of curvature represents an infinitesimal linear transformation (or a rotation in the case of a Riemannian connection.)

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 she experiences no acceleration. Suppose that in addition the observer carries with herself 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 the 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

- $left.nabla\_\{partial/partial\; tau\}frac\{partial\}\{partial\; x\}right|\_\{x=0\}\; =\; 0.$

Consequently, the torsion is given by

- $left.Tleft(frac\{partial\}\{partial\; x\},frac\{partial\}\{partial\; tau\}right)right|\_\{x=0\}\; =\; left.nabla\_\{frac\{partial\}\{partial\; x\}\}frac\{partial\}\{partial\; tau\}right|\_\{x=0\}.$

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.

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

- $nabla\_\{dot\{gamma\}(t)\}dot\{gamma\}(t)\; =\; 0$

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 geodesics (i.e., the same geodesic spray) differ only by torsion.

More precisely, if X and Y are a pair of tangent vectors at p ∈ M, then let

- $Delta(X,Y)=nabla\_Xtilde\{Y\}-nabla\text{'}\_Xtilde\{Y\}$

- $S(X,Y)=tfrac12left(Delta(X,Y)+Delta(Y,X)right)$

- $A(X,Y)=tfrac12left(Delta(X,Y)-Delta(Y,X)right)$

- $A(X,Y)\; =\; tfrac12left(T(X,Y)\; -\; T\text{'}(X,Y)right)$ is the difference of the torsion tensors.
- ∇ and ∇′ define the same families of affinely parametrized geodesics if and only if S(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.

This is a generalization of the fundamental theorem of Riemannian geometry to general affine (possibly non-metric) connections. Picking out the unique 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.

- Cartan, Elie (1923). "Sur les variétés à connexion affine, et la théorie de la relativité généralisée (première partie)".
*Annales Scientifiques de l'École Normale Supérieure*40 325–412. - Cartan, Elie (1924). "Sur les variétés à connexion affine, et la théorie de la relativité généralisée (première partie) (Suite)".
*Annales Scientifiques de l'École Normale Supérieure*41 1-25.

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Last updated on Sunday August 10, 2008 at 18:23:35 PDT (GMT -0700)

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