In differential geometry, the torsion tensor is a tensor that is associated to any affine connection. The torsion tensor is bilinear map of two input vectors , that produces an output vector representing the displacement within a tangent space when the tangent space is developed (or "rolled") along an infinitesimal parallelogram whose sides are . It is skew symmetric in its inputs, because developing over the parallelogram in the opposite sense produces the opposite displacement, similarly to how a screw moves in opposite ways when it is twisted in two directions.
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 LeviCivita connection to other, possibly nonmetric 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 Gstructures 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.
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Tensor Calculus 20: The Abstract Covariant Derivative (LeviCivita Connection)
Transcription
Definition
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 vectorvalued 2form defined on vector fields X and Y by^{[1]}
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 connection which is a first order differential operator: it gives a 2form 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 (e_{1}, ..., 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}_{ij}e_{k} := [e_{i}, e_{j}]. The components of the torsion are then^{[2]}
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 oneform 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 oneform θ, with values in R^{n}, defined at a frame u ∈ F_{x}M (regarded as a linear function u : R^{n} → T_{x}M) by^{[3]}
where π : FM → M is the projection mapping for the principal bundle and π∗ is its pushforward. The torsion form is then^{[4]}
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 R^{n}, meaning that under the right action of g ∈ GL(n) it transforms equivariantly:
where g acts on the righthand side through its adjoint representation on R^{n}.
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 (e_{1}, ..., e_{n}). 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 θ^{i} ∈ T^{∗}M of the e_{i}, so that θ^{i}(e_{j}) = δ^{i}_{j} (the Kronecker delta). Then the torsion 2form has components
In the rightmost expression,
are the framecomponents of the torsion tensor, as given in the previous definition.
It can be easily shown that Θ^{i} transforms tensorially in the sense that if a different frame
for some invertible matrixvalued function (g^{j}_{i}), 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 frameindependent fashion as the TMvalued oneform θ on M corresponding to the identity endomorphism of the tangent bundle under the duality isomorphism End(TM) ≈ TM ⊗ T^{∗}M. Then the torsion 2form is a section
given by
where D is the exterior covariant derivative. (See connection form for further details.)
Irreducible decomposition
The torsion tensor can be decomposed into two irreducible parts: a tracefree part and another part which contains the trace terms. Using the index notation, the trace of T is given by
and the tracefree part is
where δ^{i}_{j} is the Kronecker delta.
Intrinsically, one has
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
Then (tr T)(X) is defined as the trace of this endomorphism. That is,
The tracefree part of T is then
where ι denotes the interior product.
Curvature and the Bianchi identities
The curvature tensor of ∇ is a mapping TM × TM → End(TM) defined on vector fields X, Y, and Z by
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).
The Bianchi identities relate the curvature and torsion as follows.^{[6]} Let denote the cyclic sum over X, Y, and Z. For instance,
Then the following identities hold
 Bianchi's first identity:
 Bianchi's second identity:
The curvature form and Bianchi identities
The curvature form is the gl(n)valued 2form
where, again, D denotes the exterior covariant derivative. In terms of the curvature form and torsion form, the corresponding Bianchi identities are^{[7]}
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^{[8]}
where again u : R^{n} → T_{x}M 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).
Characterizations and interpretations
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.^{[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 the curvature 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 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 . Thus the torsion tensor is a tensor: a (bilinear) function of two input vectors v and w that produces an output vector . It is skew symmetric in the arguments v and w, a reflection of the fact that traversing the circuit in the opposite sense undoes the original displacement, in much the same way that twisting a screw in opposite directions displaces the screw in opposite ways. The torsion tensor thus is related to, although distinct from, the torsion of a curve, as it appears in the Frenet–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 its osculating plane. 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 slipping or twisting.
Example
Consider the (flat) Euclidean space . On it, we put a connection that is flat, but with nonzero torsion, defined on the standard Euclidean frame by the (Euclidean) cross product:
Now the tip of the vector , as it is transported along the axis traces out the helix
Development
One interpretation of the torsion involves the development of a curve.^{[10]} Suppose that a piecewise smooth closed loop is given, based at the point , where . We assume that is homotopic to zero. The curve can be developed into the tangent space at in the following manner. Let be a parallel coframe along , and let be the coordinates on induced by . A development of is a curve in whose coordinates sastify the differential equation
The foregoing considerations can be made more quantitative by considering a small parallelogram, originating at the point , with sides . Then the tangent bivector to the parallelogram is . 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 , where is the torsion tensor, up to higher order terms in . This displacement is directly analogous to the Burgers vector of crystallography.^{[12]}^{[13]}
More generally, one can also transport a moving frame along the curve . The linear transformation that the frame undergoes between is then determined by the curvature of the connection. Together, the linear transformation of the frame and the translation of the starting point from to comprise the holonomy of the connection.
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.^{[14]} The vine itself is modeled as a pair of elastic filaments twisted around one another. In its energyminimizing 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 lengthmaximizing (geodesic) configuration of the vine and its energyminimizing configuration.
Torsion and vorticity
In fluid dynamics, torsion is naturally associated to vortex lines.
Suppose that a connection is given in three dimensions, with curvature 2form and torsion 2form . Let be the skewsymmetric LeviCivita tensor, and
Geodesics and the absorption of torsion
Suppose that γ(t) is a curve on M. Then γ is an affinely parametrized geodesic provided that
for all time t in the domain of γ. (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, .
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.^{[16]}
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). Let S and A be the symmetric and alternating parts of Δ:
Then
 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 torsionfree connection ∇′ with the same family of affinely parametrized geodesics. The difference between these two connections is in fact a tensor, the contorsion tensor.
This is a generalization of the fundamental theorem of Riemannian geometry to general affine (possibly nonmetric) connections. Picking out the unique torsionfree 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.
See also
 Contorsion tensor
 Curtright field
 Curvature tensor
 LeviCivita connection
 Torsion coefficient
 Torsion of curves
Notes
 ^ Kobayashi & Nomizu (1963), Chapter III, Theorem 5.1
 ^ Kobayashi & Nomizu (1963), Chapter III, Proposition 7.6
 ^ Kobayashi & Nomizu (1963), Chapter III, Section 2
 ^ Kobayashi & Nomizu (1963), Chapter III, Theorem 2.4
 ^ Kobayashi & Nomizu (1963), Chapter III, Section 7
 ^ Kobayashi & Nomizu 1963, Volume 1, Proposition III.5.2.
 ^ Kobayashi & Nomizu 1963, Volume 1, III.2.
 ^ Kobayashi & Nomizu 1963, Volume 1, III.5.
 ^ Hehl, F. W., & Obukhov, Y. N. (2007). Elie Cartan's torsion in geometry and in field theory, an essay. arXiv preprint arXiv:0711.1535.
 ^ Kobayashi & Nomizu (1963), Chapter III, Section 4
 ^ Bilby, B. A., Bullough, R., & Smith, E. (1955). Continuous distributions of dislocations: a new application of the methods of nonRiemannian geometry. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 231(1185), 263273.
 ^ "Torsion", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
 ^ Ozakin, A., & Yavari, A. (2014). Affine development of closed curves in Weitzenböck manifolds and the Burgers vector of dislocation mechanics. Mathematics and Mechanics of Solids, 19(3), 299307.
 ^ Goriely et al. 2006.
 ^ Trautman (1980) Comments on the paper by Elie Cartan: Sur une generalisation de la notion de courbure de Riemann et les espaces a torsion. In Bergmann, P. G., & De Sabbata, V. Cosmology and Gravitation: Spin, Torsion, Rotation, and Supergravity (Vol. 58). Springer Science & Business Media.
 ^ See Spivak (1999) Volume II, Addendum 1 to Chapter 6. See also Bishop and Goldberg (1980), section 5.10.
References
 Bishop, R.L.; Goldberg, S.I. (1980), Tensor analysis on manifolds, Dover Publications
 Cartan, É. (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, doi:10.24033/asens.751
 Cartan, É. (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, doi:10.24033/asens.753
 Elzanowski, M.; Epstein, M. (1985), "Geometric characterization of hyperelastic uniformity", Archive for Rational Mechanics and Analysis, 88 (4): 347–357, Bibcode:1985ArRMA..88..347E, doi:10.1007/BF00250871, S2CID 120127682
 Goriely, A.; RobertsonTessi, M.; Tabor, M.; Vandiver, R. (2006), "Elastic growth models" (PDF), BIOMAT2006, SpringerVerlag, archived from the original (PDF) on 20061229
 Hehl, F.W.; von der Heyde, P.; Kerlick, G.D.; Nester, J.M. (1976), "General relativity with spin and torsion: Foundations and prospects", Rev. Mod. Phys., 48 (3): 393–416, Bibcode:1976RvMP...48..393H, doi:10.1103/revmodphys.48.393, 393.
 Kibble, T.W.B. (1961), "Lorentz invariance and the gravitational field", J. Math. Phys., 2 (2): 212–221, Bibcode:1961JMP.....2..212K, doi:10.1063/1.1703702, 212.
 Kobayashi, S.; Nomizu, K. (1963), Foundations of Differential Geometry, vol. 1 & 2 (New ed.), WileyInterscience (published 1996), ISBN 0471157333
 Poplawski, N.J. (2009), Spacetime and fields, arXiv:0911.0334, Bibcode:2009arXiv0911.0334P
 Schouten, J.A. (1954), Ricci Calculus, SpringerVerlag
 Schrödinger, E. (1950), SpaceTime Structure, Cambridge University Press
 Sciama, D.W. (1964), "The physical structure of general relativity", Rev. Mod. Phys., 36 (1): 463, Bibcode:1964RvMP...36..463S, doi:10.1103/RevModPhys.36.463
 Spivak, M. (1999), A comprehensive introduction to differential geometry, Volume II, Houston, Texas: Publish or Perish, ISBN 0914098713
External links
 Bill Thurston (2011) Rolling without slipping interpretation of torsion, URL (version: 20110127).