42:
6126:. 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.
4049:
4830:
6122:, 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. 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
4087:
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
3476:
4092:
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
3782:
1945:
4577:
3610:
2553:
7330:
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:
1772:
933:
2652:
2245:
2859:
6511:
3304:
3117:
6952:
4108:
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
3320:
2130:
7196:
7096:
4044:{\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}}}
6635:
2942:
2026:
1783:
7306:
6352:
4071:
is the function specifying the frame in the fibre, and the choice of lift of the vectors via π is irrelevant since the curvature and torsion forms are horizontal (they vanish on the ambiguous vertical vectors).
1587:
7335:
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, the
6768:
5734:
3685:
146:
5672:
4825:{\displaystyle {\begin{array}{rl}0={\dot {X}}&=\nabla _{e_{1}}X={\dot {a}}e_{2}+{\dot {b}}e_{3}+ae_{1}\times e_{2}+be_{1}\times e_{3}\\&=({\dot {a}}-b)e_{2}+({\dot {b}}+a)e_{3}.\end{array}}}
4397:
1357:
6419:
5085:
1458:
3787:
582:
3489:
5861:
2391:
5581:
3730:
1648:
1287:
1101:
4955:
2763:
4889:
6238:
4530:
778:
5217:
405:. 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
210:
6100:
6062:
4093:
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.
5087:
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 classical
1028:
6828:
2380:
5982:
5610:
5446:
4267:
975:
4320:
2283:
3122:
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).
72:
6424:
6195:
3151:
5525:
5397:
2578:
5809:
5141:
5896:
5863:. 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
695:
6271:
5284:
3760:
2175:
4572:
6547:
6024:
5916:
5417:
5304:
5237:
2782:
6837:
of the connection: roughly the family of all affinely parametrized geodesics. Torsion is the ambiguity of classifying connections in terms of their geodesic sprays:
6665:
5767:
5476:
5361:
5167:
4185:
4142:
362:
5331:
5005:
4451:
4424:
273:
233:
166:
7348:
to general affine (possibly non-metric) connections. Picking out the unique torsion-free connection subordinate to a family of parametrized geodesics is known as
5942:
388:
327:
7596:. In Bergmann, P. G., & De Sabbata, V. Cosmology and Gravitation: Spin, Torsion, Rotation, and Supergravity (Vol. 58). Springer Science & Business Media.
6163:
5257:
4978:
253:
6552:
3175:
364:
representing the displacement within a tangent space when the tangent space is developed (or "rolled") along an infinitesimal parallelogram whose sides are
3471:{\displaystyle {\mathfrak {S}}\left(R\left(X,Y\right)Z\right)={\mathfrak {S}}\left(T\left(T(X,Y),Z\right)+\left(\nabla _{X}T\right)\left(Y,Z\right)\right)}
6280:
4211:: 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
3001:
7345:
6026:
is then determined by the curvature of the connection. Together, the linear transformation of the frame and the translation of the starting point from
2046:
650:
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.
8571:
4199:, 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
7567:
Ozakin, A., & Yavari, A. (2014). Affine development of closed curves in
Weitzenböck manifolds and the Burgers vector of dislocation mechanics.
6869:
7102:
7002:
4080:
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
592:
1940:{\displaystyle {T^{k}}_{ij}=\theta ^{k}\left(\nabla _{\mathbf {e} _{i}}\mathbf {e} _{j}-\nabla _{\mathbf {e} _{j}}\mathbf {e} _{i}-\left\right)}
4329:
2874:
1959:
1167:
6785:, which associates with γ the tangent vector pointing along it.) Each geodesic is uniquely determined by its initial tangent vector at time
6356:
7822:
5010:
6841:
Two connections ∇ and ∇′ which have the same affinely parametrized geodesics (i.e., the same geodesic spray) differ only by torsion.
1523:
7993:
7568:
6701:
7207:
8755:
8434:
8040:
5677:
3694:
denotes the exterior covariant derivative. In terms of the curvature form and torsion form, the corresponding
Bianchi identities are
394:
in its inputs, because developing over the parallelogram in the opposite sense produces the opposite displacement, similarly to how a
7729:
3640:
8863:
8214:
8128:
8065:
7946:
7831:
5530:
77:
8868:
8636:
5615:
4203:
in opposite directions displaces the screw in opposite ways. The torsion tensor thus is related to, although distinct from, the
8015:
5088:
8858:
7385:
1318:
35:
7353:
426:
417:). The difference between a connection with torsion, and a corresponding connection without torsion is a tensor, called the
8487:
8419:
7554:
5743:
The foregoing considerations can be made more quantitative by considering a small parallelogram, originating at the point
4096:
Thus the torsion tensor can be intuitively understood by taking a small parallelogram circuit with sides given by vectors
2293:
1406:
1371:
1136:
7594:
Comments on the paper by Elie Cartan: Sur une generalisation de la notion de courbure de
Riemann et les espaces a torsion
3605:{\displaystyle {\mathfrak {S}}\left(\left(\nabla _{X}R\right)\left(Y,Z\right)+R\left(T\left(X,Y\right),Z\right)\right)=0}
8512:
3765:
Moreover, one can recover the curvature and torsion tensors from the curvature and torsion forms as follows. At a point
2548:{\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},}
588:
502:
8095:
1767:{\displaystyle \Theta ^{k}=d\theta ^{k}+{\omega ^{k}}_{j}\wedge \theta ^{j}={T^{k}}_{ij}\theta ^{i}\wedge \theta ^{j}.}
8750:
7549:
8035:
5814:
4208:
438:
5099:
One interpretation of the torsion involves the development of a curve. Suppose that a piecewise smooth closed loop
3700:
1222:
1033:
8561:
8381:
7892:
4894:
2309:
647:
8233:
2709:
928:{\displaystyle T^{k}{}_{ij}:=\Gamma ^{k}{}_{ij}-\Gamma ^{k}{}_{ji}-\gamma ^{k}{}_{ij},\quad i,j,k=1,2,\ldots ,n.}
8735:
7971:
4835:
8817:
8689:
8396:
8158:
8133:
8055:
7534:. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 231(1185), 263-273.
7375:
4269:. On it, we put a connection that is flat, but with non-zero torsion, defined on the standard Euclidean frame
2964:
31:
6200:
4456:
5172:
8787:
8474:
8391:
8361:
8105:
7986:
698:
171:
7668:"Sur les variétés à connexion affine, et la théorie de la relativité généralisée (première partie) (Suite)"
6067:
6029:
8745:
8601:
8556:
8110:
8100:
7380:
2313:
1468:
988:
978:
721:
410:
7605:
See Spivak (1999) Volume II, Addendum 1 to
Chapter 6. See also Bishop and Goldberg (1980), section 5.10.
6795:
2330:
8827:
8782:
8262:
8207:
8007:
7869:
5958:
5586:
5422:
4237:
941:
430:
284:
5674:). On the other hand, if the torsion is non-zero, then the developed curve may not be closed, so that
4272:
2647:{\displaystyle T\in \operatorname {Hom} \left({\textstyle \bigwedge }^{2}{\rm {T}}M,{\rm {T}}M\right).}
2256:
1107:
determine the symmetric part of the connection, the torsion tensor determines the antisymmetric part.
48:
8802:
8730:
8616:
8482:
8444:
8376:
7915:
7884:
7858:
7800:
7768:
7701:
7532:
Continuous distributions of dislocations: a new application of the methods of non-Riemannian geometry
6168:
5736:. Thus the development of a loop in the presence of torsion can become dislocated, analogously to a
3132:
462:
7531:
5481:
5366:
2240:{\displaystyle \Theta \in {\text{Hom}}\left({\textstyle \bigwedge }^{2}{\rm {T}}M,{\rm {T}}M\right)}
8679:
8502:
8492:
8341:
8326:
8282:
8148:
8120:
8075:
7639:"Sur les variétés à connexion affine, et la théorie de la relativité généralisée (première partie)"
5772:
5102:
41:
5866:
661:
429:. Torsion is also useful in the study of unparametrized families of geodesics, via the associated
8812:
8669:
8522:
8336:
8272:
8080:
8030:
7979:
7848:
7717:
7627:
6274:
6243:
5262:
3736:
3154:
482:
4535:
2854:{\displaystyle (\operatorname {tr} \,T)(X){\stackrel {\text{def}}{=}}\operatorname {tr} (T(X)).}
2154:
corresponding to the identity endomorphism of the tangent bundle under the duality isomorphism
17:
8807:
8715:
8576:
8551:
8366:
8277:
8257:
7942:
7840:
7827:
7619:
7390:
7365:
7337:
6516:
6119:
6115:
5737:
4204:
454:
434:
418:
296:
7692:
Elzanowski, M.; Epstein, M. (1985), "Geometric characterization of hyperelastic uniformity",
7544:
5991:
5901:
5402:
5289:
5222:
74:, with four different choices of flat connection preserving the Euclidean metric, defined by
8873:
8822:
8720:
8497:
8464:
8449:
8331:
8200:
8143:
8045:
7923:
7899:
7808:
7788:
7776:
7709:
7679:
7650:
6640:
5746:
5451:
5336:
5146:
4212:
4155:
4112:
2952:
1953:
It can be easily shown that Θ transforms tensorially in the sense that if a different frame
1132:
414:
332:
5309:
4983:
4429:
4402:
2142:
Alternatively, the solder form can be characterized in a frame-independent fashion as the T
1517:. The connection form expresses the exterior covariant derivative of these basic sections:
258:
215:
151:
8792:
8740:
8684:
8664:
8566:
8454:
8321:
8292:
8138:
8050:
7877:
7744:
7740:
7370:
7311:∇ and ∇′ define the same families of affinely parametrized geodesics if and only if
6637:
These are the equations satisfied by an equilibrium continuous medium with moment density
4232:
4081:
2566:
2297:
1639:
1490:
1484:
982:
632:
6957:
be the difference of the two connections, calculated in terms of arbitrary extensions of
5921:
367:
306:
7919:
7862:
7804:
7772:
7705:
6506:{\displaystyle D\Omega _{b}^{a}=0,\quad D\Theta ^{a}=\Omega _{b}^{a}\wedge \theta ^{b}.}
3299:{\displaystyle {\mathfrak {S}}\left(R\left(X,Y\right)Z\right):=R(X,Y)Z+R(Y,Z)X+R(Z,X)Y.}
8832:
8797:
8777:
8527:
8517:
8507:
8429:
8401:
8386:
8371:
8287:
8166:
8085:
7934:
7663:
7638:
7634:
6834:
6148:
6135:
5945:
5242:
4963:
4188:
3623:
2317:
1378:
1150:)-valued one-form which maps vertical vectors to the generators of the right action in
458:
391:
238:
8852:
8769:
8674:
8586:
8459:
8090:
7961:
7721:
7667:
4323:
4219:
describes how a surface twists about a curve on the surface. The companion notion of
4105:
1950:
are the frame-components of the torsion tensor, as given in the previous definition.
276:
3112:{\displaystyle R(X,Y)Z=\nabla _{X}\nabla _{Y}Z-\nabla _{Y}\nabla _{X}Z-\nabla _{}Z.}
8837:
8641:
8626:
8591:
8439:
8424:
6970:
5952:
4144:. Thus the torsion tensor is a tensor: a (bilinear) function of two input vectors
1120:
486:
300:
8725:
8699:
8621:
8310:
8249:
8176:
7903:
7756:
6139:
2125:{\displaystyle {\tilde {\Theta }}^{i}={\left(g^{-1}\right)^{i}}_{j}\Theta ^{j}.}
1593:
1397:
1179:
422:
4582:
8606:
7927:
7780:
1597:
6947:{\displaystyle \Delta (X,Y)=\nabla _{X}{\tilde {Y}}-\nabla '_{X}{\tilde {Y}}}
8581:
8532:
8025:
8003:
7191:{\displaystyle A(X,Y)={\tfrac {1}{2}}\left(\Delta (X,Y)-\Delta (Y,X)\right)}
7091:{\displaystyle S(X,Y)={\tfrac {1}{2}}\left(\Delta (X,Y)+\Delta (Y,X)\right)}
5239:
is homotopic to zero. The curve can be developed into the tangent space at
4223:
measures how moving frames roll along a curve without slipping or twisting.
4220:
4089:
643:
7826:, vol. 1 & 2 (New ed.), Wiley-Interscience (published 1996),
27:
Manner of characterizing a twist or screw of a moving frame around a curve
8611:
8596:
8181:
6103:
1104:
402:
6630:{\displaystyle Ds_{ab}=\theta _{a}\wedge t_{b}-\theta _{b}\wedge t_{a}.}
8305:
8267:
7713:
7684:
7655:
2937:{\displaystyle T_{0}=T-{\frac {1}{n-1}}\iota (\operatorname {tr} \,T),}
2021:{\displaystyle {\tilde {\mathbf {e} }}_{i}=\mathbf {e} _{j}{g^{j}}_{i}}
421:. Absorption of torsion also plays a fundamental role in the study of
7812:
8631:
8223:
7757:"General relativity with spin and torsion: Foundations and prospects"
292:
6347:{\displaystyle t_{a}={\tfrac {1}{2}}\eta _{abc}\wedge \Omega ^{bc},}
1582:{\displaystyle D\mathbf {e} _{i}=\mathbf {e} _{j}{\omega ^{j}}_{i}.}
7965:
7755:
Hehl, F.W.; von der Heyde, P.; Kerlick, G.D.; Nester, J.M. (1976),
7580:
7853:
7728:
Goriely, A.; Robertson-Tessi, M.; Tabor, M.; Vandiver, R. (2006),
7506:
6763:{\displaystyle \nabla _{{\dot {\gamma }}(t)}{\dot {\gamma }}(t)=0}
6123:
4200:
395:
40:
7301:{\displaystyle A(X,Y)={\tfrac {1}{2}}\left(T(X,Y)-T'(X,Y)\right)}
7939:
A comprehensive introduction to differential geometry, Volume II
5729:{\displaystyle {\tilde {\gamma }}(0)\not ={\tilde {\gamma }}(1)}
2316:
part and another part which contains the trace terms. Using the
8196:
7975:
7507:
Elie Cartan's torsion in geometry and in field theory, an essay
3680:{\displaystyle \Omega =D\omega =d\omega +\omega \wedge \omega }
401:
Torsion is particularly useful in the study of the geometry of
1119:, an alternative characterization of torsion, applies to the
398:
moves in opposite ways when it is twisted in two directions.
141:{\displaystyle \nabla _{e_{i}}e_{j}=\tau \,e_{i}\times e_{j}}
6833:
One application of the torsion of a connection involves the
5667:{\displaystyle {\tilde {\gamma }}(0)={\tilde {\gamma }}(1)}
8192:
7791:(1961), "Lorentz invariance and the gravitational field",
6781:. (Here the dot denotes differentiation with respect to
4453:
axis, starting at the origin. The parallel vector field
2776:) is defined as the trace of this endomorphism. That is,
2139:(carrying one contravariant and two covariant indices).
6421:
Then the
Bianchi identities The Bianchi identities are
4392:{\displaystyle \nabla _{e_{i}}e_{j}=e_{i}\times e_{j}.}
1596:
for the tangent bundle (relative to this frame) is the
1352:{\displaystyle \Theta =d\theta +\omega \wedge \theta .}
1306:
is the projection mapping for the principal bundle and
1158:) and equivariantly intertwines the right action of GL(
7233:
7128:
7028:
6414:{\displaystyle s_{ab}=-\eta _{abc}\wedge \Theta ^{c}.}
6298:
2601:
2197:
1497:, written in a particular frame of the tangent bundle
45:
Development of the unit circle in the
Euclidean space
7210:
7105:
7005:
6872:
6798:
6704:
6643:
6555:
6519:
6427:
6359:
6283:
6246:
6203:
6171:
6151:
6070:
6032:
5994:
5961:
5924:
5904:
5869:
5817:
5811:. Then the tangent bivector to the parallelogram is
5775:
5749:
5680:
5618:
5589:
5533:
5484:
5454:
5425:
5405:
5369:
5339:
5312:
5292:
5265:
5245:
5225:
5175:
5149:
5105:
5080:{\displaystyle x\,e_{1}+\cos x\,e_{2}-\sin x\,e_{3}.}
5013:
4986:
4966:
4897:
4838:
4580:
4538:
4459:
4432:
4405:
4332:
4275:
4240:
4158:
4115:
3785:
3739:
3703:
3643:
3492:
3323:
3178:
3135:
3004:
2877:
2785:
2712:
2581:
2394:
2333:
2259:
2178:
2049:
1962:
1786:
1651:
1526:
1409:
1321:
1225:
1036:
991:
944:
781:
664:
505:
370:
335:
309:
261:
241:
218:
174:
154:
80:
51:
7530:
Bilby, B. A., Bullough, R., & Smith, E. (1955).
6165:
is given in three dimensions, with curvature 2-form
1453:{\displaystyle R_{g}^{*}\Theta =g^{-1}\cdot \Theta }
8768:
8708:
8657:
8650:
8542:
8473:
8410:
8354:
8301:
8248:
8241:
8157:
8119:
8064:
8014:
7672:
7643:
5918:is the torsion tensor, up to higher order terms in
7966:Rolling without slipping interpretation of torsion
7300:
7190:
7090:
6973:, one sees that Δ does not actually depend on how
6946:
6822:
6762:
6659:
6629:
6541:
6505:
6413:
6346:
6265:
6232:
6189:
6157:
6094:
6056:
6018:
5976:
5944:. This displacement is directly analogous to the
5936:
5910:
5890:
5855:
5803:
5761:
5728:
5666:
5604:
5575:
5519:
5470:
5440:
5411:
5391:
5355:
5325:
5298:
5278:
5251:
5231:
5211:
5161:
5135:
5079:
4999:
4972:
4949:
4883:
4824:
4566:
4524:
4445:
4418:
4399:Consider now the parallel transport of the vector
4391:
4314:
4261:
4179:
4136:
4043:
3754:
3724:
3679:
3604:
3470:
3298:
3145:
3111:
2936:
2853:
2757:
2646:
2547:
2374:
2277:
2239:
2124:
2020:
1939:
1766:
1581:
1452:
1351:
1281:
1095:
1022:
969:
927:
689:
577:{\displaystyle T(X,Y):=\nabla _{X}Y-\nabla _{Y}X-}
576:
437:, such ideas have been implemented in the form of
413:to other, possibly non-metric situations (such as
382:
356:
321:
267:
247:
227:
204:
160:
140:
66:
5583:If the torsion is zero, then the developed curve
3129:relate the curvature and torsion as follows. Let
5988:transformation that the frame undergoes between
1489:The torsion form may be expressed in terms of a
724:of the tangent bundle can be derived by setting
7518:
7493:
7481:
7469:
7457:
7445:
7433:
7421:
7409:
5856:{\displaystyle v\wedge w\in \Lambda ^{2}T_{p}M}
754:and by introducing the commutator coefficients
7904:"The physical structure of general relativity"
5576:{\displaystyle dx^{i}=\gamma ^{*}\theta ^{i}.}
3725:{\displaystyle D\Theta =\Omega \wedge \theta }
2308:The torsion tensor can be decomposed into two
1312:is its push-forward. The torsion form is then
1282:{\displaystyle \theta (X)=u^{-1}(\pi _{*}(X))}
1096:{\displaystyle T^{k}{}_{ij}=2\Gamma ^{k}{}_{}}
8208:
7987:
6996:be the symmetric and alternating parts of Δ:
4950:{\displaystyle X=\cos x\,e_{2}-\sin x\,e_{3}}
8:
2031:for some invertible matrix-valued function (
212:. The resulting curves all have arc length
7694:Archive for Rational Mechanics and Analysis
2758:{\displaystyle T(X):Y\mapsto T(X\wedge Y).}
1642:). Then the torsion 2-form has components
8654:
8245:
8215:
8201:
8193:
7994:
7980:
7972:
7505:Hehl, F. W., & Obukhov, Y. N. (2007).
7346:fundamental theorem of Riemannian geometry
4884:{\displaystyle {\dot {a}}=b,{\dot {b}}=-a}
2669:defined as follows. For each vector fixed
772:. The components of the torsion are then
7852:
7683:
7654:
7308:is the difference of the torsion tensors.
7232:
7209:
7127:
7104:
7027:
7004:
6933:
6932:
6923:
6905:
6904:
6898:
6871:
6800:
6799:
6797:
6734:
6733:
6711:
6710:
6709:
6703:
6648:
6642:
6618:
6605:
6592:
6579:
6563:
6554:
6527:
6518:
6494:
6481:
6476:
6463:
6440:
6435:
6426:
6402:
6383:
6364:
6358:
6332:
6313:
6297:
6288:
6282:
6251:
6245:
6224:
6208:
6202:
6181:
6176:
6170:
6150:
6072:
6071:
6069:
6034:
6033:
6031:
5993:
5963:
5962:
5960:
5951:More generally, one can also transport a
5923:
5903:
5868:
5844:
5834:
5816:
5792:
5774:
5748:
5706:
5705:
5682:
5681:
5679:
5644:
5643:
5620:
5619:
5617:
5591:
5590:
5588:
5564:
5554:
5541:
5532:
5502:
5489:
5483:
5459:
5453:
5427:
5426:
5424:
5404:
5374:
5368:
5344:
5338:
5317:
5311:
5291:
5270:
5264:
5244:
5224:
5174:
5148:
5104:
5068:
5063:
5045:
5040:
5022:
5017:
5012:
4991:
4985:
4965:
4941:
4936:
4918:
4913:
4896:
4861:
4860:
4840:
4839:
4837:
4809:
4785:
4784:
4772:
4748:
4747:
4730:
4717:
4701:
4688:
4672:
4657:
4656:
4647:
4632:
4631:
4617:
4612:
4592:
4591:
4581:
4579:
4558:
4537:
4516:
4491:
4458:
4437:
4431:
4410:
4404:
4380:
4367:
4354:
4342:
4337:
4331:
4306:
4293:
4280:
4274:
4253:
4249:
4248:
4239:
4157:
4114:
4006:
3981:
3903:
3866:
3841:
3786:
3784:
3738:
3702:
3642:
3618:The curvature form and Bianchi identities
3514:
3494:
3493:
3491:
3430:
3370:
3369:
3325:
3324:
3322:
3180:
3179:
3177:
3137:
3136:
3134:
3085:
3069:
3059:
3043:
3033:
3003:
2924:
2897:
2882:
2876:
2816:
2811:
2809:
2808:
2792:
2784:
2711:
2627:
2626:
2614:
2613:
2607:
2600:
2580:
2536:
2526:
2524:
2517:
2495:
2486:
2476:
2474:
2467:
2445:
2433:
2431:
2424:
2408:
2406:
2399:
2393:
2360:
2358:
2351:
2338:
2332:
2258:
2223:
2222:
2210:
2209:
2203:
2196:
2185:
2177:
2113:
2103:
2096:
2083:
2073:
2063:
2052:
2051:
2048:
2012:
2005:
2000:
1993:
1988:
1978:
1967:
1965:
1964:
1961:
1921:
1916:
1906:
1901:
1886:
1881:
1872:
1867:
1865:
1852:
1847:
1838:
1833:
1831:
1816:
1800:
1793:
1788:
1785:
1755:
1742:
1729:
1722:
1717:
1707:
1694:
1687:
1682:
1672:
1656:
1650:
1570:
1563:
1558:
1551:
1546:
1536:
1531:
1525:
1435:
1419:
1414:
1408:
1385:, meaning that under the right action of
1320:
1261:
1245:
1224:
1078:
1076:
1069:
1050:
1048:
1041:
1035:
1005:
1003:
996:
990:
981:defining the connection. If the basis is
958:
951:
946:
943:
870:
868:
861:
845:
843:
836:
820:
818:
811:
795:
793:
786:
780:
678:
676:
669:
663:
547:
531:
504:
369:
334:
308:
260:
240:
217:
173:
153:
132:
119:
114:
102:
90:
85:
79:
58:
54:
53:
50:
8572:Covariance and contravariance of vectors
6984:are extended (so it defines a tensor on
6233:{\displaystyle \Theta ^{a}=D\theta ^{a}}
4525:{\displaystyle X(x)=a(x)e_{2}+b(x)e_{3}}
1467:acts on the right-hand side through its
646:, despite being defined in terms of the
7402:
6671:Geodesics and the absorption of torsion
5212:{\displaystyle \gamma (0)=\gamma (1)=p}
2169:. Then the torsion 2-form is a section
1103:. In particular (see below), while the
2135:In other terms, Θ is a tensor of type
205:{\displaystyle \tau =0.01,0.1,0.5,1.0}
7941:, Houston, Texas: Publish or Perish,
6138:, torsion is naturally associated to
6095:{\displaystyle {\tilde {\gamma }}(1)}
6057:{\displaystyle {\tilde {\gamma }}(0)}
4076:Characterizations and interpretations
658:The components of the torsion tensor
7:
7823:Foundations of Differential Geometry
4215:. In the geometry of surfaces, the
2959:Curvature and the Bianchi identities
1023:{\displaystyle \gamma ^{k}{}_{ij}=0}
168:is a constant scalar, respectively:
7569:Mathematics and Mechanics of Solids
6823:{\displaystyle {\dot {\gamma }}(0)}
3495:
3371:
3326:
3309:Then the following identities hold
3181:
3138:
2375:{\displaystyle a_{i}=T^{k}{}_{ik},}
1377:The torsion form is a (horizontal)
1178:). The frame bundle also carries a
8435:Tensors in curvilinear coordinates
8041:Radius of curvature (applications)
7820:Kobayashi, S.; Nomizu, K. (1963),
7165:
7144:
7065:
7044:
6920:
6895:
6873:
6706:
6473:
6460:
6432:
6399:
6329:
6205:
6173:
5977:{\displaystyle {\tilde {\gamma }}}
5905:
5870:
5831:
5605:{\displaystyle {\tilde {\gamma }}}
5527:sastify the differential equation
5441:{\displaystyle {\tilde {\gamma }}}
4609:
4334:
4262:{\displaystyle M=\mathbb {R} ^{3}}
4088:rolling it), an effect due to the
3969:
3829:
3743:
3713:
3707:
3644:
3511:
3427:
3082:
3066:
3056:
3040:
3030:
2628:
2615:
2260:
2224:
2211:
2179:
2110:
2054:
1862:
1828:
1653:
1447:
1425:
1322:
1066:
970:{\displaystyle {\Gamma ^{k}}_{ij}}
948:
833:
808:
544:
528:
82:
25:
8129:Curvature of Riemannian manifolds
7509:. arXiv preprint arXiv:0711.1535.
7352:, and it is one of the stages of
6853:are a pair of tangent vectors at
4980:, as it is transported along the
4315:{\displaystyle e_{1},e_{2},e_{3}}
2278:{\displaystyle \Theta =D\theta ,}
329:, that produces an output vector
7472:, Volume 1, Proposition III.5.2.
7344:This is a generalization of the
4574:, and the differential equation
1989:
1968:
1917:
1902:
1882:
1868:
1848:
1834:
1547:
1532:
654:Components of the torsion tensor
67:{\displaystyle \mathbb {R} ^{3}}
6455:
6190:{\displaystyle \Omega _{a}^{b}}
5612:is also a closed loop (so that
5089:differential geometry of curves
4152:that produces an output vector
3146:{\displaystyle {\mathfrak {S}}}
1199:(regarded as a linear function
882:
18:Torsion (differential geometry)
7424:, Chapter III, Proposition 7.6
7290:
7278:
7264:
7252:
7226:
7214:
7180:
7168:
7159:
7147:
7121:
7109:
7080:
7068:
7059:
7047:
7021:
7009:
6938:
6910:
6888:
6876:
6817:
6811:
6751:
6745:
6728:
6722:
6693:affinely parametrized geodesic
6089:
6083:
6077:
6051:
6045:
6039:
5968:
5885:
5873:
5723:
5717:
5711:
5699:
5693:
5687:
5661:
5655:
5649:
5637:
5631:
5625:
5596:
5520:{\displaystyle x^{i}=x^{i}(t)}
5514:
5508:
5432:
5392:{\displaystyle \theta ^{i}(p)}
5386:
5380:
5259:in the following manner. Let
5200:
5194:
5185:
5179:
5127:
5124:
5112:
4802:
4781:
4765:
4744:
4548:
4542:
4509:
4503:
4484:
4478:
4469:
4463:
4174:
4162:
4131:
4119:
4021:
4015:
3996:
3990:
3948:
3936:
3918:
3912:
3881:
3875:
3856:
3850:
3805:
3793:
3404:
3392:
3287:
3275:
3263:
3251:
3239:
3227:
3098:
3086:
3020:
3008:
2928:
2918:
2845:
2842:
2836:
2830:
2805:
2799:
2796:
2786:
2749:
2737:
2731:
2722:
2716:
2057:
1972:
1374:determined by the connection.
1276:
1273:
1267:
1254:
1235:
1229:
1088:
1079:
985:then the Lie brackets vanish,
591:of two vector fields. By the
571:
559:
521:
509:
351:
339:
36:Torsion field (disambiguation)
1:
8488:Exterior covariant derivative
8420:Tensor (intrinsic definition)
7519:Kobayashi & Nomizu (1963)
7458:Kobayashi & Nomizu (1963)
7446:Kobayashi & Nomizu (1963)
7434:Kobayashi & Nomizu (1963)
7422:Kobayashi & Nomizu (1963)
7410:Kobayashi & Nomizu (1963)
5804:{\displaystyle v,w\in T_{p}M}
5143:is given, based at the point
5136:{\displaystyle \gamma :\to M}
4104:, in a space and rolling the
2294:exterior covariant derivative
1777:In the rightmost expression,
1372:exterior covariant derivative
8513:Raising and lowering indices
7968:, URL (version: 2011-01-27).
7624:Tensor analysis on manifolds
5891:{\displaystyle \Theta (v,w)}
5286:be a parallel coframe along
1162:) on the tangent bundle of F
690:{\displaystyle T^{c}{}_{ab}}
8751:Gluon field strength tensor
7550:Encyclopedia of Mathematics
7494:Kobayashi & Nomizu 1963
7482:Kobayashi & Nomizu 1963
7470:Kobayashi & Nomizu 1963
7354:Cartan's equivalence method
6266:{\displaystyle \eta _{abc}}
5279:{\displaystyle \theta ^{i}}
3755:{\displaystyle D\Omega =0.}
2385:and the trace-free part is
427:Cartan's equivalence method
299:. The torsion tensor is a
8890:
8562:Cartan formalism (physics)
8382:Penrose graphical notation
7893:Cambridge University Press
7448:, Chapter III, Theorem 2.4
7412:, Chapter III, Theorem 5.1
6145:Suppose that a connection
5007:axis traces out the helix
4960:Now the tip of the vector
4567:{\displaystyle X(0)=e_{2}}
3483:Bianchi's second identity:
1482:
295:that is associated to any
29:
8234:Glossary of tensor theory
8230:
7928:10.1103/RevModPhys.36.463
7781:10.1103/revmodphys.48.393
7622:; Goldberg, S.I. (1980),
6110:The torsion of a filament
3314:Bianchi's first identity:
2983:defined on vector fields
2304:Irreducible decomposition
255:, and respective torsion
8864:Connection (mathematics)
8818:Gregorio Ricci-Curbastro
8690:Riemann curvature tensor
8397:Van der Waerden notation
8159:Curvature of connections
8134:Riemann curvature tensor
8056:Total absolute curvature
7521:, Chapter III, Section 4
7460:, Chapter III, Section 7
7436:, Chapter III, Section 2
6542:{\displaystyle Dt_{a}=0}
32:Torsion (disambiguation)
8869:Curvature (mathematics)
8788:Elwin Bruno Christoffel
8721:Angular momentum tensor
8392:Tetrad (index notation)
8362:Abstract index notation
8106:Second fundamental form
8096:Gauss–Codazzi equations
7730:"Elastic growth models"
6019:{\displaystyle t=0,t=1}
5911:{\displaystyle \Theta }
5412:{\displaystyle \gamma }
5299:{\displaystyle \gamma }
5232:{\displaystyle \gamma }
4207:, as it appears in the
2864:The trace-free part of
2572:Intrinsically, one has
1479:Torsion form in a frame
979:connection coefficients
8602:Levi-Civita connection
8111:Third fundamental form
8101:First fundamental form
8066:Differential geometry
8036:Frenet–Serret formulas
8016:Differential geometry
7381:Levi-Civita connection
7302:
7192:
7092:
6948:
6824:
6764:
6661:
6660:{\displaystyle s_{ab}}
6631:
6543:
6507:
6415:
6348:
6273:be the skew-symmetric
6267:
6234:
6191:
6159:
6096:
6058:
6020:
5978:
5938:
5912:
5892:
5857:
5805:
5763:
5762:{\displaystyle p\in M}
5730:
5668:
5606:
5577:
5521:
5472:
5471:{\displaystyle T_{p}M}
5442:
5413:
5393:
5357:
5356:{\displaystyle T_{p}M}
5333:be the coordinates on
5327:
5300:
5280:
5253:
5233:
5213:
5163:
5162:{\displaystyle p\in M}
5137:
5081:
5001:
4974:
4951:
4891:, and the solution is
4885:
4826:
4568:
4526:
4447:
4420:
4393:
4316:
4263:
4209:Frenet–Serret formulas
4181:
4180:{\displaystyle T(v,w)}
4138:
4137:{\displaystyle T(v,w)}
4045:
3756:
3726:
3681:
3606:
3472:
3300:
3147:
3113:
2938:
2855:
2759:
2648:
2549:
2376:
2300:for further details.)
2279:
2241:
2126:
2022:
1941:
1768:
1583:
1469:adjoint representation
1454:
1353:
1283:
1168:adjoint representation
1097:
1024:
971:
929:
691:
578:
469:(sometimes called the
453:be a manifold with an
439:Einstein–Cartan theory
411:Levi-Civita connection
384:
358:
357:{\displaystyle T(X,Y)}
323:
280:
269:
249:
229:
206:
162:
142:
68:
8859:Differential geometry
8828:Jan Arnoldus Schouten
8783:Augustin-Louis Cauchy
8263:Differential geometry
8008:differential geometry
7350:absorption of torsion
7303:
7193:
7093:
6949:
6825:
6765:
6662:
6632:
6544:
6508:
6416:
6349:
6268:
6235:
6192:
6160:
6130:Torsion and vorticity
6097:
6059:
6021:
5979:
5939:
5913:
5893:
5858:
5806:
5764:
5731:
5669:
5607:
5578:
5522:
5473:
5443:
5414:
5394:
5358:
5328:
5326:{\displaystyle x^{i}}
5301:
5281:
5254:
5234:
5214:
5164:
5138:
5082:
5002:
5000:{\displaystyle e_{1}}
4975:
4952:
4886:
4827:
4569:
4527:
4448:
4446:{\displaystyle e_{1}}
4421:
4419:{\displaystyle e_{2}}
4394:
4317:
4264:
4182:
4139:
4046:
3757:
3727:
3682:
3607:
3473:
3301:
3148:
3114:
2939:
2856:
2760:
2649:
2550:
2377:
2280:
2242:
2127:
2023:
1942:
1769:
1584:
1493:on the base manifold
1455:
1354:
1284:
1186:, defined at a frame
1098:
1025:
972:
930:
692:
579:
431:projective connection
385:
359:
324:
303:of two input vectors
285:differential geometry
270:
268:{\displaystyle \tau }
250:
230:
228:{\displaystyle 2\pi }
207:
163:
161:{\displaystyle \tau }
143:
69:
44:
8803:Carl Friedrich Gauss
8736:stress–energy tensor
8731:Cauchy stress tensor
8483:Covariant derivative
8445:Antisymmetric tensor
8377:Multi-index notation
8076:Principal curvatures
7889:Space-Time Structure
7845:Spacetime and fields
7208:
7103:
7003:
6971:Leibniz product rule
6870:
6796:
6702:
6641:
6553:
6517:
6425:
6357:
6281:
6244:
6201:
6169:
6149:
6068:
6030:
5992:
5959:
5948:of crystallography.
5922:
5902:
5867:
5815:
5773:
5747:
5678:
5616:
5587:
5531:
5482:
5452:
5423:
5403:
5399:. A development of
5367:
5337:
5310:
5290:
5263:
5243:
5223:
5173:
5147:
5103:
5011:
4984:
4964:
4895:
4836:
4578:
4536:
4457:
4430:
4403:
4330:
4273:
4238:
4231:Consider the (flat)
4156:
4113:
3783:
3737:
3701:
3641:
3490:
3321:
3176:
3133:
3002:
2875:
2783:
2710:
2665:, is an element of T
2579:
2392:
2331:
2257:
2176:
2047:
1960:
1784:
1649:
1524:
1407:
1319:
1223:
1034:
989:
942:
779:
697:in terms of a local
662:
503:
483:vector-valued 2-form
463:covariant derivative
368:
333:
307:
259:
239:
216:
172:
152:
78:
49:
30:For other uses, see
8680:Nonmetricity tensor
8535:(2nd-order tensors)
8503:Hodge star operator
8493:Exterior derivative
8342:Transport phenomena
8327:Continuum mechanics
8283:Multilinear algebra
8149:Sectional curvature
8121:Riemannian geometry
8002:Various notions of
7920:1964RvMP...36..463S
7863:2009arXiv0911.0334P
7805:1961JMP.....2..212K
7773:1976RvMP...48..393H
7706:1985ArRMA..88..347E
7581:Goriely et al. 2006
7386:Torsion coefficient
6931:
6845:More precisely, if
6486:
6445:
6197:and torsion 2-form
6186:
6106:of the connection.
5937:{\displaystyle v,w}
4322:by the (Euclidean)
2683:defines an element
1424:
1135:is equipped with a
409:, generalizing the
407:absorbs the torsion
383:{\displaystyle X,Y}
322:{\displaystyle X,Y}
8813:Tullio Levi-Civita
8756:Metric tensor (GR)
8670:Levi-Civita symbol
8523:Tensor contraction
8337:General relativity
8273:Euclidean geometry
8081:Gaussian curvature
8031:Torsion of a curve
7714:10.1007/BF00250871
7685:10.24033/asens.753
7656:10.24033/asens.751
7628:Dover Publications
7496:, Volume 1, III.5.
7484:, Volume 1, III.2.
7298:
7242:
7188:
7137:
7088:
7037:
6944:
6919:
6820:
6760:
6657:
6627:
6539:
6503:
6472:
6431:
6411:
6344:
6307:
6275:Levi-Civita tensor
6263:
6230:
6187:
6172:
6155:
6092:
6054:
6016:
5974:
5934:
5908:
5888:
5853:
5801:
5759:
5726:
5664:
5602:
5573:
5517:
5478:whose coordinates
5468:
5438:
5409:
5389:
5353:
5323:
5296:
5276:
5249:
5229:
5219:. We assume that
5209:
5159:
5133:
5077:
4997:
4970:
4947:
4881:
4822:
4820:
4564:
4522:
4443:
4416:
4389:
4312:
4259:
4205:torsion of a curve
4177:
4134:
4041:
4039:
3752:
3722:
3677:
3602:
3468:
3296:
3143:
3127:Bianchi identities
3109:
2967:of ∇ is a mapping
2934:
2851:
2755:
2644:
2605:
2545:
2372:
2275:
2237:
2201:
2122:
2018:
1937:
1764:
1579:
1450:
1410:
1362:Equivalently, Θ =
1349:
1279:
1182:θ, with values in
1180:canonical one-form
1105:geodesic equations
1093:
1020:
967:
925:
687:
574:
380:
354:
319:
281:
265:
245:
225:
202:
158:
138:
64:
8846:
8845:
8808:Hermann Grassmann
8764:
8763:
8716:Moment of inertia
8577:Differential form
8552:Affine connection
8367:Einstein notation
8350:
8349:
8278:Exterior calculus
8258:Coordinate system
8190:
8189:
7813:10.1063/1.1703702
7571:, 19(3), 299-307.
7391:Torsion of curves
7366:Contorsion tensor
7338:contorsion tensor
7241:
7136:
7036:
6941:
6913:
6808:
6777:in the domain of
6742:
6719:
6306:
6158:{\displaystyle D}
6120:elasticity theory
6118:, and especially
6116:materials science
6080:
6042:
5971:
5738:screw dislocation
5714:
5690:
5652:
5628:
5599:
5435:
5252:{\displaystyle p}
4973:{\displaystyle X}
4869:
4848:
4793:
4756:
4665:
4640:
4600:
4191:in the arguments
3169:. For instance,
2913:
2821:
2819:
2511:
2461:
2188:
2146:-valued one-form
2060:
1975:
455:affine connection
435:relativity theory
419:contorsion tensor
297:affine connection
275:(in the sense of
248:{\displaystyle 1}
16:(Redirected from
8881:
8823:Bernhard Riemann
8655:
8498:Exterior product
8465:Two-point tensor
8450:Symmetric tensor
8332:Electromagnetism
8246:
8217:
8210:
8203:
8194:
8144:Scalar curvature
8046:Affine curvature
7996:
7989:
7982:
7973:
7951:
7930:
7895:
7880:
7865:
7856:
7836:
7815:
7783:
7751:
7749:
7743:, archived from
7734:
7724:
7688:
7687:
7659:
7658:
7630:
7606:
7603:
7597:
7592:Trautman (1980)
7590:
7584:
7578:
7572:
7565:
7559:
7558:
7541:
7535:
7528:
7522:
7516:
7510:
7503:
7497:
7491:
7485:
7479:
7473:
7467:
7461:
7455:
7449:
7443:
7437:
7431:
7425:
7419:
7413:
7407:
7376:Curvature tensor
7325:
7307:
7305:
7304:
7299:
7297:
7293:
7277:
7243:
7234:
7197:
7195:
7194:
7189:
7187:
7183:
7138:
7129:
7097:
7095:
7094:
7089:
7087:
7083:
7038:
7029:
6983:
6953:
6951:
6950:
6945:
6943:
6942:
6934:
6927:
6915:
6914:
6906:
6903:
6902:
6862:
6829:
6827:
6826:
6821:
6810:
6809:
6801:
6791:
6769:
6767:
6766:
6761:
6744:
6743:
6735:
6732:
6731:
6721:
6720:
6712:
6683:) is a curve on
6666:
6664:
6663:
6658:
6656:
6655:
6636:
6634:
6633:
6628:
6623:
6622:
6610:
6609:
6597:
6596:
6584:
6583:
6571:
6570:
6548:
6546:
6545:
6540:
6532:
6531:
6512:
6510:
6509:
6504:
6499:
6498:
6485:
6480:
6468:
6467:
6444:
6439:
6420:
6418:
6417:
6412:
6407:
6406:
6394:
6393:
6372:
6371:
6353:
6351:
6350:
6345:
6340:
6339:
6324:
6323:
6308:
6299:
6293:
6292:
6272:
6270:
6269:
6264:
6262:
6261:
6239:
6237:
6236:
6231:
6229:
6228:
6213:
6212:
6196:
6194:
6193:
6188:
6185:
6180:
6164:
6162:
6161:
6156:
6101:
6099:
6098:
6093:
6082:
6081:
6073:
6063:
6061:
6060:
6055:
6044:
6043:
6035:
6025:
6023:
6022:
6017:
5983:
5981:
5980:
5975:
5973:
5972:
5964:
5955:along the curve
5943:
5941:
5940:
5935:
5917:
5915:
5914:
5909:
5897:
5895:
5894:
5889:
5862:
5860:
5859:
5854:
5849:
5848:
5839:
5838:
5810:
5808:
5807:
5802:
5797:
5796:
5768:
5766:
5765:
5760:
5735:
5733:
5732:
5727:
5716:
5715:
5707:
5692:
5691:
5683:
5673:
5671:
5670:
5665:
5654:
5653:
5645:
5630:
5629:
5621:
5611:
5609:
5608:
5603:
5601:
5600:
5592:
5582:
5580:
5579:
5574:
5569:
5568:
5559:
5558:
5546:
5545:
5526:
5524:
5523:
5518:
5507:
5506:
5494:
5493:
5477:
5475:
5474:
5469:
5464:
5463:
5447:
5445:
5444:
5439:
5437:
5436:
5428:
5418:
5416:
5415:
5410:
5398:
5396:
5395:
5390:
5379:
5378:
5362:
5360:
5359:
5354:
5349:
5348:
5332:
5330:
5329:
5324:
5322:
5321:
5305:
5303:
5302:
5297:
5285:
5283:
5282:
5277:
5275:
5274:
5258:
5256:
5255:
5250:
5238:
5236:
5235:
5230:
5218:
5216:
5215:
5210:
5168:
5166:
5165:
5160:
5142:
5140:
5139:
5134:
5086:
5084:
5083:
5078:
5073:
5072:
5050:
5049:
5027:
5026:
5006:
5004:
5003:
4998:
4996:
4995:
4979:
4977:
4976:
4971:
4956:
4954:
4953:
4948:
4946:
4945:
4923:
4922:
4890:
4888:
4887:
4882:
4871:
4870:
4862:
4850:
4849:
4841:
4831:
4829:
4828:
4823:
4821:
4814:
4813:
4795:
4794:
4786:
4777:
4776:
4758:
4757:
4749:
4739:
4735:
4734:
4722:
4721:
4706:
4705:
4693:
4692:
4677:
4676:
4667:
4666:
4658:
4652:
4651:
4642:
4641:
4633:
4624:
4623:
4622:
4621:
4602:
4601:
4593:
4573:
4571:
4570:
4565:
4563:
4562:
4531:
4529:
4528:
4523:
4521:
4520:
4496:
4495:
4452:
4450:
4449:
4444:
4442:
4441:
4425:
4423:
4422:
4417:
4415:
4414:
4398:
4396:
4395:
4390:
4385:
4384:
4372:
4371:
4359:
4358:
4349:
4348:
4347:
4346:
4321:
4319:
4318:
4313:
4311:
4310:
4298:
4297:
4285:
4284:
4268:
4266:
4265:
4260:
4258:
4257:
4252:
4217:geodesic torsion
4213:osculating plane
4186:
4184:
4183:
4178:
4143:
4141:
4140:
4135:
4070:
4050:
4048:
4047:
4042:
4040:
4033:
4029:
4028:
4024:
4014:
4013:
3989:
3988:
3925:
3921:
3911:
3910:
3893:
3889:
3888:
3884:
3874:
3873:
3849:
3848:
3761:
3759:
3758:
3753:
3731:
3729:
3728:
3723:
3686:
3684:
3683:
3678:
3634:)-valued 2-form
3611:
3609:
3608:
3603:
3595:
3591:
3590:
3586:
3579:
3575:
3546:
3542:
3527:
3523:
3519:
3518:
3499:
3498:
3477:
3475:
3474:
3469:
3467:
3463:
3462:
3458:
3443:
3439:
3435:
3434:
3417:
3413:
3375:
3374:
3365:
3361:
3357:
3353:
3330:
3329:
3305:
3303:
3302:
3297:
3220:
3216:
3212:
3208:
3185:
3184:
3152:
3150:
3149:
3144:
3142:
3141:
3118:
3116:
3115:
3110:
3102:
3101:
3074:
3073:
3064:
3063:
3048:
3047:
3038:
3037:
2982:
2965:curvature tensor
2953:interior product
2943:
2941:
2940:
2935:
2914:
2912:
2898:
2887:
2886:
2860:
2858:
2857:
2852:
2823:
2822:
2820:
2817:
2815:
2810:
2764:
2762:
2761:
2756:
2702:
2678:
2653:
2651:
2650:
2645:
2640:
2636:
2632:
2631:
2619:
2618:
2612:
2611:
2606:
2554:
2552:
2551:
2546:
2541:
2540:
2531:
2530:
2525:
2522:
2521:
2512:
2510:
2496:
2491:
2490:
2481:
2480:
2475:
2472:
2471:
2462:
2460:
2446:
2441:
2440:
2432:
2429:
2428:
2416:
2415:
2407:
2404:
2403:
2381:
2379:
2378:
2373:
2368:
2367:
2359:
2356:
2355:
2343:
2342:
2284:
2282:
2281:
2276:
2246:
2244:
2243:
2238:
2236:
2232:
2228:
2227:
2215:
2214:
2208:
2207:
2202:
2189:
2186:
2168:
2138:
2131:
2129:
2128:
2123:
2118:
2117:
2108:
2107:
2102:
2101:
2100:
2095:
2091:
2090:
2068:
2067:
2062:
2061:
2053:
2027:
2025:
2024:
2019:
2017:
2016:
2011:
2010:
2009:
1998:
1997:
1992:
1983:
1982:
1977:
1976:
1971:
1966:
1946:
1944:
1943:
1938:
1936:
1932:
1931:
1927:
1926:
1925:
1920:
1911:
1910:
1905:
1891:
1890:
1885:
1879:
1878:
1877:
1876:
1871:
1857:
1856:
1851:
1845:
1844:
1843:
1842:
1837:
1821:
1820:
1808:
1807:
1799:
1798:
1797:
1773:
1771:
1770:
1765:
1760:
1759:
1747:
1746:
1737:
1736:
1728:
1727:
1726:
1712:
1711:
1699:
1698:
1693:
1692:
1691:
1677:
1676:
1661:
1660:
1637:
1608:
1588:
1586:
1585:
1580:
1575:
1574:
1569:
1568:
1567:
1556:
1555:
1550:
1541:
1540:
1535:
1516:
1459:
1457:
1456:
1451:
1443:
1442:
1423:
1418:
1395:
1358:
1356:
1355:
1350:
1311:
1305:
1288:
1286:
1285:
1280:
1266:
1265:
1253:
1252:
1215:
1198:
1133:principal bundle
1127:of the manifold
1111:The torsion form
1102:
1100:
1099:
1094:
1092:
1091:
1077:
1074:
1073:
1058:
1057:
1049:
1046:
1045:
1029:
1027:
1026:
1021:
1013:
1012:
1004:
1001:
1000:
976:
974:
973:
968:
966:
965:
957:
956:
955:
934:
932:
931:
926:
878:
877:
869:
866:
865:
853:
852:
844:
841:
840:
828:
827:
819:
816:
815:
803:
802:
794:
791:
790:
771:
753:
738:
719:
696:
694:
693:
688:
686:
685:
677:
674:
673:
583:
581:
580:
575:
552:
551:
536:
535:
415:Finsler geometry
389:
387:
386:
381:
363:
361:
360:
355:
328:
326:
325:
320:
274:
272:
271:
266:
254:
252:
251:
246:
234:
232:
231:
226:
211:
209:
208:
203:
167:
165:
164:
159:
147:
145:
144:
139:
137:
136:
124:
123:
107:
106:
97:
96:
95:
94:
73:
71:
70:
65:
63:
62:
57:
21:
8889:
8888:
8884:
8883:
8882:
8880:
8879:
8878:
8849:
8848:
8847:
8842:
8793:Albert Einstein
8760:
8741:Einstein tensor
8704:
8685:Ricci curvature
8665:Kronecker delta
8651:Notable tensors
8646:
8567:Connection form
8544:
8538:
8469:
8455:Tensor operator
8412:
8406:
8346:
8322:Computer vision
8315:
8297:
8293:Tensor calculus
8237:
8226:
8221:
8191:
8186:
8153:
8139:Ricci curvature
8115:
8067:
8060:
8051:Total curvature
8017:
8010:
8000:
7958:
7949:
7933:
7908:Rev. Mod. Phys.
7898:
7885:Schrödinger, E.
7883:
7878:Springer-Verlag
7868:
7841:Poplawski, N.J.
7839:
7834:
7819:
7787:
7761:Rev. Mod. Phys.
7754:
7747:
7741:Springer-Verlag
7732:
7727:
7691:
7662:
7633:
7618:
7615:
7610:
7609:
7604:
7600:
7591:
7587:
7579:
7575:
7566:
7562:
7543:
7542:
7538:
7529:
7525:
7517:
7513:
7504:
7500:
7492:
7488:
7480:
7476:
7468:
7464:
7456:
7452:
7444:
7440:
7432:
7428:
7420:
7416:
7408:
7404:
7399:
7371:Curtright field
7362:
7312:
7270:
7248:
7244:
7206:
7205:
7143:
7139:
7101:
7100:
7043:
7039:
7001:
7000:
6981:
6894:
6868:
6867:
6854:
6794:
6793:
6786:
6705:
6700:
6699:
6673:
6644:
6639:
6638:
6614:
6601:
6588:
6575:
6559:
6551:
6550:
6523:
6515:
6514:
6490:
6459:
6423:
6422:
6398:
6379:
6360:
6355:
6354:
6328:
6309:
6284:
6279:
6278:
6247:
6242:
6241:
6220:
6204:
6199:
6198:
6167:
6166:
6147:
6146:
6132:
6112:
6066:
6065:
6028:
6027:
5990:
5989:
5957:
5956:
5920:
5919:
5900:
5899:
5865:
5864:
5840:
5830:
5813:
5812:
5788:
5771:
5770:
5745:
5744:
5676:
5675:
5614:
5613:
5585:
5584:
5560:
5550:
5537:
5529:
5528:
5498:
5485:
5480:
5479:
5455:
5450:
5449:
5421:
5420:
5401:
5400:
5370:
5365:
5364:
5340:
5335:
5334:
5313:
5308:
5307:
5288:
5287:
5266:
5261:
5260:
5241:
5240:
5221:
5220:
5171:
5170:
5145:
5144:
5101:
5100:
5097:
5064:
5041:
5018:
5009:
5008:
4987:
4982:
4981:
4962:
4961:
4937:
4914:
4893:
4892:
4834:
4833:
4819:
4818:
4805:
4768:
4737:
4736:
4726:
4713:
4697:
4684:
4668:
4643:
4613:
4608:
4603:
4576:
4575:
4554:
4534:
4533:
4532:thus satisfies
4512:
4487:
4455:
4454:
4433:
4428:
4427:
4406:
4401:
4400:
4376:
4363:
4350:
4338:
4333:
4328:
4327:
4302:
4289:
4276:
4271:
4270:
4247:
4236:
4235:
4233:Euclidean space
4229:
4154:
4153:
4111:
4110:
4082:affine manifold
4078:
4066:
4055:
4038:
4037:
4002:
3977:
3976:
3972:
3965:
3961:
3951:
3930:
3929:
3899:
3898:
3894:
3862:
3837:
3836:
3832:
3825:
3821:
3811:
3781:
3780:
3772:
3735:
3734:
3699:
3698:
3639:
3638:
3620:
3565:
3561:
3557:
3553:
3532:
3528:
3510:
3509:
3505:
3504:
3500:
3488:
3487:
3448:
3444:
3426:
3425:
3421:
3388:
3384:
3380:
3376:
3343:
3339:
3335:
3331:
3319:
3318:
3198:
3194:
3190:
3186:
3174:
3173:
3131:
3130:
3081:
3065:
3055:
3039:
3029:
3000:
2999:
2968:
2961:
2902:
2878:
2873:
2872:
2781:
2780:
2708:
2707:
2692:
2670:
2599:
2598:
2594:
2577:
2576:
2567:Kronecker delta
2563:
2532:
2523:
2513:
2500:
2482:
2473:
2463:
2450:
2430:
2420:
2405:
2395:
2390:
2389:
2357:
2347:
2334:
2329:
2328:
2320:, the trace of
2306:
2298:connection form
2255:
2254:
2195:
2194:
2190:
2174:
2173:
2155:
2136:
2109:
2079:
2075:
2074:
2072:
2050:
2045:
2044:
2039:
2001:
1999:
1987:
1963:
1958:
1957:
1915:
1900:
1899:
1895:
1880:
1866:
1861:
1846:
1832:
1827:
1826:
1822:
1812:
1789:
1787:
1782:
1781:
1751:
1738:
1718:
1716:
1703:
1683:
1681:
1668:
1652:
1647:
1646:
1640:Kronecker delta
1635:
1629:
1619:
1617:
1600:
1559:
1557:
1545:
1530:
1522:
1521:
1514:
1505:
1498:
1491:connection form
1487:
1485:connection form
1481:
1431:
1405:
1404:
1386:
1381:with values in
1317:
1316:
1307:
1293:
1257:
1241:
1221:
1220:
1211:
1200:
1194:
1187:
1137:connection form
1113:
1075:
1065:
1047:
1037:
1032:
1031:
1002:
992:
987:
986:
947:
945:
940:
939:
867:
857:
842:
832:
817:
807:
792:
782:
777:
776:
769:
760:
755:
752:
740:
737:
725:
717:
708:
701:
675:
665:
660:
659:
656:
633:smooth function
543:
527:
501:
500:
481:) of ∇ is the
447:
366:
365:
331:
330:
305:
304:
257:
256:
237:
236:
214:
213:
170:
169:
150:
149:
128:
115:
98:
86:
81:
76:
75:
52:
47:
46:
39:
28:
23:
22:
15:
12:
11:
5:
8887:
8885:
8877:
8876:
8871:
8866:
8861:
8851:
8850:
8844:
8843:
8841:
8840:
8835:
8833:Woldemar Voigt
8830:
8825:
8820:
8815:
8810:
8805:
8800:
8798:Leonhard Euler
8795:
8790:
8785:
8780:
8774:
8772:
8770:Mathematicians
8766:
8765:
8762:
8761:
8759:
8758:
8753:
8748:
8743:
8738:
8733:
8728:
8723:
8718:
8712:
8710:
8706:
8705:
8703:
8702:
8697:
8695:Torsion tensor
8692:
8687:
8682:
8677:
8672:
8667:
8661:
8659:
8652:
8648:
8647:
8645:
8644:
8639:
8634:
8629:
8624:
8619:
8614:
8609:
8604:
8599:
8594:
8589:
8584:
8579:
8574:
8569:
8564:
8559:
8554:
8548:
8546:
8540:
8539:
8537:
8536:
8530:
8528:Tensor product
8525:
8520:
8518:Symmetrization
8515:
8510:
8508:Lie derivative
8505:
8500:
8495:
8490:
8485:
8479:
8477:
8471:
8470:
8468:
8467:
8462:
8457:
8452:
8447:
8442:
8437:
8432:
8430:Tensor density
8427:
8422:
8416:
8414:
8408:
8407:
8405:
8404:
8402:Voigt notation
8399:
8394:
8389:
8387:Ricci calculus
8384:
8379:
8374:
8372:Index notation
8369:
8364:
8358:
8356:
8352:
8351:
8348:
8347:
8345:
8344:
8339:
8334:
8329:
8324:
8318:
8316:
8314:
8313:
8308:
8302:
8299:
8298:
8296:
8295:
8290:
8288:Tensor algebra
8285:
8280:
8275:
8270:
8268:Dyadic algebra
8265:
8260:
8254:
8252:
8243:
8239:
8238:
8231:
8228:
8227:
8222:
8220:
8219:
8212:
8205:
8197:
8188:
8187:
8185:
8184:
8179:
8174:
8172:Torsion tensor
8169:
8167:Curvature form
8163:
8161:
8155:
8154:
8152:
8151:
8146:
8141:
8136:
8131:
8125:
8123:
8117:
8116:
8114:
8113:
8108:
8103:
8098:
8093:
8088:
8086:Mean curvature
8083:
8078:
8072:
8070:
8062:
8061:
8059:
8058:
8053:
8048:
8043:
8038:
8033:
8028:
8022:
8020:
8012:
8011:
8001:
7999:
7998:
7991:
7984:
7976:
7970:
7969:
7957:
7956:External links
7954:
7953:
7952:
7947:
7931:
7896:
7881:
7874:Ricci Calculus
7870:Schouten, J.A.
7866:
7837:
7832:
7817:
7799:(2): 212–221,
7793:J. Math. Phys.
7789:Kibble, T.W.B.
7785:
7767:(3): 393–416,
7752:
7725:
7700:(4): 347–357,
7689:
7660:
7631:
7614:
7611:
7608:
7607:
7598:
7585:
7573:
7560:
7536:
7523:
7511:
7498:
7486:
7474:
7462:
7450:
7438:
7426:
7414:
7401:
7400:
7398:
7395:
7394:
7393:
7388:
7383:
7378:
7373:
7368:
7361:
7358:
7342:
7341:
7328:
7327:
7309:
7296:
7292:
7289:
7286:
7283:
7280:
7276:
7273:
7269:
7266:
7263:
7260:
7257:
7254:
7251:
7247:
7240:
7237:
7231:
7228:
7225:
7222:
7219:
7216:
7213:
7199:
7198:
7186:
7182:
7179:
7176:
7173:
7170:
7167:
7164:
7161:
7158:
7155:
7152:
7149:
7146:
7142:
7135:
7132:
7126:
7123:
7120:
7117:
7114:
7111:
7108:
7098:
7086:
7082:
7079:
7076:
7073:
7070:
7067:
7064:
7061:
7058:
7055:
7052:
7049:
7046:
7042:
7035:
7032:
7026:
7023:
7020:
7017:
7014:
7011:
7008:
6955:
6954:
6940:
6937:
6930:
6926:
6922:
6918:
6912:
6909:
6901:
6897:
6893:
6890:
6887:
6884:
6881:
6878:
6875:
6843:
6842:
6835:geodesic spray
6819:
6816:
6813:
6807:
6804:
6771:
6770:
6759:
6756:
6753:
6750:
6747:
6741:
6738:
6730:
6727:
6724:
6718:
6715:
6708:
6695:provided that
6672:
6669:
6654:
6651:
6647:
6626:
6621:
6617:
6613:
6608:
6604:
6600:
6595:
6591:
6587:
6582:
6578:
6574:
6569:
6566:
6562:
6558:
6538:
6535:
6530:
6526:
6522:
6502:
6497:
6493:
6489:
6484:
6479:
6475:
6471:
6466:
6462:
6458:
6454:
6451:
6448:
6443:
6438:
6434:
6430:
6410:
6405:
6401:
6397:
6392:
6389:
6386:
6382:
6378:
6375:
6370:
6367:
6363:
6343:
6338:
6335:
6331:
6327:
6322:
6319:
6316:
6312:
6305:
6302:
6296:
6291:
6287:
6260:
6257:
6254:
6250:
6227:
6223:
6219:
6216:
6211:
6207:
6184:
6179:
6175:
6154:
6136:fluid dynamics
6131:
6128:
6111:
6108:
6091:
6088:
6085:
6079:
6076:
6053:
6050:
6047:
6041:
6038:
6015:
6012:
6009:
6006:
6003:
6000:
5997:
5970:
5967:
5946:Burgers vector
5933:
5930:
5927:
5907:
5887:
5884:
5881:
5878:
5875:
5872:
5852:
5847:
5843:
5837:
5833:
5829:
5826:
5823:
5820:
5800:
5795:
5791:
5787:
5784:
5781:
5778:
5758:
5755:
5752:
5725:
5722:
5719:
5713:
5710:
5704:
5701:
5698:
5695:
5689:
5686:
5663:
5660:
5657:
5651:
5648:
5642:
5639:
5636:
5633:
5627:
5624:
5598:
5595:
5572:
5567:
5563:
5557:
5553:
5549:
5544:
5540:
5536:
5516:
5513:
5510:
5505:
5501:
5497:
5492:
5488:
5467:
5462:
5458:
5434:
5431:
5408:
5388:
5385:
5382:
5377:
5373:
5352:
5347:
5343:
5320:
5316:
5295:
5273:
5269:
5248:
5228:
5208:
5205:
5202:
5199:
5196:
5193:
5190:
5187:
5184:
5181:
5178:
5158:
5155:
5152:
5132:
5129:
5126:
5123:
5120:
5117:
5114:
5111:
5108:
5096:
5093:
5076:
5071:
5067:
5062:
5059:
5056:
5053:
5048:
5044:
5039:
5036:
5033:
5030:
5025:
5021:
5016:
4994:
4990:
4969:
4944:
4940:
4935:
4932:
4929:
4926:
4921:
4917:
4912:
4909:
4906:
4903:
4900:
4880:
4877:
4874:
4868:
4865:
4859:
4856:
4853:
4847:
4844:
4817:
4812:
4808:
4804:
4801:
4798:
4792:
4789:
4783:
4780:
4775:
4771:
4767:
4764:
4761:
4755:
4752:
4746:
4743:
4740:
4738:
4733:
4729:
4725:
4720:
4716:
4712:
4709:
4704:
4700:
4696:
4691:
4687:
4683:
4680:
4675:
4671:
4664:
4661:
4655:
4650:
4646:
4639:
4636:
4630:
4627:
4620:
4616:
4611:
4607:
4604:
4599:
4596:
4590:
4587:
4584:
4583:
4561:
4557:
4553:
4550:
4547:
4544:
4541:
4519:
4515:
4511:
4508:
4505:
4502:
4499:
4494:
4490:
4486:
4483:
4480:
4477:
4474:
4471:
4468:
4465:
4462:
4440:
4436:
4413:
4409:
4388:
4383:
4379:
4375:
4370:
4366:
4362:
4357:
4353:
4345:
4341:
4336:
4309:
4305:
4301:
4296:
4292:
4288:
4283:
4279:
4256:
4251:
4246:
4243:
4228:
4225:
4189:skew symmetric
4176:
4173:
4170:
4167:
4164:
4161:
4133:
4130:
4127:
4124:
4121:
4118:
4077:
4074:
4064:
4052:
4051:
4036:
4032:
4027:
4023:
4020:
4017:
4012:
4009:
4005:
4001:
3998:
3995:
3992:
3987:
3984:
3980:
3975:
3971:
3968:
3964:
3960:
3957:
3954:
3952:
3950:
3947:
3944:
3941:
3938:
3935:
3932:
3931:
3928:
3924:
3920:
3917:
3914:
3909:
3906:
3902:
3897:
3892:
3887:
3883:
3880:
3877:
3872:
3869:
3865:
3861:
3858:
3855:
3852:
3847:
3844:
3840:
3835:
3831:
3828:
3824:
3820:
3817:
3814:
3812:
3810:
3807:
3804:
3801:
3798:
3795:
3792:
3789:
3788:
3770:
3763:
3762:
3751:
3748:
3745:
3742:
3732:
3721:
3718:
3715:
3712:
3709:
3706:
3690:where, again,
3688:
3687:
3676:
3673:
3670:
3667:
3664:
3661:
3658:
3655:
3652:
3649:
3646:
3624:curvature form
3619:
3616:
3615:
3614:
3613:
3612:
3601:
3598:
3594:
3589:
3585:
3582:
3578:
3574:
3571:
3568:
3564:
3560:
3556:
3552:
3549:
3545:
3541:
3538:
3535:
3531:
3526:
3522:
3517:
3513:
3508:
3503:
3497:
3480:
3479:
3478:
3466:
3461:
3457:
3454:
3451:
3447:
3442:
3438:
3433:
3429:
3424:
3420:
3416:
3412:
3409:
3406:
3403:
3400:
3397:
3394:
3391:
3387:
3383:
3379:
3373:
3368:
3364:
3360:
3356:
3352:
3349:
3346:
3342:
3338:
3334:
3328:
3307:
3306:
3295:
3292:
3289:
3286:
3283:
3280:
3277:
3274:
3271:
3268:
3265:
3262:
3259:
3256:
3253:
3250:
3247:
3244:
3241:
3238:
3235:
3232:
3229:
3226:
3223:
3219:
3215:
3211:
3207:
3204:
3201:
3197:
3193:
3189:
3183:
3140:
3120:
3119:
3108:
3105:
3100:
3097:
3094:
3091:
3088:
3084:
3080:
3077:
3072:
3068:
3062:
3058:
3054:
3051:
3046:
3042:
3036:
3032:
3028:
3025:
3022:
3019:
3016:
3013:
3010:
3007:
2960:
2957:
2945:
2944:
2933:
2930:
2927:
2923:
2920:
2917:
2911:
2908:
2905:
2901:
2896:
2893:
2890:
2885:
2881:
2862:
2861:
2850:
2847:
2844:
2841:
2838:
2835:
2832:
2829:
2826:
2814:
2807:
2804:
2801:
2798:
2795:
2791:
2788:
2766:
2765:
2754:
2751:
2748:
2745:
2742:
2739:
2736:
2733:
2730:
2727:
2724:
2721:
2718:
2715:
2655:
2654:
2643:
2639:
2635:
2630:
2625:
2622:
2617:
2610:
2604:
2597:
2593:
2590:
2587:
2584:
2561:
2556:
2555:
2544:
2539:
2535:
2529:
2520:
2516:
2509:
2506:
2503:
2499:
2494:
2489:
2485:
2479:
2470:
2466:
2459:
2456:
2453:
2449:
2444:
2439:
2436:
2427:
2423:
2419:
2414:
2411:
2402:
2398:
2383:
2382:
2371:
2366:
2363:
2354:
2350:
2346:
2341:
2337:
2318:index notation
2305:
2302:
2286:
2285:
2274:
2271:
2268:
2265:
2262:
2248:
2247:
2235:
2231:
2226:
2221:
2218:
2213:
2206:
2200:
2193:
2184:
2181:
2133:
2132:
2121:
2116:
2112:
2106:
2099:
2094:
2089:
2086:
2082:
2078:
2071:
2066:
2059:
2056:
2035:
2029:
2028:
2015:
2008:
2004:
1996:
1991:
1986:
1981:
1974:
1970:
1948:
1947:
1935:
1930:
1924:
1919:
1914:
1909:
1904:
1898:
1894:
1889:
1884:
1875:
1870:
1864:
1860:
1855:
1850:
1841:
1836:
1830:
1825:
1819:
1815:
1811:
1806:
1803:
1796:
1792:
1775:
1774:
1763:
1758:
1754:
1750:
1745:
1741:
1735:
1732:
1725:
1721:
1715:
1710:
1706:
1702:
1697:
1690:
1686:
1680:
1675:
1671:
1667:
1664:
1659:
1655:
1633:
1627:
1613:
1590:
1589:
1578:
1573:
1566:
1562:
1554:
1549:
1544:
1539:
1534:
1529:
1510:
1503:
1480:
1477:
1461:
1460:
1449:
1446:
1441:
1438:
1434:
1430:
1427:
1422:
1417:
1413:
1396:it transforms
1379:tensorial form
1360:
1359:
1348:
1345:
1342:
1339:
1336:
1333:
1330:
1327:
1324:
1290:
1289:
1278:
1275:
1272:
1269:
1264:
1260:
1256:
1251:
1248:
1244:
1240:
1237:
1234:
1231:
1228:
1209:
1192:
1112:
1109:
1090:
1087:
1084:
1081:
1072:
1068:
1064:
1061:
1056:
1053:
1044:
1040:
1019:
1016:
1011:
1008:
999:
995:
964:
961:
954:
950:
936:
935:
924:
921:
918:
915:
912:
909:
906:
903:
900:
897:
894:
891:
888:
885:
881:
876:
873:
864:
860:
856:
851:
848:
839:
835:
831:
826:
823:
814:
810:
806:
801:
798:
789:
785:
765:
758:
748:
733:
713:
706:
684:
681:
672:
668:
655:
652:
587:where is the
585:
584:
573:
570:
567:
564:
561:
558:
555:
550:
546:
542:
539:
534:
530:
526:
523:
520:
517:
514:
511:
508:
467:torsion tensor
459:tangent bundle
446:
443:
392:skew symmetric
379:
376:
373:
353:
350:
347:
344:
341:
338:
318:
315:
312:
289:torsion tensor
264:
244:
224:
221:
201:
198:
195:
192:
189:
186:
183:
180:
177:
157:
135:
131:
127:
122:
118:
113:
110:
105:
101:
93:
89:
84:
61:
56:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
8886:
8875:
8872:
8870:
8867:
8865:
8862:
8860:
8857:
8856:
8854:
8839:
8836:
8834:
8831:
8829:
8826:
8824:
8821:
8819:
8816:
8814:
8811:
8809:
8806:
8804:
8801:
8799:
8796:
8794:
8791:
8789:
8786:
8784:
8781:
8779:
8776:
8775:
8773:
8771:
8767:
8757:
8754:
8752:
8749:
8747:
8744:
8742:
8739:
8737:
8734:
8732:
8729:
8727:
8724:
8722:
8719:
8717:
8714:
8713:
8711:
8707:
8701:
8698:
8696:
8693:
8691:
8688:
8686:
8683:
8681:
8678:
8676:
8675:Metric tensor
8673:
8671:
8668:
8666:
8663:
8662:
8660:
8656:
8653:
8649:
8643:
8640:
8638:
8635:
8633:
8630:
8628:
8625:
8623:
8620:
8618:
8615:
8613:
8610:
8608:
8605:
8603:
8600:
8598:
8595:
8593:
8590:
8588:
8587:Exterior form
8585:
8583:
8580:
8578:
8575:
8573:
8570:
8568:
8565:
8563:
8560:
8558:
8555:
8553:
8550:
8549:
8547:
8541:
8534:
8531:
8529:
8526:
8524:
8521:
8519:
8516:
8514:
8511:
8509:
8506:
8504:
8501:
8499:
8496:
8494:
8491:
8489:
8486:
8484:
8481:
8480:
8478:
8476:
8472:
8466:
8463:
8461:
8460:Tensor bundle
8458:
8456:
8453:
8451:
8448:
8446:
8443:
8441:
8438:
8436:
8433:
8431:
8428:
8426:
8423:
8421:
8418:
8417:
8415:
8409:
8403:
8400:
8398:
8395:
8393:
8390:
8388:
8385:
8383:
8380:
8378:
8375:
8373:
8370:
8368:
8365:
8363:
8360:
8359:
8357:
8353:
8343:
8340:
8338:
8335:
8333:
8330:
8328:
8325:
8323:
8320:
8319:
8317:
8312:
8309:
8307:
8304:
8303:
8300:
8294:
8291:
8289:
8286:
8284:
8281:
8279:
8276:
8274:
8271:
8269:
8266:
8264:
8261:
8259:
8256:
8255:
8253:
8251:
8247:
8244:
8240:
8236:
8235:
8229:
8225:
8218:
8213:
8211:
8206:
8204:
8199:
8198:
8195:
8183:
8180:
8178:
8175:
8173:
8170:
8168:
8165:
8164:
8162:
8160:
8156:
8150:
8147:
8145:
8142:
8140:
8137:
8135:
8132:
8130:
8127:
8126:
8124:
8122:
8118:
8112:
8109:
8107:
8104:
8102:
8099:
8097:
8094:
8092:
8091:Darboux frame
8089:
8087:
8084:
8082:
8079:
8077:
8074:
8073:
8071:
8069:
8063:
8057:
8054:
8052:
8049:
8047:
8044:
8042:
8039:
8037:
8034:
8032:
8029:
8027:
8024:
8023:
8021:
8019:
8013:
8009:
8005:
7997:
7992:
7990:
7985:
7983:
7978:
7977:
7974:
7967:
7963:
7962:Bill Thurston
7960:
7959:
7955:
7950:
7948:0-914098-71-3
7944:
7940:
7936:
7932:
7929:
7925:
7921:
7917:
7913:
7909:
7905:
7901:
7897:
7894:
7890:
7886:
7882:
7879:
7875:
7871:
7867:
7864:
7860:
7855:
7850:
7846:
7842:
7838:
7835:
7833:0-471-15733-3
7829:
7825:
7824:
7818:
7814:
7810:
7806:
7802:
7798:
7794:
7790:
7786:
7782:
7778:
7774:
7770:
7766:
7762:
7758:
7753:
7750:on 2006-12-29
7746:
7742:
7738:
7731:
7726:
7723:
7719:
7715:
7711:
7707:
7703:
7699:
7695:
7690:
7686:
7681:
7677:
7673:
7669:
7665:
7661:
7657:
7652:
7648:
7644:
7640:
7636:
7632:
7629:
7625:
7621:
7617:
7616:
7612:
7602:
7599:
7595:
7589:
7586:
7582:
7577:
7574:
7570:
7564:
7561:
7556:
7552:
7551:
7546:
7540:
7537:
7533:
7527:
7524:
7520:
7515:
7512:
7508:
7502:
7499:
7495:
7490:
7487:
7483:
7478:
7475:
7471:
7466:
7463:
7459:
7454:
7451:
7447:
7442:
7439:
7435:
7430:
7427:
7423:
7418:
7415:
7411:
7406:
7403:
7396:
7392:
7389:
7387:
7384:
7382:
7379:
7377:
7374:
7372:
7369:
7367:
7364:
7363:
7359:
7357:
7355:
7351:
7347:
7339:
7334:
7333:
7332:
7323:
7319:
7315:
7310:
7294:
7287:
7284:
7281:
7274:
7271:
7267:
7261:
7258:
7255:
7249:
7245:
7238:
7235:
7229:
7223:
7220:
7217:
7211:
7204:
7203:
7202:
7184:
7177:
7174:
7171:
7162:
7156:
7153:
7150:
7140:
7133:
7130:
7124:
7118:
7115:
7112:
7106:
7099:
7084:
7077:
7074:
7071:
7062:
7056:
7053:
7050:
7040:
7033:
7030:
7024:
7018:
7015:
7012:
7006:
6999:
6998:
6997:
6995:
6991:
6987:
6980:
6976:
6972:
6968:
6964:
6960:
6935:
6928:
6924:
6916:
6907:
6899:
6891:
6885:
6882:
6879:
6866:
6865:
6864:
6861:
6857:
6852:
6848:
6840:
6839:
6838:
6836:
6831:
6814:
6805:
6802:
6789:
6784:
6780:
6776:
6773:for all time
6757:
6754:
6748:
6739:
6736:
6725:
6716:
6713:
6698:
6697:
6696:
6694:
6690:
6686:
6682:
6678:
6675:Suppose that
6670:
6668:
6652:
6649:
6645:
6624:
6619:
6615:
6611:
6606:
6602:
6598:
6593:
6589:
6585:
6580:
6576:
6572:
6567:
6564:
6560:
6556:
6536:
6533:
6528:
6524:
6520:
6500:
6495:
6491:
6487:
6482:
6477:
6469:
6464:
6456:
6452:
6449:
6446:
6441:
6436:
6428:
6408:
6403:
6395:
6390:
6387:
6384:
6380:
6376:
6373:
6368:
6365:
6361:
6341:
6336:
6333:
6325:
6320:
6317:
6314:
6310:
6303:
6300:
6294:
6289:
6285:
6276:
6258:
6255:
6252:
6248:
6225:
6221:
6217:
6214:
6209:
6182:
6177:
6152:
6143:
6141:
6137:
6129:
6127:
6125:
6121:
6117:
6109:
6107:
6105:
6102:comprise the
6086:
6074:
6048:
6036:
6013:
6010:
6007:
6004:
6001:
5998:
5995:
5987:
5965:
5954:
5949:
5947:
5931:
5928:
5925:
5882:
5879:
5876:
5850:
5845:
5841:
5835:
5827:
5824:
5821:
5818:
5798:
5793:
5789:
5785:
5782:
5779:
5776:
5769:, with sides
5756:
5753:
5750:
5741:
5739:
5720:
5708:
5702:
5696:
5684:
5658:
5646:
5640:
5634:
5622:
5593:
5570:
5565:
5561:
5555:
5551:
5547:
5542:
5538:
5534:
5511:
5503:
5499:
5495:
5490:
5486:
5465:
5460:
5456:
5429:
5406:
5383:
5375:
5371:
5350:
5345:
5341:
5318:
5314:
5293:
5271:
5267:
5246:
5226:
5206:
5203:
5197:
5191:
5188:
5182:
5176:
5156:
5153:
5150:
5130:
5121:
5118:
5115:
5109:
5106:
5094:
5092:
5090:
5074:
5069:
5065:
5060:
5057:
5054:
5051:
5046:
5042:
5037:
5034:
5031:
5028:
5023:
5019:
5014:
4992:
4988:
4967:
4958:
4942:
4938:
4933:
4930:
4927:
4924:
4919:
4915:
4910:
4907:
4904:
4901:
4898:
4878:
4875:
4872:
4866:
4863:
4857:
4854:
4851:
4845:
4842:
4815:
4810:
4806:
4799:
4796:
4790:
4787:
4778:
4773:
4769:
4762:
4759:
4753:
4750:
4741:
4731:
4727:
4723:
4718:
4714:
4710:
4707:
4702:
4698:
4694:
4689:
4685:
4681:
4678:
4673:
4669:
4662:
4659:
4653:
4648:
4644:
4637:
4634:
4628:
4625:
4618:
4614:
4605:
4597:
4594:
4588:
4585:
4559:
4555:
4551:
4545:
4539:
4517:
4513:
4506:
4500:
4497:
4492:
4488:
4481:
4475:
4472:
4466:
4460:
4438:
4434:
4411:
4407:
4386:
4381:
4377:
4373:
4368:
4364:
4360:
4355:
4351:
4343:
4339:
4325:
4324:cross product
4307:
4303:
4299:
4294:
4290:
4286:
4281:
4277:
4254:
4244:
4241:
4234:
4226:
4224:
4222:
4218:
4214:
4210:
4206:
4202:
4198:
4194:
4190:
4171:
4168:
4165:
4159:
4151:
4147:
4128:
4125:
4122:
4116:
4107:
4106:tangent space
4103:
4099:
4094:
4091:
4085:
4083:
4075:
4073:
4069:
4062:
4058:
4034:
4030:
4025:
4018:
4010:
4007:
4003:
3999:
3993:
3985:
3982:
3978:
3973:
3966:
3962:
3958:
3955:
3953:
3945:
3942:
3939:
3933:
3926:
3922:
3915:
3907:
3904:
3900:
3895:
3890:
3885:
3878:
3870:
3867:
3863:
3859:
3853:
3845:
3842:
3838:
3833:
3826:
3822:
3818:
3815:
3813:
3808:
3802:
3799:
3796:
3790:
3779:
3778:
3777:
3775:
3768:
3749:
3746:
3740:
3733:
3719:
3716:
3710:
3704:
3697:
3696:
3695:
3693:
3674:
3671:
3668:
3665:
3662:
3659:
3656:
3653:
3650:
3647:
3637:
3636:
3635:
3633:
3629:
3625:
3617:
3599:
3596:
3592:
3587:
3583:
3580:
3576:
3572:
3569:
3566:
3562:
3558:
3554:
3550:
3547:
3543:
3539:
3536:
3533:
3529:
3524:
3520:
3515:
3506:
3501:
3486:
3485:
3484:
3481:
3464:
3459:
3455:
3452:
3449:
3445:
3440:
3436:
3431:
3422:
3418:
3414:
3410:
3407:
3401:
3398:
3395:
3389:
3385:
3381:
3377:
3366:
3362:
3358:
3354:
3350:
3347:
3344:
3340:
3336:
3332:
3317:
3316:
3315:
3312:
3311:
3310:
3293:
3290:
3284:
3281:
3278:
3272:
3269:
3266:
3260:
3257:
3254:
3248:
3245:
3242:
3236:
3233:
3230:
3224:
3221:
3217:
3213:
3209:
3205:
3202:
3199:
3195:
3191:
3187:
3172:
3171:
3170:
3168:
3164:
3160:
3156:
3128:
3123:
3106:
3103:
3095:
3092:
3089:
3078:
3075:
3070:
3060:
3052:
3049:
3044:
3034:
3026:
3023:
3017:
3014:
3011:
3005:
2998:
2997:
2996:
2994:
2990:
2986:
2980:
2976:
2972:
2966:
2958:
2956:
2954:
2950:
2931:
2925:
2921:
2915:
2909:
2906:
2903:
2899:
2894:
2891:
2888:
2883:
2879:
2871:
2870:
2869:
2867:
2848:
2839:
2833:
2827:
2824:
2812:
2802:
2793:
2789:
2779:
2778:
2777:
2775:
2771:
2752:
2746:
2743:
2740:
2734:
2728:
2725:
2719:
2713:
2706:
2705:
2704:
2700:
2696:
2690:
2686:
2682:
2677:
2673:
2668:
2664:
2660:
2657:The trace of
2641:
2637:
2633:
2623:
2620:
2608:
2602:
2595:
2591:
2588:
2585:
2582:
2575:
2574:
2573:
2570:
2568:
2564:
2542:
2537:
2533:
2527:
2518:
2514:
2507:
2504:
2501:
2497:
2492:
2487:
2483:
2477:
2468:
2464:
2457:
2454:
2451:
2447:
2442:
2437:
2434:
2425:
2421:
2417:
2412:
2409:
2400:
2396:
2388:
2387:
2386:
2369:
2364:
2361:
2352:
2348:
2344:
2339:
2335:
2327:
2326:
2325:
2323:
2319:
2315:
2311:
2303:
2301:
2299:
2295:
2291:
2272:
2269:
2266:
2263:
2253:
2252:
2251:
2233:
2229:
2219:
2216:
2204:
2198:
2191:
2182:
2172:
2171:
2170:
2167:
2163:
2159:
2153:
2149:
2145:
2140:
2119:
2114:
2104:
2097:
2092:
2087:
2084:
2080:
2076:
2069:
2064:
2043:
2042:
2041:
2038:
2034:
2013:
2006:
2002:
1994:
1984:
1979:
1956:
1955:
1954:
1951:
1933:
1928:
1922:
1912:
1907:
1896:
1892:
1887:
1873:
1858:
1853:
1839:
1823:
1817:
1813:
1809:
1804:
1801:
1794:
1790:
1780:
1779:
1778:
1761:
1756:
1752:
1748:
1743:
1739:
1733:
1730:
1723:
1719:
1713:
1708:
1704:
1700:
1695:
1688:
1684:
1678:
1673:
1669:
1665:
1662:
1657:
1645:
1644:
1643:
1641:
1636:
1626:
1622:
1616:
1612:
1607:
1603:
1599:
1595:
1576:
1571:
1564:
1560:
1552:
1542:
1537:
1527:
1520:
1519:
1518:
1513:
1509:
1502:
1496:
1492:
1486:
1478:
1476:
1474:
1470:
1466:
1444:
1439:
1436:
1432:
1428:
1420:
1415:
1411:
1403:
1402:
1401:
1399:
1398:equivariantly
1393:
1389:
1384:
1380:
1375:
1373:
1369:
1365:
1346:
1343:
1340:
1337:
1334:
1331:
1328:
1325:
1315:
1314:
1313:
1310:
1304:
1300:
1296:
1270:
1262:
1258:
1249:
1246:
1242:
1238:
1232:
1226:
1219:
1218:
1217:
1214:
1207:
1203:
1197:
1190:
1185:
1181:
1177:
1173:
1169:
1165:
1161:
1157:
1153:
1149:
1145:
1141:
1138:
1134:
1130:
1126:
1122:
1118:
1110:
1108:
1106:
1085:
1082:
1070:
1062:
1059:
1054:
1051:
1042:
1038:
1017:
1014:
1009:
1006:
997:
993:
984:
980:
962:
959:
952:
922:
919:
916:
913:
910:
907:
904:
901:
898:
895:
892:
889:
886:
883:
879:
874:
871:
862:
858:
854:
849:
846:
837:
829:
824:
821:
812:
804:
799:
796:
787:
783:
775:
774:
773:
768:
764:
761:
751:
747:
743:
736:
732:
728:
723:
716:
712:
705:
700:
682:
679:
670:
666:
653:
651:
649:
645:
641:
637:
634:
630:
626:
622:
618:
614:
610:
606:
602:
598:
594:
590:
568:
565:
562:
556:
553:
548:
540:
537:
532:
524:
518:
515:
512:
506:
499:
498:
497:
495:
491:
488:
487:vector fields
484:
480:
476:
472:
468:
464:
460:
456:
452:
444:
442:
440:
436:
432:
428:
424:
420:
416:
412:
408:
404:
399:
397:
393:
377:
374:
371:
348:
345:
342:
336:
316:
313:
310:
302:
298:
294:
290:
286:
278:
277:Frenet-Serret
262:
242:
222:
219:
199:
196:
193:
190:
187:
184:
181:
178:
175:
155:
133:
129:
125:
120:
116:
111:
108:
103:
99:
91:
87:
59:
43:
37:
33:
19:
8838:Hermann Weyl
8694:
8642:Vector space
8627:Pseudotensor
8592:Fiber bundle
8545:abstractions
8440:Mixed tensor
8425:Tensor field
8232:
8171:
7938:
7911:
7907:
7900:Sciama, D.W.
7888:
7873:
7844:
7821:
7796:
7792:
7764:
7760:
7745:the original
7736:
7697:
7693:
7675:
7671:
7646:
7642:
7623:
7620:Bishop, R.L.
7601:
7593:
7588:
7576:
7563:
7548:
7539:
7526:
7514:
7501:
7489:
7477:
7465:
7453:
7441:
7429:
7417:
7405:
7349:
7343:
7329:
7321:
7317:
7313:
7200:
6993:
6989:
6985:
6978:
6974:
6966:
6962:
6958:
6956:
6859:
6855:
6850:
6846:
6844:
6832:
6787:
6782:
6778:
6774:
6772:
6692:
6688:
6684:
6680:
6676:
6674:
6144:
6140:vortex lines
6133:
6113:
5985:
5953:moving frame
5950:
5742:
5098:
4959:
4230:
4216:
4196:
4192:
4149:
4145:
4101:
4097:
4095:
4086:
4079:
4067:
4060:
4056:
4054:where again
4053:
3773:
3766:
3764:
3691:
3689:
3631:
3627:
3621:
3482:
3313:
3308:
3166:
3162:
3158:
3126:
3124:
3121:
2992:
2988:
2984:
2978:
2974:
2970:
2962:
2951:denotes the
2948:
2946:
2865:
2863:
2773:
2769:
2767:
2698:
2694:
2688:
2684:
2680:
2675:
2671:
2666:
2662:
2658:
2656:
2571:
2559:
2557:
2384:
2324:is given by
2321:
2307:
2289:
2287:
2249:
2165:
2161:
2157:
2151:
2147:
2143:
2141:
2134:
2036:
2032:
2030:
1952:
1949:
1776:
1631:
1624:
1620:
1614:
1610:
1605:
1601:
1591:
1511:
1507:
1500:
1494:
1488:
1472:
1464:
1462:
1391:
1387:
1382:
1376:
1367:
1363:
1361:
1308:
1302:
1298:
1294:
1291:
1212:
1205:
1201:
1195:
1188:
1183:
1175:
1171:
1163:
1159:
1155:
1151:
1147:
1143:
1139:
1128:
1124:
1121:frame bundle
1117:torsion form
1116:
1114:
937:
766:
762:
756:
749:
745:
741:
734:
730:
726:
714:
710:
703:
657:
639:
635:
628:
624:
620:
616:
612:
608:
604:
600:
596:
593:Leibniz rule
586:
493:
489:
478:
474:
470:
466:
450:
448:
423:G-structures
406:
400:
301:bilinear map
288:
282:
235:, curvature
8778:Élie Cartan
8726:Spin tensor
8700:Weyl tensor
8658:Mathematics
8622:Multivector
8413:definitions
8311:Engineering
8250:Mathematics
8177:Cocurvature
8068:of surfaces
8006:defined in
7737:BIOMAT-2006
7649:: 325–412,
6863:, then let
6513:imply that
5419:is a curve
5363:induced by
5095:Development
3153:denote the
2310:irreducible
1594:solder form
589:Lie bracket
485:defined on
8853:Categories
8607:Linear map
8475:Operations
7935:Spivak, M.
7914:(1): 463,
7664:Cartan, É.
7635:Cartan, É.
7613:References
6965:away from
5306:, and let
4426:along the
3776:, one has
3155:cyclic sum
2314:trace-free
1618:, so that
1598:dual basis
1483:See also:
648:connection
631:) for any
465:) ∇. The
445:Definition
8746:EM tensor
8582:Dimension
8533:Transpose
8026:Curvature
8018:of curves
8004:curvature
7854:0911.0334
7722:120127682
7555:EMS Press
7545:"Torsion"
7268:−
7166:Δ
7163:−
7145:Δ
7066:Δ
7045:Δ
6969:. By the
6939:~
6921:∇
6917:−
6911:~
6896:∇
6874:Δ
6806:˙
6803:γ
6740:˙
6737:γ
6717:˙
6714:γ
6707:∇
6612:∧
6603:θ
6599:−
6586:∧
6577:θ
6492:θ
6488:∧
6474:Ω
6461:Θ
6433:Ω
6400:Θ
6396:∧
6381:η
6377:−
6330:Ω
6326:∧
6311:η
6249:η
6222:θ
6206:Θ
6174:Ω
6078:~
6075:γ
6040:~
6037:γ
5969:~
5966:γ
5906:Θ
5871:Θ
5832:Λ
5828:∈
5822:∧
5786:∈
5754:∈
5712:~
5709:γ
5688:~
5685:γ
5650:~
5647:γ
5626:~
5623:γ
5597:~
5594:γ
5562:θ
5556:∗
5552:γ
5433:~
5430:γ
5407:γ
5372:θ
5294:γ
5268:θ
5227:γ
5192:γ
5177:γ
5154:∈
5128:→
5107:γ
5058:
5052:−
5035:
4931:
4925:−
4908:
4876:−
4867:˙
4846:˙
4791:˙
4760:−
4754:˙
4724:×
4695:×
4663:˙
4638:˙
4610:∇
4598:˙
4374:×
4335:∇
4221:curvature
4187:. It is
4090:curvature
4008:−
4004:π
3983:−
3979:π
3970:Θ
3905:−
3868:−
3864:π
3843:−
3839:π
3830:Ω
3744:Ω
3720:θ
3717:∧
3714:Ω
3708:Θ
3675:ω
3672:∧
3669:ω
3663:ω
3654:ω
3645:Ω
3512:∇
3428:∇
3083:∇
3079:−
3067:∇
3057:∇
3053:−
3041:∇
3031:∇
2916:ι
2907:−
2895:−
2828:
2768:Then (tr
2744:∧
2732:↦
2603:⋀
2592:
2586:∈
2515:δ
2505:−
2493:−
2465:δ
2455:−
2312:parts: a
2270:θ
2261:Θ
2250:given by
2199:⋀
2183:∈
2180:Θ
2111:Θ
2085:−
2058:~
2055:Θ
1973:~
1893:−
1863:∇
1859:−
1829:∇
1814:θ
1753:θ
1749:∧
1740:θ
1705:θ
1701:∧
1685:ω
1670:θ
1654:Θ
1561:ω
1448:Θ
1445:⋅
1437:−
1426:Θ
1421:∗
1344:θ
1341:∧
1338:ω
1332:θ
1323:Θ
1297: : F
1263:∗
1259:π
1247:−
1227:θ
1166:with the
1067:Γ
994:γ
983:holonomic
949:Γ
914:…
859:γ
855:−
834:Γ
830:−
809:Γ
770: :=
644:tensorial
557:−
545:∇
541:−
529:∇
403:geodesics
390:. It is
263:τ
223:π
176:τ
156:τ
126:×
112:τ
83:∇
8612:Manifold
8597:Geodesic
8355:Notation
8182:Holonomy
7937:(1999),
7902:(1964),
7887:(1950),
7872:(1954),
7843:(2009),
7678:: 1–25,
7666:(1924),
7637:(1923),
7360:See also
7275:′
6988:). Let
6929:′
6687:. Then
6104:holonomy
5898:, where
5703:≠
5169:, where
4059: :
2868:is then
2296:. (See
2040:), then
1366:, where
1204: :
1131:. This
977:are the
722:sections
148:, where
8874:Tensors
8709:Physics
8543:Related
8306:Physics
8224:Tensors
7964:(2011)
7916:Bibcode
7859:Bibcode
7801:Bibcode
7769:Bibcode
7702:Bibcode
7557:, 2001
5984:. The
4227:Example
3626:is the
2977:→ End(T
2565:is the
2292:is the
1609:of the
1506:, ...,
1370:is the
1030:. So
709:, ...,
475:torsion
457:on the
8637:Vector
8632:Spinor
8617:Matrix
8411:Tensor
7945:
7830:
7816:, 212.
7784:, 393.
7720:
6691:is an
6277:, and
6240:. Let
5986:linear
3165:, and
2991:, and
2947:where
2558:where
2288:where
2137:(1, 2)
1463:where
1292:where
638:. So
479:tensor
471:Cartan
293:tensor
287:, the
8557:Basis
8242:Scope
7849:arXiv
7748:(PDF)
7733:(PDF)
7718:S2CID
7397:Notes
7324:) = 0
7201:Then
6124:helix
4832:Thus
4201:screw
3157:over
2693:Hom(T
2691:) of
2661:, tr
2160:) ≈ T
2156:End(T
1638:(the
1390:∈ GL(
1216:) by
938:Here
699:basis
461:(aka
433:. In
396:screw
291:is a
7943:ISBN
7828:ISBN
6992:and
6977:and
6961:and
6849:and
6549:and
6142:.
4195:and
4148:and
4100:and
3769:of F
3622:The
3125:The
2963:The
2703:via
1630:) =
1592:The
1142:, a
1115:The
619:) =
607:) =
492:and
449:Let
425:and
182:0.01
34:and
7924:doi
7809:doi
7777:doi
7710:doi
7680:doi
7651:doi
6790:= 0
6134:In
6114:In
6064:to
5448:in
5055:sin
5032:cos
4928:sin
4905:cos
4084:.
4063:→ T
2995:by
2973:× T
2818:def
2697:, T
2674:∈ T
2589:Hom
2187:Hom
2164:⊗ T
2150:on
1604:∈ T
1471:on
1208:→ T
1191:∈ F
1170:on
720:of
642:is
496:by
283:In
200:1.0
194:0.5
188:0.1
8855::
7922:,
7912:36
7910:,
7906:,
7891:,
7876:,
7857:,
7847:,
7807:,
7795:,
7775:,
7765:48
7763:,
7759:,
7739:,
7735:,
7716:,
7708:,
7698:88
7696:,
7676:41
7674:,
7670:,
7647:40
7645:,
7641:,
7626:,
7553:,
7547:,
7356:.
7320:,
6858:∈
6830:.
6792:,
6667:.
5740:.
5091:.
4957:.
4326::
3750:0.
3628:gl
3222::=
3161:,
2987:,
2955:.
2922:tr
2825:tr
2790:tr
2772:)(
2679:,
2569:.
1475:.
1400::
1364:Dθ
1309:π∗
1301:→
1172:gl
1152:gl
1144:gl
805::=
759:ij
744:=
739:,
729:=
627:,
621:fT
617:fY
615:,
603:,
601:fX
595:,
525::=
477:)
441:.
279:).
8216:e
8209:t
8202:v
7995:e
7988:t
7981:v
7926::
7918::
7861::
7851::
7811::
7803::
7797:2
7779::
7771::
7712::
7704::
7682::
7653::
7583:.
7340:.
7326:.
7322:Y
7318:X
7316:(
7314:S
7295:)
7291:)
7288:Y
7285:,
7282:X
7279:(
7272:T
7265:)
7262:Y
7259:,
7256:X
7253:(
7250:T
7246:(
7239:2
7236:1
7230:=
7227:)
7224:Y
7221:,
7218:X
7215:(
7212:A
7185:)
7181:)
7178:X
7175:,
7172:Y
7169:(
7160:)
7157:Y
7154:,
7151:X
7148:(
7141:(
7134:2
7131:1
7125:=
7122:)
7119:Y
7116:,
7113:X
7110:(
7107:A
7085:)
7081:)
7078:X
7075:,
7072:Y
7069:(
7063:+
7060:)
7057:Y
7054:,
7051:X
7048:(
7041:(
7034:2
7031:1
7025:=
7022:)
7019:Y
7016:,
7013:X
7010:(
7007:S
6994:A
6990:S
6986:M
6982:′
6979:Y
6975:X
6967:p
6963:Y
6959:X
6936:Y
6925:X
6908:Y
6900:X
6892:=
6889:)
6886:Y
6883:,
6880:X
6877:(
6860:M
6856:p
6851:Y
6847:X
6818:)
6815:0
6812:(
6788:t
6783:t
6779:γ
6775:t
6758:0
6755:=
6752:)
6749:t
6746:(
6729:)
6726:t
6723:(
6689:γ
6685:M
6681:t
6679:(
6677:γ
6653:b
6650:a
6646:s
6625:.
6620:a
6616:t
6607:b
6594:b
6590:t
6581:a
6573:=
6568:b
6565:a
6561:s
6557:D
6537:0
6534:=
6529:a
6525:t
6521:D
6501:.
6496:b
6483:a
6478:b
6470:=
6465:a
6457:D
6453:,
6450:0
6447:=
6442:a
6437:b
6429:D
6409:.
6404:c
6391:c
6388:b
6385:a
6374:=
6369:b
6366:a
6362:s
6342:,
6337:c
6334:b
6321:c
6318:b
6315:a
6304:2
6301:1
6295:=
6290:a
6286:t
6259:c
6256:b
6253:a
6226:a
6218:D
6215:=
6210:a
6183:b
6178:a
6153:D
6090:)
6087:1
6084:(
6052:)
6049:0
6046:(
6014:1
6011:=
6008:t
6005:,
6002:0
5999:=
5996:t
5932:w
5929:,
5926:v
5886:)
5883:w
5880:,
5877:v
5874:(
5851:M
5846:p
5842:T
5836:2
5825:w
5819:v
5799:M
5794:p
5790:T
5783:w
5780:,
5777:v
5757:M
5751:p
5724:)
5721:1
5718:(
5700:)
5697:0
5694:(
5662:)
5659:1
5656:(
5641:=
5638:)
5635:0
5632:(
5571:.
5566:i
5548:=
5543:i
5539:x
5535:d
5515:)
5512:t
5509:(
5504:i
5500:x
5496:=
5491:i
5487:x
5466:M
5461:p
5457:T
5387:)
5384:p
5381:(
5376:i
5351:M
5346:p
5342:T
5319:i
5315:x
5272:i
5247:p
5207:p
5204:=
5201:)
5198:1
5195:(
5189:=
5186:)
5183:0
5180:(
5157:M
5151:p
5131:M
5125:]
5122:1
5119:,
5116:0
5113:[
5110::
5075:.
5070:3
5066:e
5061:x
5047:2
5043:e
5038:x
5029:+
5024:1
5020:e
5015:x
4993:1
4989:e
4968:X
4943:3
4939:e
4934:x
4920:2
4916:e
4911:x
4902:=
4899:X
4879:a
4873:=
4864:b
4858:,
4855:b
4852:=
4843:a
4816:.
4811:3
4807:e
4803:)
4800:a
4797:+
4788:b
4782:(
4779:+
4774:2
4770:e
4766:)
4763:b
4751:a
4745:(
4742:=
4732:3
4728:e
4719:1
4715:e
4711:b
4708:+
4703:2
4699:e
4690:1
4686:e
4682:a
4679:+
4674:3
4670:e
4660:b
4654:+
4649:2
4645:e
4635:a
4629:=
4626:X
4619:1
4615:e
4606:=
4595:X
4589:=
4586:0
4560:2
4556:e
4552:=
4549:)
4546:0
4543:(
4540:X
4518:3
4514:e
4510:)
4507:x
4504:(
4501:b
4498:+
4493:2
4489:e
4485:)
4482:x
4479:(
4476:a
4473:=
4470:)
4467:x
4464:(
4461:X
4439:1
4435:e
4412:2
4408:e
4387:.
4382:j
4378:e
4369:i
4365:e
4361:=
4356:j
4352:e
4344:i
4340:e
4308:3
4304:e
4300:,
4295:2
4291:e
4287:,
4282:1
4278:e
4255:3
4250:R
4245:=
4242:M
4197:w
4193:v
4175:)
4172:w
4169:,
4166:v
4163:(
4160:T
4150:w
4146:v
4132:)
4129:w
4126:,
4123:v
4120:(
4117:T
4102:w
4098:v
4068:M
4065:x
4061:R
4057:u
4035:,
4031:)
4026:)
4022:)
4019:Y
4016:(
4011:1
4000:,
3997:)
3994:X
3991:(
3986:1
3974:(
3967:2
3963:(
3959:u
3956:=
3949:)
3946:Y
3943:,
3940:X
3937:(
3934:T
3927:,
3923:)
3919:)
3916:Z
3913:(
3908:1
3901:u
3896:(
3891:)
3886:)
3882:)
3879:Y
3876:(
3871:1
3860:,
3857:)
3854:X
3851:(
3846:1
3834:(
3827:2
3823:(
3819:u
3816:=
3809:Z
3806:)
3803:Y
3800:,
3797:X
3794:(
3791:R
3774:M
3771:x
3767:u
3747:=
3741:D
3711:=
3705:D
3692:D
3666:+
3660:d
3657:=
3651:D
3648:=
3632:n
3630:(
3600:0
3597:=
3593:)
3588:)
3584:Z
3581:,
3577:)
3573:Y
3570:,
3567:X
3563:(
3559:T
3555:(
3551:R
3548:+
3544:)
3540:Z
3537:,
3534:Y
3530:(
3525:)
3521:R
3516:X
3507:(
3502:(
3496:S
3465:)
3460:)
3456:Z
3453:,
3450:Y
3446:(
3441:)
3437:T
3432:X
3423:(
3419:+
3415:)
3411:Z
3408:,
3405:)
3402:Y
3399:,
3396:X
3393:(
3390:T
3386:(
3382:T
3378:(
3372:S
3367:=
3363:)
3359:Z
3355:)
3351:Y
3348:,
3345:X
3341:(
3337:R
3333:(
3327:S
3294:.
3291:Y
3288:)
3285:X
3282:,
3279:Z
3276:(
3273:R
3270:+
3267:X
3264:)
3261:Z
3258:,
3255:Y
3252:(
3249:R
3246:+
3243:Z
3240:)
3237:Y
3234:,
3231:X
3228:(
3225:R
3218:)
3214:Z
3210:)
3206:Y
3203:,
3200:X
3196:(
3192:R
3188:(
3182:S
3167:Z
3163:Y
3159:X
3139:S
3107:.
3104:Z
3099:]
3096:Y
3093:,
3090:X
3087:[
3076:Z
3071:X
3061:Y
3050:Z
3045:Y
3035:X
3027:=
3024:Z
3021:)
3018:Y
3015:,
3012:X
3009:(
3006:R
2993:Z
2989:Y
2985:X
2981:)
2979:M
2975:M
2971:M
2969:T
2949:ι
2932:,
2929:)
2926:T
2919:(
2910:1
2904:n
2900:1
2892:T
2889:=
2884:0
2880:T
2866:T
2849:.
2846:)
2843:)
2840:X
2837:(
2834:T
2831:(
2813:=
2806:)
2803:X
2800:(
2797:)
2794:T
2787:(
2774:X
2770:T
2753:.
2750:)
2747:Y
2741:X
2738:(
2735:T
2729:Y
2726::
2723:)
2720:X
2717:(
2714:T
2701:)
2699:M
2695:M
2689:X
2687:(
2685:T
2681:T
2676:M
2672:X
2667:M
2663:T
2659:T
2642:.
2638:)
2634:M
2629:T
2624:,
2621:M
2616:T
2609:2
2596:(
2583:T
2562:j
2560:δ
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2418:=
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2397:B
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2345:=
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2264:=
2234:)
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2220:,
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2166:M
2162:M
2158:M
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2148:θ
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2120:.
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2105:j
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2088:1
2081:g
2077:(
2070:=
2065:i
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2014:i
2007:j
2003:g
1995:j
1990:e
1985:=
1980:i
1969:e
1934:)
1929:]
1923:j
1918:e
1913:,
1908:i
1903:e
1897:[
1888:i
1883:e
1874:j
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1854:j
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1818:k
1810:=
1805:j
1802:i
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1762:.
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1734:j
1731:i
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1720:T
1714:=
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1663:=
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1632:δ
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1326:=
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1156:n
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1140:ω
1129:M
1125:M
1123:F
1089:]
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1080:[
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507:T
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20:)
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