166:
to one that is simpler or more suitable for calculations. It is frequently the case that the vielbein coordinate system is orthonormal, as that is generally the easiest to use. Most tensors become simple or even trivial in this coordinate system; thus the complexity of most expressions is revealed to
1396:
Changing tetrads is a routine operation in the standard formalism, as it is involved in every coordinate transformation (i.e., changing from one coordinate tetrad basis to another). Switching between multiple coordinate charts is necessary because, except in trivial cases, it is not possible for a
4025:
2783:
2376:
The manipulation with tetrad coefficients shows that abstract index formulas can, in principle, be obtained from tensor formulas with respect to a coordinate tetrad by "replacing greek by latin indices". However care must be taken that a coordinate tetrad formula defines a genuine tensor when
1560:
3359:)-component of) a tensor. Since it agrees with the coordinate expression for the curvature when specialised to a coordinate tetrad it is clear, even without using the abstract definition of the curvature, that it defines the same tensor as the coordinate basis expression.
1953:
3001:
is, in general, a first order differential operator rather than a zeroth order operator which defines a tensor coefficient. Substituting a general tetrad basis in the abstract formula we find the proper definition of the curvature in abstract index notation, however:
2279:
3161:
2918:
194:
convert between the tetradic and metric formulations of the fermionic actions despite this being possible for bosonic actions . This is effectively because Weyl spinors can be very naturally defined on a
Riemannian manifold and their natural setting leads to the
1694:
1261:
929:
All tensors of the theory can be expressed in the vector and covector basis, by expressing them as linear combinations of members of the (co)vielbein. For example, the spacetime metric tensor can be transformed from a coordinate basis to the
3350:
221:
sites such that the higher dimensional metric is replaced by a set of interacting metrics that depend only on the 4D components. Vielbeins commonly appear in other general settings in physics and mathematics. Vielbeins can be understood as
3615:
2647:
4353:
2439:
3777:
2441:), naive substitutions of formulas that correctly compute tensor coefficients with respect to a coordinate tetrad may not correctly define a tensor with respect to a general tetrad because the Lie bracket is non-vanishing:
2659:
2999:
621:
1439:
174:
The advantage of the tetrad formalism over the standard coordinate-based approach to general relativity lies in the ability to choose the tetrad basis to reflect important physical aspects of the spacetime. The
2366:
2040:
1111:
1332:. The involvement of the coordinate tetrad is not usually made explicit in the standard formalism. In the tetrad formalism, instead of writing tensor equations out fully (including tetrad elements and
1778:
2110:
356:
4750:
4204:
3008:
2794:
877:
4447:
2102:
1770:
510:
1310:
1575:
794:
1007:
4109:
3245:
1144:
656:
696:
1043:
397:
3739:
1431:
1175:
831:
3250:
1353:
723:
3402:
4783:
4643:
4553:
4497:
3769:
3414:
2491:
1387:
4524:
3696:
2536:
2528:
137:
4807:
4663:
4613:
4593:
4573:
4467:
4376:
3665:
3638:
1330:
1167:
924:
904:
441:
417:
291:
271:
111:
4215:
4020:{\displaystyle e^{-X}de^{X}=dX^{i}e_{i}-{\frac {1}{2!}}X^{i}dX^{j}{f_{ij}}^{k}e_{k}+{\frac {1}{3!}}X^{i}X^{j}dX^{k}{f_{jk}}^{l}{f_{il}}^{m}e_{m}-\cdots }
2778:{\displaystyle R_{\ \nu \sigma \tau }^{\mu }=dx^{\mu }\left((\nabla _{\sigma }\nabla _{\tau }-\nabla _{\tau }\nabla _{\sigma })\partial _{\nu }\right).}
2384:
423:
at each point in the set. Dually, a vielbein (or tetrad in 4 dimensions) determines (and is determined by) a dual co-vielbein (co-tetrad) — a set of
2934:
1397:
single coordinate chart to cover the entire manifold. Changing to and between general tetrads is much similar and equally necessary (except for
521:
1555:{\displaystyle \mathbf {g} =g_{\mu \nu }dx^{\mu }dx^{\nu }\qquad {\text{where}}~g_{\mu \nu }=\mathbf {g} (\partial _{\mu },\partial _{\nu }).}
82:. This article as currently written makes frequent mention of general relativity; however, almost everything it says is equally applicable to
3368:
5130:
4879:
941:
31:
5214:
2287:
1961:
5199:
5181:
4965:
4941:
1056:
190:
formulation of general relativity. The tetradic formalism of the theory is more fundamental than its metric formulation as one can
1948:{\displaystyle \mathbf {g} =g_{ab}e^{a}e^{b}=g_{ab}e^{a}{}_{\mu }e^{b}{}_{\nu }dx^{\mu }dx^{\nu }=g_{\mu \nu }dx^{\mu }dx^{\nu }}
2274:{\displaystyle \mathbf {g} =g_{\mu \nu }dx^{\mu }dx^{\nu }=g_{\mu \nu }e^{\mu }{}_{a}e^{\nu }{}_{b}e^{a}e^{b}=g_{ab}e^{a}e^{b}}
5239:
5229:
3156:{\displaystyle R_{\ bcd}^{a}=e^{a}\left((\nabla _{c}\nabla _{d}-\nabla _{d}\nabla _{c}-f_{cd}{}^{e}\nabla _{e})e_{b}\right)}
2913:{\displaystyle R_{\ bcd}^{a}=e^{a}\left((\nabla _{c}\nabla _{d}-\nabla _{d}\nabla _{c})e_{b}\right)\qquad {\text{(wrong!)}}}
5234:
1566:
804:
302:
4671:
4121:
949:
2378:
1393:. It allows to easily specify contraction between tensors by repeating indices as in the Einstein summation convention.
836:
3668:
931:
179:
denotes tensors as if they were represented by their coefficients with respect to a fixed local tetrad. Compared to a
63:
187:
162:, one on the left, and one on the right. The effect of the vielbeins is to change the coordinate system used on the
4381:
2045:
1713:
453:
83:
76:
183:, which is often conceptually clearer, it allows an easy and computationally explicit way to denote contractions.
1269:
4859:
4844:
2501:
1689:{\displaystyle \mathbf {g} =g_{ab}e^{a}e^{b}\qquad {\text{where}}~g_{ab}=\mathbf {g} \left(e_{a},e_{b}\right).}
242:=4. Make note of the spelling: in German, "viel" means "many", not to be confused with "vier", meaning "four".
210:
167:
be an artifact of the choice of coordinates, rather than a innate property or physical effect. That is, as a
1398:
1390:
747:
176:
168:
1050:
979:
46:
57:
to the less restrictive choice of a local basis, i.e. a locally defined set of four linearly independent
4033:
3169:
965:
733:
199:. Those spinors take form in the vielbein coordinate system, and not in the manifold coordinate system.
1120:
5076:
5003:
629:
218:
1389:". When the tetrad is unspecified this becomes a matter of specifying the type of the tensor called
1256:{\displaystyle \partial _{\mu }\equiv {\frac {\partial (f\circ \varphi ^{-1})}{\partial x^{\mu }}}.}
665:
4810:
1012:
972:
in the tetrad formalism. The coordinate tetrad is the canonical set of vectors associated with the
934:
364:
86:
79:
3701:
3345:{\displaystyle \left(\nabla _{c}\nabla _{d}-\nabla _{d}\nabla _{c}-f_{cd}{}^{e}\nabla _{e}\right)}
1414:
5066:
5035:
4993:
4112:
3405:
234:
The tetrad formulation is a special case of a more general formulation, known as the vielbein or
42:
1710:
We can translate from a general co-tetrad to the coordinate co-tetrad by expanding the covector
810:
3610:{\displaystyle e^{-X}de^{X}=dX-{\frac {1}{2!}}\left+{\frac {1}{3!}}]-{\frac {1}{4!}}]]+\cdots }
879:. Since not every manifold is parallelizable, a vielbein can generally only be chosen locally (
5195:
5177:
5112:
5094:
5027:
5019:
4961:
4937:
4814:
1338:
701:
4899:
The same approach can be used for a spacetime of arbitrary dimension, where the frame of the
3378:
5149:
Nejat Tevfik Yilmaz, (2007) "On the
Symmetric Space Sigma-Model Kinematics" arXiv:0707.2150
5102:
5084:
5011:
4849:
4758:
4618:
4532:
4472:
3744:
2642:{\displaystyle R(X,Y)=\left(\nabla _{X}\nabla _{Y}-\nabla _{Y}\nabla _{X}-\nabla _{}\right)}
2444:
1362:
973:
884:
163:
54:
4502:
3674:
4818:
3671:, and group elements correspond to the geodesics of the tangent vector. Choosing a basis
2494:
1333:
659:
214:
196:
180:
2507:
116:
5080:
5007:
4348:{\displaystyle W=\sum _{n=0}^{\infty }{\frac {(-1)^{n}M^{n}}{(n+1)!}}=(I-e^{-M})M^{-1}.}
5107:
5054:
4981:
4874:
4792:
4648:
4598:
4578:
4558:
4452:
4361:
3650:
3623:
1704:
1700:
1315:
1152:
909:
889:
698:
with respect to a coordinate basis, despite the choice of a set of (local) coordinates
426:
420:
402:
276:
256:
96:
50:
948:, so are used frequently in problems dealing with radiation, and are the basis of the
5223:
4869:
1409:
1046:
953:
737:
155:
90:
1569:). Likewise, the metric can be expressed with respect to an arbitrary (co)tetrad as
5039:
4900:
4839:
4826:
4786:
1405:
can locally be written in terms of this coordinate tetrad or a general (co)tetrad.
1147:
797:
294:
58:
2434:{\displaystyle \partial _{\mu }\partial _{\nu }=\partial _{\nu }\partial _{\mu }}
5015:
4864:
4854:
4822:
3644:
945:
726:
223:
2377:
differentiation is involved. Since the coordinate vector fields have vanishing
246:
5098:
5023:
250:
5116:
5031:
2994:{\displaystyle \left(\nabla _{c}\nabla _{d}-\nabla _{d}\nabla _{c}\right)}
616:{\displaystyle e^{a}(e_{b})=e^{a}{}_{\mu }e_{b}{}^{\mu }=\delta _{b}^{a},}
3771:
the commutators can be explicitly written out. One readily computes that
3372:
3367:
Given a vector (or covector) in the tangent (or cotangent) manifold, the
1114:
217:
theories, in which the extra-dimension(s) is/are replaced by series of N
171:, it does not alter predictions; it is rather a calculational technique.
5089:
725:
being unnecessary for the specification of a tetrad. Each covector is a
17:
5158:
Arjan
Keurentjes (2003) "The group theory of oxidation", arXiv:0210178
4998:
3667:
can be taken to be an element of the algebra, the exponential is the
1402:
1359:
of the tensors are mentioned. For example, the metric is written as "
444:
5192:
An introduction to
Spinors and Geometry with Applications in Physics
30:
This article is about general tetrads. For orthonormal tetrads, see
5071:
4115:
of the Lie algebra. The series can be written more compactly as
273:
and a local basis for each of those open sets is chosen: a set of
4980:
Arkani-Hamed, Nima; Cohen, Andrew G.; Georgi, Howard (May 2001).
2788:
The naive "Greek to Latin" substitution of the latter expression
1266:
The definition of the cotetrad uses the usual abuse of notation
2493:. Thus, it is sometimes said that tetrad coordinates provide a
2361:{\displaystyle g_{ab}=g_{\mu \nu }e^{\mu }{}_{a}e^{\nu }{}_{b}}
2035:{\displaystyle g_{\mu \nu }=g_{ab}e^{a}{}_{\mu }e^{b}{}_{\nu }}
1707:) for the index variables to distinguish the applicable basis.
1106:{\displaystyle {\varphi =(\varphi ^{1},\ldots ,\varphi ^{n})}}
5176:(first published 1990 ed.), Cambridge University Press,
4789:. Note that, as a matrix, the second W is the transpose. For
4499:
in terms of the "flat coordinates" (orthonormal, at that)
186:
The significance of the tetradic formalism appear in the
93:. Most statements hold simply by substituting arbitrary
4821:. These vielbeins are used to perform calculations in
2653:
In a coordinate tetrad this gives tensor coefficients
968:(and general relativity) consists simply of using the
662:. A vielbein is usually specified by its coefficients
202:
The privileged tetradic formalism also appears in the
5131:
Gravitation, Gauge
Theories and Differential Geometry
5129:
Tohru Eguchi, Peter B. Gilkey and Andrew J. Hanson, "
4795:
4761:
4674:
4651:
4621:
4601:
4581:
4561:
4535:
4505:
4475:
4455:
4384:
4364:
4218:
4124:
4036:
3780:
3747:
3704:
3677:
3653:
3626:
3417:
3381:
3253:
3172:
3011:
2937:
2797:
2662:
2539:
2510:
2447:
2387:
2290:
2113:
2048:
1964:
1781:
1716:
1578:
1442:
1417:
1365:
1341:
1318:
1272:
1178:
1155:
1123:
1059:
1015:
982:
912:
892:
839:
813:
750:
704:
668:
632:
524:
456:
429:
405:
367:
305:
279:
259:
119:
99:
71:. It is a special case of the more general idea of a
4469:
is then the vielbein; it expresses the differential
944:
and null tetrads. Null tetrads are composed of four
351:{\displaystyle e_{a}=e_{a}{}^{\mu }\partial _{\mu }}
27:
Relativity with a basis not derived from coordinates
4745:{\displaystyle g_{ij}={W_{i}}^{m}B_{mn}{W^{n}}_{j}}
4199:{\displaystyle e^{-X}de^{X}=e_{i}{W^{i}}_{j}dX^{j}}
3620:The above can be readily verified simply by taking
940:Popular tetrad bases in general relativity include
4801:
4777:
4744:
4657:
4637:
4607:
4587:
4567:
4547:
4518:
4491:
4461:
4441:
4370:
4347:
4198:
4103:
4019:
3763:
3733:
3690:
3659:
3632:
3609:
3396:
3344:
3239:
3155:
2993:
2912:
2777:
2641:
2522:
2485:
2433:
2360:
2273:
2096:
2034:
1947:
1764:
1688:
1554:
1425:
1381:
1347:
1324:
1304:
1255:
1161:
1138:
1105:
1037:
1001:
918:
898:
872:{\displaystyle TU\cong U\times {\mathbb {R} ^{n}}}
871:
825:
788:
717:
690:
650:
615:
504:
435:
411:
391:
350:
285:
265:
131:
105:
4785:on the Lie group is the Cartan metric, aka the
3352:is indeed a zeroth order operator, hence (the (
5194:(first published 1987 ed.), Adam Hilger,
4442:{\displaystyle {M_{j}}^{k}=X^{i}{f_{ij}}^{k}}
2097:{\displaystyle dx^{\mu }=e^{\mu }{}_{a}e^{a}}
1765:{\displaystyle e^{a}=e^{a}{}_{\mu }dx^{\mu }}
505:{\displaystyle e^{a}=e^{a}{}_{\mu }dx^{\mu }}
8:
1032:
1016:
996:
983:
976:. The coordinate tetrad is commonly denoted
765:
751:
146:
140:
4936:, Cambridge University Press, p. 133,
2104:with respect to the general tetrad, we get
4958:Riemannian Geometry and Geometric Analysis
4615:becomes the pullback of the metric tensor
1305:{\displaystyle dx^{\mu }=d\varphi ^{\mu }}
5106:
5088:
5070:
4997:
4794:
4766:
4760:
4736:
4729:
4724:
4714:
4704:
4697:
4692:
4679:
4673:
4650:
4626:
4620:
4600:
4580:
4560:
4534:
4510:
4504:
4483:
4474:
4454:
4433:
4423:
4418:
4411:
4398:
4391:
4386:
4383:
4363:
4333:
4317:
4272:
4262:
4246:
4240:
4229:
4217:
4190:
4177:
4170:
4165:
4158:
4145:
4129:
4123:
4095:
4085:
4075:
4070:
4057:
4044:
4035:
4005:
3995:
3985:
3980:
3973:
3963:
3958:
3951:
3938:
3928:
3909:
3900:
3890:
3880:
3875:
3868:
3855:
3836:
3827:
3817:
3801:
3785:
3779:
3752:
3746:
3725:
3715:
3703:
3682:
3676:
3652:
3625:
3544:
3496:
3456:
3438:
3422:
3416:
3380:
3331:
3321:
3319:
3309:
3296:
3286:
3273:
3263:
3252:
3231:
3221:
3219:
3209:
3193:
3180:
3171:
3142:
3129:
3119:
3117:
3107:
3094:
3084:
3071:
3061:
3043:
3030:
3016:
3010:
2980:
2970:
2957:
2947:
2936:
2905:
2893:
2880:
2870:
2857:
2847:
2829:
2816:
2802:
2796:
2761:
2748:
2738:
2725:
2715:
2697:
2681:
2667:
2661:
2616:
2603:
2593:
2580:
2570:
2538:
2509:
2468:
2455:
2446:
2425:
2415:
2402:
2392:
2386:
2352:
2350:
2343:
2333:
2331:
2324:
2311:
2295:
2289:
2265:
2255:
2242:
2229:
2219:
2209:
2207:
2200:
2190:
2188:
2181:
2168:
2155:
2142:
2126:
2114:
2112:
2088:
2078:
2076:
2069:
2056:
2047:
2026:
2024:
2017:
2007:
2005:
1998:
1985:
1969:
1963:
1939:
1926:
1910:
1897:
1884:
1871:
1869:
1862:
1852:
1850:
1843:
1830:
1817:
1807:
1794:
1782:
1780:
1756:
1743:
1741:
1734:
1721:
1715:
1672:
1659:
1645:
1633:
1621:
1614:
1604:
1591:
1579:
1577:
1540:
1527:
1515:
1503:
1491:
1484:
1471:
1455:
1443:
1441:
1418:
1416:
1370:
1364:
1340:
1317:
1296:
1280:
1271:
1241:
1220:
1201:
1183:
1177:
1154:
1130:
1126:
1125:
1122:
1093:
1074:
1060:
1058:
1026:
1014:
990:
981:
911:
891:
862:
858:
857:
855:
838:
812:
768:
758:
749:
709:
703:
682:
680:
673:
667:
642:
637:
631:
604:
599:
586:
584:
577:
567:
565:
558:
542:
529:
523:
496:
483:
481:
474:
461:
455:
428:
404:
366:
342:
332:
330:
323:
310:
304:
278:
258:
118:
98:
4932:De Felice, F.; Clarke, C. J. S. (1990),
4817:. The above generalizes to the case of
1169:, the coordinate vectors are such that:
4924:
4892:
5172:De Felice, F.; Clarke, C.J.S. (1990),
4378:is a matrix whose matrix elements are
833:which is equivalent to an isomorphism
789:{\displaystyle \{e_{a}\}_{a=1\dots n}}
2504:is defined for general vector fields
1009:whereas the dual cotetrad is denoted
7:
4595:, the metric tensor on the manifold
1002:{\displaystyle \{\partial _{\mu }\}}
181:completely coordinate free notation
5053:de Rham, Claudia (December 2014).
4880:Dirac equation in curved spacetime
4241:
4104:{\displaystyle ={f_{ij}}^{k}e_{k}}
3328:
3293:
3283:
3270:
3260:
3240:{\displaystyle =f_{ab}{}^{c}e_{c}}
3126:
3091:
3081:
3068:
3058:
2977:
2967:
2954:
2944:
2877:
2867:
2854:
2844:
2758:
2745:
2735:
2722:
2712:
2613:
2600:
2590:
2577:
2567:
2422:
2412:
2399:
2389:
1537:
1524:
1234:
1204:
1180:
987:
339:
32:Frame fields in general relativity
25:
5190:Benn, I.M.; Tucker, R.W. (1987),
3408:of a differential corresponds to
3375:of that tangent vector. Writing
1699:Here, we use choice of alphabet (
1312:to define covectors (1-forms) on
154:The general idea is to write the
3698:for the Lie algebra and writing
2115:
1783:
1646:
1580:
1516:
1444:
1419:
1139:{\displaystyle \mathbb {R} ^{n}}
5215:General Relativity with Tetrads
2923:is incorrect because for fixed
2904:
1620:
1490:
651:{\displaystyle \delta _{b}^{a}}
45:that generalizes the choice of
5174:Relativity on Curved Manifolds
4934:Relativity on Curved Manifolds
4539:
4326:
4304:
4292:
4280:
4259:
4249:
4063:
4037:
3669:exponential map of a Lie group
3598:
3595:
3592:
3577:
3568:
3559:
3538:
3535:
3520:
3511:
3199:
3173:
3135:
3054:
2886:
2840:
2754:
2708:
2629:
2617:
2555:
2543:
2474:
2448:
1546:
1520:
1229:
1207:
1195:
1189:
1099:
1067:
960:Relation to standard formalism
732:From the point of view of the
691:{\displaystyle e^{\mu }{}_{a}}
548:
535:
245:In the vielbein formalism, an
1:
4982:"(De)Constructing Dimensions"
1567:Einstein summation convention
1038:{\displaystyle \{dx^{\mu }\}}
392:{\displaystyle a=1,\ldots ,n}
145:" translates to "four", and "
5059:Living Reviews in Relativity
4813:, the metric is a (pseudo-)
3734:{\displaystyle X=X^{i}e_{i}}
3371:describes the corresponding
1426:{\displaystyle \mathbf {g} }
5016:10.1103/PhysRevLett.86.4757
3247:. Note that the expression
1958:from which it follows that
1113:which maps a subset of the
5256:
4209:with the infinite series
3643:For the special case of a
964:The standard formalism of
826:{\displaystyle U\subset M}
29:
1053:operators: given a chart
4860:Connection (mathematics)
4845:Orthonormal frame bundle
2502:Riemann curvature tensor
1399:parallelizable manifolds
1348:{\displaystyle \otimes }
950:Newman–Penrose formalism
796:define a section of the
718:{\displaystyle x^{\mu }}
238:-bein formulation, with
230:Mathematical formulation
89:in general, and even to
4986:Physical Review Letters
3397:{\displaystyle X\in TM}
2372:Manipulation of indices
1391:abstract index notation
1049:are usually defined as
399:that together span the
177:abstract index notation
4803:
4779:
4778:{\displaystyle B_{mn}}
4746:
4659:
4639:
4638:{\displaystyle B_{mn}}
4609:
4589:
4569:
4549:
4548:{\displaystyle N\to G}
4520:
4493:
4492:{\displaystyle dX^{j}}
4463:
4443:
4372:
4349:
4245:
4200:
4105:
4021:
3765:
3764:{\displaystyle X^{i},}
3735:
3692:
3661:
3634:
3611:
3398:
3346:
3241:
3157:
2995:
2914:
2779:
2643:
2524:
2487:
2486:{\displaystyle \neq 0}
2435:
2362:
2275:
2098:
2036:
1949:
1766:
1690:
1556:
1427:
1383:
1382:{\displaystyle g_{ab}}
1349:
1326:
1306:
1257:
1163:
1140:
1117:into coordinate space
1107:
1051:directional derivative
1039:
1003:
920:
900:
873:
827:
790:
719:
692:
652:
617:
506:
437:
413:
393:
352:
287:
267:
158:as the product of two
147:
141:
133:
107:
5240:Mathematical notation
5230:Differential geometry
4956:Jost, JĂĽrgen (1995),
4903:is referred to as an
4827:supergravity theories
4804:
4780:
4747:
4660:
4640:
4610:
4590:
4570:
4550:
4521:
4519:{\displaystyle e_{i}}
4494:
4464:
4444:
4373:
4350:
4225:
4201:
4106:
4022:
3766:
3736:
3693:
3691:{\displaystyle e_{i}}
3662:
3635:
3612:
3399:
3347:
3242:
3158:
2996:
2915:
2780:
2644:
2525:
2488:
2436:
2363:
2276:
2099:
2042:. Likewise expanding
2037:
1950:
1767:
1691:
1557:
1433:can be expressed as:
1428:
1384:
1350:
1327:
1307:
1258:
1164:
1141:
1108:
1040:
1004:
966:differential geometry
921:
901:
874:
828:
791:
734:differential geometry
720:
693:
653:
618:
507:
438:
414:
394:
353:
288:
268:
213:gravity theories and
134:
108:
5235:Theory of relativity
4829:are a special case.
4793:
4759:
4672:
4649:
4619:
4599:
4579:
4559:
4533:
4503:
4473:
4453:
4382:
4362:
4216:
4122:
4034:
3778:
3745:
3702:
3675:
3651:
3624:
3415:
3379:
3251:
3170:
3009:
2935:
2795:
2660:
2537:
2508:
2445:
2385:
2288:
2111:
2046:
1962:
1779:
1714:
1576:
1440:
1415:
1363:
1339:
1316:
1270:
1176:
1153:
1121:
1057:
1013:
980:
910:
890:
837:
811:
748:
702:
666:
630:
522:
454:
427:
403:
365:
303:
277:
257:
117:
97:
87:Riemannian manifolds
5090:10.12942/lrr-2014-7
5081:2014LRR....17....7D
5008:2001PhRvL..86.4757A
4811:Riemannian manifold
4555:from some manifold
4113:structure constants
3741:for some functions
3363:Example: Lie groups
3035:
2821:
2686:
2523:{\displaystyle X,Y}
2495:non-holonomic basis
942:orthonormal tetrads
647:
609:
132:{\displaystyle n=4}
80:Riemannian geometry
5140:(1980) pp 213-393.
4799:
4775:
4755:The metric tensor
4742:
4655:
4635:
4605:
4585:
4575:to some Lie group
4565:
4545:
4516:
4489:
4459:
4439:
4368:
4345:
4196:
4101:
4017:
3761:
3731:
3688:
3657:
3630:
3607:
3406:parallel transport
3394:
3342:
3237:
3153:
3012:
2991:
2910:
2798:
2775:
2663:
2639:
2520:
2483:
2431:
2358:
2271:
2094:
2032:
1945:
1762:
1686:
1552:
1423:
1379:
1345:
1322:
1302:
1253:
1159:
1136:
1103:
1035:
999:
916:
896:
869:
823:
786:
715:
688:
648:
633:
613:
595:
502:
433:
409:
389:
348:
283:
263:
208:higher dimensional
129:
103:
75:, which is set in
73:vielbein formalism
43:general relativity
41:is an approach to
5055:"Massive Gravity"
4992:(21): 4757–4761.
4815:Riemannian metric
4802:{\displaystyle N}
4658:{\displaystyle G}
4645:on the Lie group
4608:{\displaystyle N}
4588:{\displaystyle G}
4568:{\displaystyle N}
4462:{\displaystyle W}
4371:{\displaystyle M}
4299:
3922:
3849:
3660:{\displaystyle X}
3633:{\displaystyle X}
3557:
3509:
3469:
3019:
2908:
2805:
2670:
2500:For example, the
2284:which shows that
1628:
1624:
1565:(Here we use the
1498:
1494:
1408:For example, the
1325:{\displaystyle M}
1248:
1162:{\displaystyle f}
970:coordinate tetrad
919:{\displaystyle M}
899:{\displaystyle U}
436:{\displaystyle n}
412:{\displaystyle n}
286:{\displaystyle n}
266:{\displaystyle M}
106:{\displaystyle n}
16:(Redirected from
5247:
5204:
5186:
5159:
5156:
5150:
5147:
5141:
5127:
5121:
5120:
5110:
5092:
5074:
5050:
5044:
5043:
5001:
4977:
4971:
4970:
4953:
4947:
4946:
4929:
4912:
4897:
4850:Principal bundle
4819:symmetric spaces
4808:
4806:
4805:
4800:
4784:
4782:
4781:
4776:
4774:
4773:
4751:
4749:
4748:
4743:
4741:
4740:
4735:
4734:
4733:
4722:
4721:
4709:
4708:
4703:
4702:
4701:
4687:
4686:
4664:
4662:
4661:
4656:
4644:
4642:
4641:
4636:
4634:
4633:
4614:
4612:
4611:
4606:
4594:
4592:
4591:
4586:
4574:
4572:
4571:
4566:
4554:
4552:
4551:
4546:
4525:
4523:
4522:
4517:
4515:
4514:
4498:
4496:
4495:
4490:
4488:
4487:
4468:
4466:
4465:
4460:
4448:
4446:
4445:
4440:
4438:
4437:
4432:
4431:
4430:
4416:
4415:
4403:
4402:
4397:
4396:
4395:
4377:
4375:
4374:
4369:
4354:
4352:
4351:
4346:
4341:
4340:
4325:
4324:
4300:
4298:
4278:
4277:
4276:
4267:
4266:
4247:
4244:
4239:
4205:
4203:
4202:
4197:
4195:
4194:
4182:
4181:
4176:
4175:
4174:
4163:
4162:
4150:
4149:
4137:
4136:
4110:
4108:
4107:
4102:
4100:
4099:
4090:
4089:
4084:
4083:
4082:
4062:
4061:
4049:
4048:
4026:
4024:
4023:
4018:
4010:
4009:
4000:
3999:
3994:
3993:
3992:
3978:
3977:
3972:
3971:
3970:
3956:
3955:
3943:
3942:
3933:
3932:
3923:
3921:
3910:
3905:
3904:
3895:
3894:
3889:
3888:
3887:
3873:
3872:
3860:
3859:
3850:
3848:
3837:
3832:
3831:
3822:
3821:
3806:
3805:
3793:
3792:
3770:
3768:
3767:
3762:
3757:
3756:
3740:
3738:
3737:
3732:
3730:
3729:
3720:
3719:
3697:
3695:
3694:
3689:
3687:
3686:
3666:
3664:
3663:
3658:
3640:to be a matrix.
3639:
3637:
3636:
3631:
3616:
3614:
3613:
3608:
3558:
3556:
3545:
3510:
3508:
3497:
3492:
3488:
3470:
3468:
3457:
3443:
3442:
3430:
3429:
3403:
3401:
3400:
3395:
3351:
3349:
3348:
3343:
3341:
3337:
3336:
3335:
3326:
3325:
3320:
3317:
3316:
3301:
3300:
3291:
3290:
3278:
3277:
3268:
3267:
3246:
3244:
3243:
3238:
3236:
3235:
3226:
3225:
3220:
3217:
3216:
3198:
3197:
3185:
3184:
3162:
3160:
3159:
3154:
3152:
3148:
3147:
3146:
3134:
3133:
3124:
3123:
3118:
3115:
3114:
3099:
3098:
3089:
3088:
3076:
3075:
3066:
3065:
3048:
3047:
3034:
3029:
3017:
3000:
2998:
2997:
2992:
2990:
2986:
2985:
2984:
2975:
2974:
2962:
2961:
2952:
2951:
2919:
2917:
2916:
2911:
2909:
2906:
2903:
2899:
2898:
2897:
2885:
2884:
2875:
2874:
2862:
2861:
2852:
2851:
2834:
2833:
2820:
2815:
2803:
2784:
2782:
2781:
2776:
2771:
2767:
2766:
2765:
2753:
2752:
2743:
2742:
2730:
2729:
2720:
2719:
2702:
2701:
2685:
2680:
2668:
2648:
2646:
2645:
2640:
2638:
2634:
2633:
2632:
2608:
2607:
2598:
2597:
2585:
2584:
2575:
2574:
2529:
2527:
2526:
2521:
2492:
2490:
2489:
2484:
2473:
2472:
2460:
2459:
2440:
2438:
2437:
2432:
2430:
2429:
2420:
2419:
2407:
2406:
2397:
2396:
2367:
2365:
2364:
2359:
2357:
2356:
2351:
2348:
2347:
2338:
2337:
2332:
2329:
2328:
2319:
2318:
2303:
2302:
2280:
2278:
2277:
2272:
2270:
2269:
2260:
2259:
2250:
2249:
2234:
2233:
2224:
2223:
2214:
2213:
2208:
2205:
2204:
2195:
2194:
2189:
2186:
2185:
2176:
2175:
2160:
2159:
2147:
2146:
2134:
2133:
2118:
2103:
2101:
2100:
2095:
2093:
2092:
2083:
2082:
2077:
2074:
2073:
2061:
2060:
2041:
2039:
2038:
2033:
2031:
2030:
2025:
2022:
2021:
2012:
2011:
2006:
2003:
2002:
1993:
1992:
1977:
1976:
1954:
1952:
1951:
1946:
1944:
1943:
1931:
1930:
1918:
1917:
1902:
1901:
1889:
1888:
1876:
1875:
1870:
1867:
1866:
1857:
1856:
1851:
1848:
1847:
1838:
1837:
1822:
1821:
1812:
1811:
1802:
1801:
1786:
1771:
1769:
1768:
1763:
1761:
1760:
1748:
1747:
1742:
1739:
1738:
1726:
1725:
1695:
1693:
1692:
1687:
1682:
1678:
1677:
1676:
1664:
1663:
1649:
1641:
1640:
1626:
1625:
1622:
1619:
1618:
1609:
1608:
1599:
1598:
1583:
1561:
1559:
1558:
1553:
1545:
1544:
1532:
1531:
1519:
1511:
1510:
1496:
1495:
1492:
1489:
1488:
1476:
1475:
1463:
1462:
1447:
1432:
1430:
1429:
1424:
1422:
1388:
1386:
1385:
1380:
1378:
1377:
1354:
1352:
1351:
1346:
1331:
1329:
1328:
1323:
1311:
1309:
1308:
1303:
1301:
1300:
1285:
1284:
1262:
1260:
1259:
1254:
1249:
1247:
1246:
1245:
1232:
1228:
1227:
1202:
1188:
1187:
1168:
1166:
1165:
1160:
1145:
1143:
1142:
1137:
1135:
1134:
1129:
1112:
1110:
1109:
1104:
1102:
1098:
1097:
1079:
1078:
1044:
1042:
1041:
1036:
1031:
1030:
1008:
1006:
1005:
1000:
995:
994:
974:coordinate chart
925:
923:
922:
917:
905:
903:
902:
897:
885:coordinate chart
878:
876:
875:
870:
868:
867:
866:
861:
832:
830:
829:
824:
795:
793:
792:
787:
785:
784:
763:
762:
743:
724:
722:
721:
716:
714:
713:
697:
695:
694:
689:
687:
686:
681:
678:
677:
657:
655:
654:
649:
646:
641:
622:
620:
619:
614:
608:
603:
591:
590:
585:
582:
581:
572:
571:
566:
563:
562:
547:
546:
534:
533:
511:
509:
508:
503:
501:
500:
488:
487:
482:
479:
478:
466:
465:
442:
440:
439:
434:
418:
416:
415:
410:
398:
396:
395:
390:
357:
355:
354:
349:
347:
346:
337:
336:
331:
328:
327:
315:
314:
292:
290:
289:
284:
272:
270:
269:
264:
241:
237:
164:tangent manifold
150:
144:
138:
136:
135:
130:
112:
110:
109:
104:
55:coordinate basis
39:tetrad formalism
21:
5255:
5254:
5250:
5249:
5248:
5246:
5245:
5244:
5220:
5219:
5211:
5202:
5189:
5184:
5171:
5168:
5163:
5162:
5157:
5153:
5148:
5144:
5135:Physics Reports
5128:
5124:
5052:
5051:
5047:
4979:
4978:
4974:
4968:
4955:
4954:
4950:
4944:
4931:
4930:
4926:
4921:
4916:
4915:
4898:
4894:
4889:
4884:
4835:
4825:, of which the
4791:
4790:
4762:
4757:
4756:
4725:
4723:
4710:
4693:
4691:
4675:
4670:
4669:
4647:
4646:
4622:
4617:
4616:
4597:
4596:
4577:
4576:
4557:
4556:
4531:
4530:
4529:Given some map
4506:
4501:
4500:
4479:
4471:
4470:
4451:
4450:
4419:
4417:
4407:
4387:
4385:
4380:
4379:
4360:
4359:
4329:
4313:
4279:
4268:
4258:
4248:
4214:
4213:
4186:
4166:
4164:
4154:
4141:
4125:
4120:
4119:
4091:
4071:
4069:
4053:
4040:
4032:
4031:
4001:
3981:
3979:
3959:
3957:
3947:
3934:
3924:
3914:
3896:
3876:
3874:
3864:
3851:
3841:
3823:
3813:
3797:
3781:
3776:
3775:
3748:
3743:
3742:
3721:
3711:
3700:
3699:
3678:
3673:
3672:
3649:
3648:
3622:
3621:
3549:
3501:
3475:
3471:
3461:
3434:
3418:
3413:
3412:
3377:
3376:
3369:exponential map
3365:
3327:
3318:
3305:
3292:
3282:
3269:
3259:
3258:
3254:
3249:
3248:
3227:
3218:
3205:
3189:
3176:
3168:
3167:
3138:
3125:
3116:
3103:
3090:
3080:
3067:
3057:
3053:
3049:
3039:
3007:
3006:
2976:
2966:
2953:
2943:
2942:
2938:
2933:
2932:
2889:
2876:
2866:
2853:
2843:
2839:
2835:
2825:
2793:
2792:
2757:
2744:
2734:
2721:
2711:
2707:
2703:
2693:
2658:
2657:
2612:
2599:
2589:
2576:
2566:
2565:
2561:
2535:
2534:
2506:
2505:
2464:
2451:
2443:
2442:
2421:
2411:
2398:
2388:
2383:
2382:
2381:(i.e. commute:
2374:
2349:
2339:
2330:
2320:
2307:
2291:
2286:
2285:
2261:
2251:
2238:
2225:
2215:
2206:
2196:
2187:
2177:
2164:
2151:
2138:
2122:
2109:
2108:
2084:
2075:
2065:
2052:
2044:
2043:
2023:
2013:
2004:
1994:
1981:
1965:
1960:
1959:
1935:
1922:
1906:
1893:
1880:
1868:
1858:
1849:
1839:
1826:
1813:
1803:
1790:
1777:
1776:
1772:. We then get
1752:
1740:
1730:
1717:
1712:
1711:
1668:
1655:
1654:
1650:
1629:
1610:
1600:
1587:
1574:
1573:
1536:
1523:
1499:
1480:
1467:
1451:
1438:
1437:
1413:
1412:
1366:
1361:
1360:
1355:as above) only
1337:
1336:
1334:tensor products
1314:
1313:
1292:
1276:
1268:
1267:
1237:
1233:
1216:
1203:
1179:
1174:
1173:
1151:
1150:
1124:
1119:
1118:
1089:
1070:
1055:
1054:
1047:tangent vectors
1022:
1011:
1010:
986:
978:
977:
962:
908:
907:
906:and not all of
888:
887:
856:
835:
834:
809:
808:
805:parallelization
764:
754:
746:
745:
741:
705:
700:
699:
679:
669:
664:
663:
660:Kronecker delta
628:
627:
583:
573:
564:
554:
538:
525:
520:
519:
492:
480:
470:
457:
452:
451:
425:
424:
401:
400:
363:
362:
338:
329:
319:
306:
301:
300:
275:
274:
255:
254:
239:
235:
232:
215:massive gravity
197:spin connection
188:Einstein–Cartan
115:
114:
95:
94:
35:
28:
23:
22:
15:
12:
11:
5:
5253:
5251:
5243:
5242:
5237:
5232:
5222:
5221:
5218:
5217:
5210:
5209:External links
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4875:Spin structure
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204:deconstruction
139:. In German, "
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91:spin manifolds
51:tangent bundle
26:
24:
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5183:0-521-26639-4
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4967:3-540-57113-2
4963:
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4943:0-521-26639-4
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4910:
4906:
4902:
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4881:
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4873:
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4870:Spin manifold
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4858:
4856:
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4698:
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4676:
4668:
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4630:
4627:
4623:
4602:
4582:
4562:
4542:
4536:
4527:
4511:
4507:
4484:
4480:
4476:
4456:
4449:. The matrix
4434:
4427:
4424:
4420:
4412:
4408:
4404:
4399:
4392:
4388:
4365:
4342:
4337:
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4330:
4321:
4318:
4314:
4310:
4307:
4301:
4295:
4289:
4286:
4283:
4273:
4269:
4263:
4255:
4252:
4236:
4233:
4230:
4226:
4222:
4219:
4212:
4211:
4210:
4191:
4187:
4183:
4178:
4171:
4167:
4159:
4155:
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4146:
4142:
4138:
4133:
4130:
4126:
4118:
4117:
4116:
4114:
4096:
4092:
4086:
4079:
4076:
4072:
4066:
4058:
4054:
4050:
4045:
4041:
4014:
4011:
4006:
4002:
3996:
3989:
3986:
3982:
3974:
3967:
3964:
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3952:
3948:
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3925:
3918:
3915:
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3897:
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3869:
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3798:
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3758:
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3683:
3679:
3670:
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3627:
3604:
3601:
3589:
3586:
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3580:
3574:
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3565:
3562:
3553:
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3517:
3514:
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3498:
3493:
3489:
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3472:
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3458:
3453:
3450:
3447:
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3426:
3423:
3419:
3411:
3410:
3409:
3407:
3391:
3388:
3385:
3382:
3374:
3370:
3362:
3360:
3358:
3355:
3338:
3332:
3322:
3313:
3310:
3306:
3302:
3297:
3287:
3279:
3274:
3264:
3255:
3232:
3228:
3222:
3213:
3210:
3206:
3202:
3194:
3190:
3186:
3181:
3177:
3149:
3143:
3139:
3130:
3120:
3111:
3108:
3104:
3100:
3095:
3085:
3077:
3072:
3062:
3050:
3044:
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3036:
3031:
3026:
3023:
3020:
3013:
3005:
3004:
3003:
2987:
2981:
2971:
2963:
2958:
2948:
2939:
2930:
2926:
2900:
2894:
2890:
2881:
2871:
2863:
2858:
2848:
2836:
2830:
2826:
2822:
2817:
2812:
2809:
2806:
2799:
2791:
2790:
2789:
2772:
2768:
2762:
2749:
2739:
2731:
2726:
2716:
2704:
2698:
2694:
2690:
2687:
2682:
2677:
2674:
2671:
2664:
2656:
2655:
2654:
2635:
2626:
2623:
2620:
2609:
2604:
2594:
2586:
2581:
2571:
2562:
2558:
2552:
2549:
2546:
2540:
2533:
2532:
2531:
2517:
2514:
2511:
2503:
2498:
2496:
2480:
2477:
2469:
2465:
2461:
2456:
2452:
2426:
2416:
2408:
2403:
2393:
2380:
2371:
2369:
2353:
2344:
2340:
2334:
2325:
2321:
2315:
2312:
2308:
2304:
2299:
2296:
2292:
2266:
2262:
2256:
2252:
2246:
2243:
2239:
2235:
2230:
2226:
2220:
2216:
2210:
2201:
2197:
2191:
2182:
2178:
2172:
2169:
2165:
2161:
2156:
2152:
2148:
2143:
2139:
2135:
2130:
2127:
2123:
2119:
2107:
2106:
2105:
2089:
2085:
2079:
2070:
2066:
2062:
2057:
2053:
2049:
2027:
2018:
2014:
2008:
1999:
1995:
1989:
1986:
1982:
1978:
1973:
1970:
1966:
1940:
1936:
1932:
1927:
1923:
1919:
1914:
1911:
1907:
1903:
1898:
1894:
1890:
1885:
1881:
1877:
1872:
1863:
1859:
1853:
1844:
1840:
1834:
1831:
1827:
1823:
1818:
1814:
1808:
1804:
1798:
1795:
1791:
1787:
1775:
1774:
1773:
1757:
1753:
1749:
1744:
1735:
1731:
1727:
1722:
1718:
1708:
1706:
1702:
1683:
1679:
1673:
1669:
1665:
1660:
1656:
1651:
1642:
1637:
1634:
1630:
1615:
1611:
1605:
1601:
1595:
1592:
1588:
1584:
1572:
1571:
1570:
1568:
1549:
1541:
1533:
1528:
1512:
1507:
1504:
1500:
1485:
1481:
1477:
1472:
1468:
1464:
1459:
1456:
1452:
1448:
1436:
1435:
1434:
1411:
1410:metric tensor
1406:
1404:
1400:
1394:
1392:
1374:
1371:
1367:
1358:
1342:
1335:
1319:
1297:
1293:
1289:
1286:
1281:
1277:
1273:
1250:
1242:
1238:
1224:
1221:
1217:
1213:
1210:
1198:
1192:
1184:
1172:
1171:
1170:
1156:
1149:
1131:
1116:
1094:
1090:
1086:
1083:
1080:
1075:
1071:
1064:
1061:
1052:
1048:
1027:
1023:
1019:
991:
975:
971:
967:
959:
957:
955:
954:GHP formalism
951:
947:
943:
938:
936:
933:
927:
913:
893:
886:
882:
863:
852:
849:
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840:
820:
817:
814:
806:
802:
799:
781:
778:
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772:
769:
759:
755:
739:
738:fiber bundles
735:
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728:
710:
706:
683:
674:
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638:
634:
610:
605:
600:
596:
592:
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578:
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543:
539:
530:
526:
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517:
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497:
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475:
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462:
458:
450:
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448:
446:
430:
422:
419:-dimensional
406:
386:
383:
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374:
371:
368:
343:
333:
324:
320:
316:
311:
307:
299:
298:
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295:vector fields
280:
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212:
209:
205:
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198:
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184:
182:
178:
172:
170:
165:
161:
157:
156:metric tensor
152:
151:" to "many".
149:
143:
126:
123:
120:
100:
92:
88:
85:
81:
78:
74:
70:
66:
65:
60:
59:vector fields
56:
52:
48:
44:
40:
33:
19:
5191:
5173:
5154:
5145:
5137:
5134:
5125:
5062:
5058:
5048:
4989:
4985:
4975:
4960:, Springer,
4957:
4951:
4933:
4927:
4908:
4904:
4901:frame bundle
4895:
4840:Frame bundle
4823:sigma models
4787:Killing form
4754:
4528:
4357:
4208:
4029:
3642:
3619:
3366:
3356:
3353:
3165:
2928:
2924:
2922:
2787:
2652:
2499:
2375:
2283:
1957:
1709:
1698:
1564:
1407:
1395:
1356:
1265:
1148:scalar field
969:
963:
946:null vectors
939:
928:
880:
800:
798:frame bundle
731:
625:
514:
443:independent
360:
293:independent
244:
233:
224:solder forms
211:Kaluza–Klein
207:
203:
201:
191:
185:
173:
159:
153:
72:
68:
62:
38:
36:
4865:G-structure
4855:Spin bundle
4809:a (pseudo-)
3645:Lie algebra
2379:Lie bracket
727:solder form
515:such that
5224:Categories
5166:References
1357:components
1146:, and any
883:only on a
247:open cover
5099:2367-3613
5072:1401.4173
5024:0031-9007
4919:Citations
4540:→
4335:−
4319:−
4311:−
4253:−
4242:∞
4227:∑
4131:−
4015:⋯
4012:−
3834:−
3787:−
3605:⋯
3542:−
3454:−
3424:−
3386:∈
3329:∇
3303:−
3294:∇
3284:∇
3280:−
3271:∇
3261:∇
3127:∇
3101:−
3092:∇
3082:∇
3078:−
3069:∇
3059:∇
2978:∇
2968:∇
2964:−
2955:∇
2945:∇
2878:∇
2868:∇
2864:−
2855:∇
2845:∇
2763:ν
2759:∂
2750:σ
2746:∇
2740:τ
2736:∇
2732:−
2727:τ
2723:∇
2717:σ
2713:∇
2699:μ
2683:μ
2678:τ
2675:σ
2672:ν
2614:∇
2610:−
2601:∇
2591:∇
2587:−
2578:∇
2568:∇
2478:≠
2427:μ
2423:∂
2417:ν
2413:∂
2404:ν
2400:∂
2394:μ
2390:∂
2345:ν
2326:μ
2316:ν
2313:μ
2202:ν
2183:μ
2173:ν
2170:μ
2157:ν
2144:μ
2131:ν
2128:μ
2071:μ
2058:μ
2028:ν
2009:μ
1974:ν
1971:μ
1941:ν
1928:μ
1915:ν
1912:μ
1899:ν
1886:μ
1873:ν
1854:μ
1758:μ
1745:μ
1542:ν
1538:∂
1529:μ
1525:∂
1508:ν
1505:μ
1486:ν
1473:μ
1460:ν
1457:μ
1343:⊗
1298:μ
1294:φ
1282:μ
1243:μ
1235:∂
1222:−
1218:φ
1214:∘
1205:∂
1199:≡
1185:μ
1181:∂
1091:φ
1084:…
1072:φ
1062:φ
1028:μ
992:μ
988:∂
853:×
847:≅
818:⊂
779:…
711:μ
675:μ
635:δ
597:δ
588:μ
569:μ
498:μ
485:μ
381:…
344:μ
340:∂
334:μ
253:manifold
251:spacetime
169:formalism
160:vielbeins
84:(pseudo-)
77:(pseudo-)
61:called a
5117:28179850
5065:(1): 7.
5032:11384341
4909:vielbein
4833:See also
3373:geodesic
2907:(wrong!)
1115:manifold
1045:. These
952:and the
69:vierbein
49:for the
18:Vierbein
5108:5256007
5077:Bibcode
5040:4540121
5004:Bibcode
1401:). Any
658:is the
445:1-forms
249:of the
219:lattice
53:from a
5198:
5180:
5115:
5105:
5097:
5038:
5030:
5022:
4964:
4940:
4905:n-bein
4358:Here,
3647:, the
3404:, the
3166:where
3018:
2804:
2669:
1627:
1497:
1403:tensor
932:tetrad
740:, the
626:where
64:tetrad
5067:arXiv
5036:S2CID
4994:arXiv
4887:Notes
1705:Greek
1701:Latin
1623:where
1493:where
935:basis
47:basis
5196:ISBN
5178:ISBN
5113:PMID
5095:ISSN
5028:PMID
5020:ISSN
4962:ISBN
4938:ISBN
4111:the
4030:for
2927:and
1703:and
881:i.e.
801:i.e.
361:for
148:viel
142:vier
113:for
37:The
5133:",
5103:PMC
5085:doi
5012:doi
4907:or
4665::
2530:by
926:.)
807:of
736:of
206:of
192:not
67:or
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5138:66
5111:.
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2497:.
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803:a
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226:.
5119:.
5087::
5079::
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5006::
4996::
4911:.
4797:N
4771:n
4768:m
4764:B
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4731:n
4727:W
4719:n
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4712:B
4706:m
4699:i
4695:W
4689:=
4684:j
4681:i
4677:g
4653:G
4631:n
4628:m
4624:B
4603:N
4583:G
4563:N
4543:G
4537:N
4512:i
4508:e
4485:j
4481:X
4477:d
4457:W
4435:k
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4425:i
4421:f
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4152:=
4147:X
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4097:k
4093:e
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4067:=
4064:]
4059:j
4055:e
4051:,
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4042:e
4038:[
4007:m
4003:e
3997:m
3990:l
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3808:=
3803:X
3799:e
3795:d
3790:X
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3759:,
3754:i
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3727:i
3723:e
3717:i
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3709:=
3706:X
3684:i
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3655:X
3628:X
3602:+
3599:]
3596:]
3593:]
3590:X
3587:d
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3581:X
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3569:[
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3536:]
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3506:!
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3490:]
3486:X
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3440:X
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3392:M
3389:T
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3357:d
3354:c
3339:)
3333:e
3323:e
3314:d
3311:c
3307:f
3298:c
3288:d
3275:d
3265:c
3256:(
3233:c
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3223:c
3214:b
3211:a
3207:f
3203:=
3200:]
3195:b
3191:e
3187:,
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3174:[
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3055:(
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3037:=
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2841:(
2837:(
2831:a
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2823:=
2818:a
2813:d
2810:c
2807:b
2800:R
2773:.
2769:)
2755:)
2709:(
2705:(
2695:x
2691:d
2688:=
2665:R
2649:.
2636:)
2630:]
2627:Y
2624:,
2621:X
2618:[
2605:X
2595:Y
2582:Y
2572:X
2563:(
2559:=
2556:)
2553:Y
2550:,
2547:X
2544:(
2541:R
2518:Y
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2481:0
2475:]
2470:b
2466:e
2462:,
2457:a
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2449:[
2409:=
2354:b
2341:e
2335:a
2322:e
2309:g
2305:=
2300:b
2297:a
2293:g
2267:b
2263:e
2257:a
2253:e
2247:b
2244:a
2240:g
2236:=
2231:b
2227:e
2221:a
2217:e
2211:b
2198:e
2192:a
2179:e
2166:g
2162:=
2153:x
2149:d
2140:x
2136:d
2124:g
2120:=
2116:g
2090:a
2086:e
2080:a
2067:e
2063:=
2054:x
2050:d
2019:b
2015:e
2000:a
1996:e
1990:b
1987:a
1983:g
1979:=
1967:g
1937:x
1933:d
1924:x
1920:d
1908:g
1904:=
1895:x
1891:d
1882:x
1878:d
1864:b
1860:e
1845:a
1841:e
1835:b
1832:a
1828:g
1824:=
1819:b
1815:e
1809:a
1805:e
1799:b
1796:a
1792:g
1788:=
1784:g
1754:x
1750:d
1736:a
1732:e
1728:=
1723:a
1719:e
1684:.
1680:)
1674:b
1670:e
1666:,
1661:a
1657:e
1652:(
1647:g
1643:=
1638:b
1635:a
1631:g
1616:b
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1606:a
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1596:b
1593:a
1589:g
1585:=
1581:g
1550:.
1547:)
1534:,
1521:(
1517:g
1513:=
1501:g
1482:x
1478:d
1469:x
1465:d
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1449:=
1445:g
1420:g
1375:b
1372:a
1368:g
1320:M
1290:d
1287:=
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1225:1
1211:f
1208:(
1196:]
1193:f
1190:[
1157:f
1132:n
1127:R
1100:)
1095:n
1087:,
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1068:(
1065:=
1033:}
1024:x
1020:d
1017:{
997:}
984:{
914:M
894:U
864:n
859:R
850:U
844:U
841:T
821:M
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782:n
776:1
773:=
770:a
766:}
760:a
756:e
752:{
742:n
707:x
684:a
671:e
644:a
639:b
611:,
606:a
601:b
593:=
579:b
575:e
560:a
556:e
552:=
549:)
544:b
540:e
536:(
531:a
527:e
494:x
490:d
476:a
472:e
468:=
463:a
459:e
431:n
407:n
387:n
384:,
378:,
375:1
372:=
369:a
325:a
321:e
317:=
312:a
308:e
281:n
261:M
240:n
236:n
127:4
124:=
121:n
101:n
34:.
20:)
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