1432:
1278:
1441:
The converse question, whether the almost complex structure implies the existence of a complex structure is much less trivial, and not true in general. On an arbitrary almost complex manifold one can always find coordinates for which the almost complex structure takes the above canonical form at any
2101:
288:. Only when this local tensor can be patched together to be defined globally does the pointwise linear complex structure yield an almost complex structure, which is then uniquely determined. The possibility of this patching, and therefore existence of an almost complex structure on a manifold
1437:
One easily checks that this map defines an almost complex structure. Thus any complex structure on a manifold yields an almost complex structure, which is said to be 'induced' by the complex structure, and the complex structure is said to be 'compatible with' the almost complex structure.
2235:
The existence of an almost complex structure is a topological question and is relatively easy to answer, as discussed above. The existence of an integrable almost complex structure, on the other hand, is a much more difficult analytic question. For example, it is still not known whether
1289:
2135: = 0. The compatible complex structure is unique, as discussed above. Since the existence of an integrable almost complex structure is equivalent to the existence of a complex structure, this is sometimes taken as the definition of a complex structure.
2138:
There are several other criteria which are equivalent to the vanishing of the
Nijenhuis tensor, and which therefore furnish methods for checking the integrability of an almost complex structure (and in fact each of these can be found in the literature):
1162:
617:, we can build exterior powers of the complexified cotangent bundle (which is canonically isomorphic to the bundle of dual spaces of the complexified tangent bundle). The almost complex structure induces the decomposition of each space of
1086:
727:
1889:
899:
1427:{\displaystyle J{\frac {\partial }{\partial z^{\mu }}}=i{\frac {\partial }{\partial z^{\mu }}}\qquad J{\frac {\partial }{\partial {\bar {z}}^{\mu }}}=-i{\frac {\partial }{\partial {\bar {z}}^{\mu }}}.}
2184:
840:
1154:
1665:
2227:
2465:
In each of these equations, the two structures on the right hand side are called compatible when the corresponding construction yields a structure of the type specified. For example,
965:
404:
463:
1273:{\displaystyle J{\frac {\partial }{\partial x^{\mu }}}={\frac {\partial }{\partial y^{\mu }}}\qquad J{\frac {\partial }{\partial y^{\mu }}}=-{\frac {\partial }{\partial x^{\mu }}}}
1853:
180:
1711:
922:
136:
984:
3814:
2705:
3005:
43:
is an almost complex manifold, but there are almost complex manifolds that are not complex manifolds. Almost complex structures have important applications in
3809:
2096:{\displaystyle -(N_{A})_{ij}^{k}=A_{i}^{m}\partial _{m}A_{j}^{k}-A_{j}^{m}\partial _{m}A_{i}^{k}-A_{m}^{k}(\partial _{i}A_{j}^{m}-\partial _{j}A_{i}^{m}).}
627:
3096:
3120:
2248:
1486:
3315:
3185:
2915:
2240:
admits an integrable almost complex structure, despite a long history of ultimately unverified claims. Smoothness issues are important. For
3411:
2559:
2629:
3464:
2992:
264:
shows that any even dimensional vector space admits a linear complex structure. Therefore, an even dimensional manifold always admits a
2656:
2107:
3748:
2895:
2872:
3513:
846:
293:
3105:
2765:
2763:; Bazzoni, Giovanni; Goertsches, Oliver; Konstantis, Panagiotis; Rollenske, Sönke (2018). "On the history of the Hopf problem".
3496:
2934:
788:). Thus we may use the almost complex structure to refine the action of the exterior derivative to the forms of definite type
85:
which squares to −1) on each tangent space of the manifold, which varies smoothly on the manifold. In other words, we have a
3708:
2613:
3693:
3416:
3190:
2641:
2590:
756:
3738:
2148:
3743:
3713:
3421:
3377:
3358:
3125:
3069:
2608:
An almost complex structure integrates to a complex structure if the half-dimensional subspace is closed under the
794:
3857:
3280:
3145:
2505:
1103:
1518:
3665:
3530:
3222:
3064:
2190:
32:
943:
351:
3362:
3332:
3256:
3202:
3032:
2985:
1451:
413:
3130:
3703:
3322:
3217:
3037:
1719:
508:, inherits an almost complex structure from the octonion multiplication; the question of whether it has a
3352:
3347:
2815:
2493:: the complex structures on the tangent fibres compatible with the restriction to the symplectic forms.
197:
admits an almost complex structure, it must be even-dimensional. This can be seen as follows. Suppose
147:
3683:
3621:
3469:
3173:
3163:
3135:
3110:
3020:
2883:
2714:
2567:
1467:
967:
is a map which increases the antiholomorphic part of the type by one. These operators are called the
2571:
3821:
3794:
3503:
3381:
3366:
3295:
3054:
2860:
2678:
774:
254:
44:
3763:
3718:
3615:
3486:
3290:
3115:
2978:
2951:
2840:
2792:
2774:
2586:
968:
321:
3300:
2661:
2537:
1677:
1081:{\displaystyle d=\sum _{r+s=p+q+1}\pi _{r,s}\circ d=\partial +{\overline {\partial }}+\cdots .}
907:
3698:
3678:
3673:
3580:
3491:
3305:
3285:
3140:
3079:
2929:
2911:
2891:
2868:
2832:
2742:
2602:
2279:
2259:, analysis is required (with more difficult techniques as the regularity hypothesis weakens).
975:
606:
105:
51:
3836:
3630:
3585:
3508:
3479:
3337:
3270:
3265:
3260:
3250:
3042:
3025:
2943:
2824:
2784:
2732:
2722:
2667:
2594:
2315:
1470:
1097:
614:
598:
on the (1, 0)-vector fields of the complexified tangent bundle, and multiplication by −
336:
admits an almost complex structure. An example for such an almost complex structure is (1 ≤
40:
2963:
2852:
3779:
3688:
3518:
3474:
3240:
2959:
2848:
2810:
2617:
2575:
2298:
2272:
86:
28:
1858:
The individual expressions on the right depend on the choice of the smooth vector fields
2718:
2232:
Any of these conditions implies the existence of a unique compatible complex structure.
1458:. Such coordinates, if they exist, are called 'local holomorphic coordinates for J'. If
3645:
3570:
3540:
3438:
3431:
3371:
3342:
3212:
3207:
3168:
2673:
2620:. If furthermore this half-dimensional space is the annihilator of a nowhere vanishing
2609:
2579:
924:
is a map which increases the holomorphic part of the type by one (takes forms of type (
610:
593:
497:
261:
183:
2737:
2700:
3851:
3831:
3655:
3650:
3635:
3625:
3575:
3552:
3426:
3386:
3327:
3275:
3074:
2796:
2760:
2555:
2544:
2241:
1493:
is induced by a complex structure, then it is induced by a unique complex structure.
722:{\displaystyle \Omega ^{r}(M)^{\mathbf {C} }=\bigoplus _{p+q=r}\Omega ^{(p,q)}(M).\,}
139:
36:
3758:
3753:
3595:
3562:
3535:
3443:
3084:
2925:
581:
89:
492:
cannot be given an almost complex structure (Ehresmann and Hopf). In the case of
2788:
2143:
The Lie bracket of any two (1, 0)-vector fields is again of type (1, 0)
3601:
3590:
3547:
3448:
3049:
2650:
2621:
142:
20:
568:(which is the vector bundle of complexified tangent spaces at each point) into
3826:
3784:
3610:
3523:
3155:
3059:
545:
517:
82:
55:
2836:
496:, the almost complex structure comes from an honest complex structure on the
3640:
3605:
3310:
3197:
2598:
2746:
2727:
2585:. A generalized almost complex structure is a choice of a half-dimensional
2110:, which generalizes the Lie bracket of vector fields, the Nijenhuis tensor
3804:
3799:
3789:
3180:
3001:
2574:. An ordinary almost complex structure is a choice of a half-dimensional
505:
272:(which is just a linear transformation on each tangent space) such that
2970:
2955:
2879:
Information on compatible triples, Kähler and
Hermitian manifolds, etc.
2844:
1100:
is itself an almost complex manifold. In local holomorphic coordinates
2701:"On the Chern numbers of certain complex and almost complex manifolds"
1446:. In general, however, it is not possible to find coordinates so that
3396:
2566:, which was elaborated in the doctoral dissertations of his students
1866:, but the left side actually depends only on the pointwise values of
473:
96:
2947:
2828:
2813:(1957). "Complex analytic coordinates in almost complex manifolds".
186:. A manifold equipped with an almost complex structure is called an
2779:
2379:
when each structure can be specified by the two others as follows:
2664: – Manifold with Riemannian, complex and symplectic structure
2974:
2485:
whose sections are the almost complex structures compatible to
253:
has an almost complex structure. One can show that it must be
2932:(1953). "Groupes de Lie et puissances réduites de Steenrod".
1883:
is a tensor. This is also clear from the component formula
894:{\displaystyle {\overline {\partial }}=\pi _{p,q+1}\circ d}
2500:, one can show that a compatible almost complex structure
564:
allows a decomposition of the complexified tangent bundle
2612:. A generalized almost complex structure integrates to a
509:
2902:
Short section which introduces standard basic material.
2670: – Mathematical structure in differential geometry
2646:
Pages displaying short descriptions of redirect targets
2597:. In both cases one demands that the direct sum of the
1466:
around every point then these patch together to form a
1670:
or, for the usual case of an almost complex structure
1477:
giving it a complex structure, which moreover induces
978:, we note that the exterior derivative can be written
2193:
2151:
1892:
1722:
1680:
1521:
1292:
1165:
1106:
987:
946:
910:
849:
797:
630:
416:
354:
150:
108:
504:, when considered as the set of unit norm imaginary
3772:
3731:
3664:
3561:
3457:
3404:
3395:
3231:
3154:
3093:
3013:
2496:Using elementary properties of the symplectic form
2247:, the Newlander–Nirenberg theorem follows from the
560:, respectively), so an almost complex structure on
2221:
2178:
2095:
1847:
1705:
1659:
1426:
1283:(just like a counterclockwise rotation of π/2) or
1272:
1148:
1080:
959:
916:
893:
834:
721:
457:
398:
174:
130:
2681: – Type of manifold in differential geometry
974:Since the sum of all the projections must be the
524:Differential topology of almost complex manifolds
2653: – Characteristic classes of vector bundles
1508:is a tensor field of rank (1, 1), then the
2706:Proceedings of the National Academy of Sciences
2363:. With this understood, the three structures (
2179:{\displaystyle d=\partial +{\bar {\partial }}}
528:Just as a complex structure on a vector space
2986:
835:{\displaystyle \partial =\pi _{p+1,q}\circ d}
8:
1149:{\displaystyle z^{\mu }=x^{\mu }+iy^{\mu }}
3401:
2993:
2979:
2971:
2888:Differential Analysis on Complex Manifolds
2766:Differential Geometry and Its Applications
2359:, is given by the analogous operation for
1660:{\displaystyle N_{A}(X,Y)=-A^{2}+A(+)-.\,}
485:
476:which admit almost complex structures are
320:. The existence question is then a purely
81:is a linear complex structure (that is, a
2778:
2736:
2726:
2481:•) is a Riemannian metric. The bundle on
2222:{\displaystyle {\bar {\partial }}^{2}=0.}
2207:
2196:
2195:
2192:
2165:
2164:
2150:
2081:
2076:
2066:
2053:
2048:
2038:
2025:
2020:
2007:
2002:
1992:
1982:
1977:
1964:
1959:
1949:
1939:
1934:
1921:
1913:
1903:
1891:
1844:
1727:
1721:
1685:
1679:
1656:
1557:
1526:
1520:
1512:is a tensor field of rank (1,2) given by
1462:admits local holomorphic coordinates for
1412:
1401:
1400:
1390:
1372:
1361:
1360:
1350:
1337:
1324:
1309:
1296:
1291:
1261:
1248:
1233:
1220:
1207:
1194:
1182:
1169:
1164:
1140:
1124:
1111:
1105:
1059:
1032:
998:
986:
947:
945:
909:
867:
850:
848:
808:
796:
736:) admits a decomposition into a sum of Ω(
718:
688:
666:
652:
651:
635:
629:
437:
421:
415:
378:
359:
353:
149:
113:
107:
2908:Algebraic Geometry, a concise dictionary
2545:2 out of 3 property of the unitary group
2124:states that an almost complex structure
2691:
960:{\displaystyle {\overline {\partial }}}
588:is a vector field of type (0, 1). Thus
399:{\displaystyle J_{ij}=-\delta _{i,j-1}}
1450:takes the canonical form on an entire
458:{\displaystyle J_{ij}=\delta _{i,j+1}}
7:
2910:. Berlin/Boston: Walter De Gruyter.
2616:if the subspace is closed under the
2560:generalized almost complex structure
2551:Generalized almost complex structure
2301:, each induces a bundle isomorphism
1848:{\displaystyle N_{J}(X,Y)=+J(+)-.\,}
1092:Integrable almost complex structures
332:For every integer n, the flat space
759:, there is a canonical projection π
584:of type (1, 0), while a section of
324:one and is fairly well understood.
219:be an almost complex structure. If
2578:of each fiber of the complexified
2285:, and an almost complex structure
2198:
2167:
2158:
2063:
2035:
1989:
1946:
1396:
1392:
1356:
1352:
1330:
1326:
1302:
1298:
1254:
1250:
1226:
1222:
1200:
1196:
1175:
1171:
1061:
1053:
949:
911:
852:
798:
685:
632:
602:on the (0, 1)-vector fields.
14:
2543:These triples are related to the
2350:, •) and the other, denoted
592:corresponds to multiplication by
2593:of the complexified tangent and
653:
294:reduction of the structure group
175:{\displaystyle J\colon TM\to TM}
2935:American Journal of Mathematics
2865:Lectures on Symplectic Geometry
2630:generalized Calabi–Yau manifold
2305:, where the first map, denoted
1346:
1216:
3033:Differentiable/Smooth manifold
2589:subspace of each fiber of the
2473:are compatible if and only if
2201:
2170:
2087:
2031:
1910:
1896:
1838:
1820:
1814:
1811:
1796:
1790:
1775:
1772:
1763:
1751:
1745:
1733:
1650:
1632:
1626:
1623:
1608:
1602:
1587:
1584:
1575:
1563:
1544:
1532:
1406:
1366:
712:
706:
701:
689:
648:
641:
163:
1:
2890:. New York: Springer-Verlag.
2614:generalized complex structure
2128:is integrable if and only if
2789:10.1016/j.difgeo.2017.10.014
2699:Van de Ven, A. (June 1966).
2642:Almost quaternionic manifold
1064:
952:
855:
3739:Classification of manifolds
2657:Frölicher–Nijenhuis bracket
2644: – Concept in geometry
2605:yield the original bundle.
2558:introduced the notion of a
2122:Newlander–Nirenberg theorem
2108:Frölicher–Nijenhuis bracket
296:of the tangent bundle from
3874:
2508:for the Riemannian metric
532:allows a decomposition of
3815:over commutative algebras
1706:{\displaystyle J^{2}=-Id}
1500:on each tangent space of
917:{\displaystyle \partial }
773:) to Ω. We also have the
238:is a real manifold, then
70:be a smooth manifold. An
3531:Riemann curvature tensor
1156:one can define the maps
486:Borel & Serre (1953)
245:is a real number – thus
131:{\displaystyle J^{2}=-1}
72:almost complex structure
33:linear complex structure
2506:almost Kähler structure
732:In other words, each Ω(
188:almost complex manifold
31:equipped with a smooth
25:almost complex manifold
3323:Manifold with boundary
3038:Differential structure
2728:10.1073/pnas.55.6.1624
2223:
2180:
2117:is just one-half of .
2097:
1849:
1707:
1661:
1428:
1274:
1150:
1082:
961:
918:
895:
836:
723:
459:
400:
205:-dimensional, and let
176:
132:
50:The concept is due to
2906:Rubei, Elena (2014).
2816:Annals of Mathematics
2524:is integrable, then (
2224:
2181:
2098:
1850:
1708:
1662:
1496:Given any linear map
1429:
1275:
1151:
1083:
962:
919:
896:
837:
724:
460:
401:
322:algebraic topological
177:
133:
3470:Covariant derivative
3021:Topological manifold
2861:Cannas da Silva, Ana
2191:
2149:
1890:
1720:
1678:
1519:
1485:is then said to be '
1290:
1163:
1104:
985:
944:
932:) to forms of type (
908:
847:
795:
628:
414:
352:
260:An easy exercise in
148:
106:
3504:Exterior derivative
3106:Atiyah–Singer index
3055:Riemannian manifold
2809:Newlander, August;
2719:1966PNAS...55.1624V
2679:Symplectic manifold
2491:contractible fibres
2271:is equipped with a
2086:
2058:
2030:
2012:
1987:
1969:
1944:
1926:
969:Dolbeault operators
775:exterior derivative
292:is equivalent to a
138:when regarded as a
45:symplectic geometry
3810:Secondary calculus
3764:Singularity theory
3719:Parallel transport
3487:De Rham cohomology
3126:Generalized Stokes
2930:Serre, Jean-Pierre
2314:, is given by the
2263:Compatible triples
2255:(and less smooth)
2219:
2176:
2093:
2072:
2044:
2016:
1998:
1973:
1955:
1930:
1909:
1845:
1703:
1657:
1424:
1270:
1146:
1078:
1027:
957:
914:
891:
832:
719:
683:
607:differential forms
552:corresponding to +
488:). In particular,
455:
396:
172:
128:
3845:
3844:
3727:
3726:
3492:Differential form
3146:Whitney embedding
3080:Differential form
2917:978-3-11-031622-3
2884:Wells, Raymond O.
2819:. Second Series.
2603:complex conjugate
2595:cotangent bundles
2377:compatible triple
2280:Riemannian metric
2249:Frobenius theorem
2204:
2173:
1419:
1409:
1379:
1369:
1344:
1316:
1268:
1240:
1214:
1189:
1067:
994:
955:
858:
662:
605:Just as we build
510:complex structure
62:Formal definition
52:Charles Ehresmann
3865:
3858:Smooth manifolds
3837:Stratified space
3795:Fréchet manifold
3509:Interior product
3402:
3099:
2995:
2988:
2981:
2972:
2967:
2921:
2901:
2878:
2856:
2811:Nirenberg, Louis
2801:
2800:
2782:
2757:
2751:
2750:
2740:
2730:
2713:(6): 1624–1627.
2696:
2668:Poisson manifold
2647:
2562:on the manifold
2316:interior product
2228:
2226:
2225:
2220:
2212:
2211:
2206:
2205:
2197:
2185:
2183:
2182:
2177:
2175:
2174:
2166:
2106:In terms of the
2102:
2100:
2099:
2094:
2085:
2080:
2071:
2070:
2057:
2052:
2043:
2042:
2029:
2024:
2011:
2006:
1997:
1996:
1986:
1981:
1968:
1963:
1954:
1953:
1943:
1938:
1925:
1920:
1908:
1907:
1854:
1852:
1851:
1846:
1732:
1731:
1712:
1710:
1709:
1704:
1690:
1689:
1666:
1664:
1663:
1658:
1562:
1561:
1531:
1530:
1510:Nijenhuis tensor
1433:
1431:
1430:
1425:
1420:
1418:
1417:
1416:
1411:
1410:
1402:
1391:
1380:
1378:
1377:
1376:
1371:
1370:
1362:
1351:
1345:
1343:
1342:
1341:
1325:
1317:
1315:
1314:
1313:
1297:
1279:
1277:
1276:
1271:
1269:
1267:
1266:
1265:
1249:
1241:
1239:
1238:
1237:
1221:
1215:
1213:
1212:
1211:
1195:
1190:
1188:
1187:
1186:
1170:
1155:
1153:
1152:
1147:
1145:
1144:
1129:
1128:
1116:
1115:
1098:complex manifold
1087:
1085:
1084:
1079:
1068:
1060:
1043:
1042:
1026:
966:
964:
963:
958:
956:
948:
923:
921:
920:
915:
900:
898:
897:
892:
884:
883:
859:
851:
841:
839:
838:
833:
825:
824:
728:
726:
725:
720:
705:
704:
682:
658:
657:
656:
640:
639:
615:cotangent bundle
512:is known as the
500:. The 6-sphere,
464:
462:
461:
456:
454:
453:
429:
428:
405:
403:
402:
397:
395:
394:
367:
366:
319:
307:
283:
267:
249:must be even if
244:
233:
225:
218:
181:
179:
178:
173:
137:
135:
134:
129:
118:
117:
101:
41:complex manifold
3873:
3872:
3868:
3867:
3866:
3864:
3863:
3862:
3848:
3847:
3846:
3841:
3780:Banach manifold
3773:Generalizations
3768:
3723:
3660:
3557:
3519:Ricci curvature
3475:Cotangent space
3453:
3391:
3233:
3227:
3186:Exponential map
3150:
3095:
3089:
3009:
2999:
2948:10.2307/2372495
2924:
2918:
2905:
2898:
2882:
2875:
2859:
2829:10.2307/1970051
2808:
2805:
2804:
2759:
2758:
2754:
2698:
2697:
2693:
2688:
2662:Kähler manifold
2645:
2638:
2618:Courant bracket
2568:Marco Gualtieri
2553:
2538:Kähler manifold
2456:
2447:
2358:
2338:
2325:
2313:
2273:symplectic form
2265:
2194:
2189:
2188:
2147:
2146:
2133:
2115:
2062:
2034:
1988:
1945:
1899:
1888:
1887:
1882:
1874:, which is why
1723:
1718:
1717:
1681:
1676:
1675:
1553:
1522:
1517:
1516:
1399:
1395:
1359:
1355:
1333:
1329:
1305:
1301:
1288:
1287:
1257:
1253:
1229:
1225:
1203:
1199:
1178:
1174:
1161:
1160:
1136:
1120:
1107:
1102:
1101:
1094:
1028:
983:
982:
942:
941:
906:
905:
863:
845:
844:
804:
793:
792:
768:
684:
647:
631:
626:
625:
611:exterior powers
576:. A section of
526:
433:
417:
412:
411:
374:
355:
350:
349:
330:
309:
297:
281:
273:
265:
239:
227:
220:
206:
146:
145:
109:
104:
103:
99:
64:
29:smooth manifold
17:
16:Smooth manifold
12:
11:
5:
3871:
3869:
3861:
3860:
3850:
3849:
3843:
3842:
3840:
3839:
3834:
3829:
3824:
3819:
3818:
3817:
3807:
3802:
3797:
3792:
3787:
3782:
3776:
3774:
3770:
3769:
3767:
3766:
3761:
3756:
3751:
3746:
3741:
3735:
3733:
3729:
3728:
3725:
3724:
3722:
3721:
3716:
3711:
3706:
3701:
3696:
3691:
3686:
3681:
3676:
3670:
3668:
3662:
3661:
3659:
3658:
3653:
3648:
3643:
3638:
3633:
3628:
3618:
3613:
3608:
3598:
3593:
3588:
3583:
3578:
3573:
3567:
3565:
3559:
3558:
3556:
3555:
3550:
3545:
3544:
3543:
3533:
3528:
3527:
3526:
3516:
3511:
3506:
3501:
3500:
3499:
3489:
3484:
3483:
3482:
3472:
3467:
3461:
3459:
3455:
3454:
3452:
3451:
3446:
3441:
3436:
3435:
3434:
3424:
3419:
3414:
3408:
3406:
3399:
3393:
3392:
3390:
3389:
3384:
3374:
3369:
3355:
3350:
3345:
3340:
3335:
3333:Parallelizable
3330:
3325:
3320:
3319:
3318:
3308:
3303:
3298:
3293:
3288:
3283:
3278:
3273:
3268:
3263:
3253:
3243:
3237:
3235:
3229:
3228:
3226:
3225:
3220:
3215:
3213:Lie derivative
3210:
3208:Integral curve
3205:
3200:
3195:
3194:
3193:
3183:
3178:
3177:
3176:
3169:Diffeomorphism
3166:
3160:
3158:
3152:
3151:
3149:
3148:
3143:
3138:
3133:
3128:
3123:
3118:
3113:
3108:
3102:
3100:
3091:
3090:
3088:
3087:
3082:
3077:
3072:
3067:
3062:
3057:
3052:
3047:
3046:
3045:
3040:
3030:
3029:
3028:
3017:
3015:
3014:Basic concepts
3011:
3010:
3000:
2998:
2997:
2990:
2983:
2975:
2969:
2968:
2942:(3): 409–448.
2922:
2916:
2903:
2896:
2880:
2873:
2857:
2823:(3): 391–404.
2803:
2802:
2761:Agricola, Ilka
2752:
2690:
2689:
2687:
2684:
2683:
2682:
2676:
2674:Rizza manifold
2671:
2665:
2659:
2654:
2648:
2637:
2634:
2580:tangent bundle
2572:Gil Cavalcanti
2552:
2549:
2463:
2462:
2452:
2443:
2430:
2407:
2354:
2334:
2330:) =
2321:
2309:
2264:
2261:
2230:
2229:
2218:
2215:
2210:
2203:
2200:
2186:
2172:
2169:
2163:
2160:
2157:
2154:
2144:
2131:
2113:
2104:
2103:
2092:
2089:
2084:
2079:
2075:
2069:
2065:
2061:
2056:
2051:
2047:
2041:
2037:
2033:
2028:
2023:
2019:
2015:
2010:
2005:
2001:
1995:
1991:
1985:
1980:
1976:
1972:
1967:
1962:
1958:
1952:
1948:
1942:
1937:
1933:
1929:
1924:
1919:
1916:
1912:
1906:
1902:
1898:
1895:
1878:
1856:
1855:
1843:
1840:
1837:
1834:
1831:
1828:
1825:
1822:
1819:
1816:
1813:
1810:
1807:
1804:
1801:
1798:
1795:
1792:
1789:
1786:
1783:
1780:
1777:
1774:
1771:
1768:
1765:
1762:
1759:
1756:
1753:
1750:
1747:
1744:
1741:
1738:
1735:
1730:
1726:
1702:
1699:
1696:
1693:
1688:
1684:
1668:
1667:
1655:
1652:
1649:
1646:
1643:
1640:
1637:
1634:
1631:
1628:
1625:
1622:
1619:
1616:
1613:
1610:
1607:
1604:
1601:
1598:
1595:
1592:
1589:
1586:
1583:
1580:
1577:
1574:
1571:
1568:
1565:
1560:
1556:
1552:
1549:
1546:
1543:
1540:
1537:
1534:
1529:
1525:
1435:
1434:
1423:
1415:
1408:
1405:
1398:
1394:
1389:
1386:
1383:
1375:
1368:
1365:
1358:
1354:
1349:
1340:
1336:
1332:
1328:
1323:
1320:
1312:
1308:
1304:
1300:
1295:
1281:
1280:
1264:
1260:
1256:
1252:
1247:
1244:
1236:
1232:
1228:
1224:
1219:
1210:
1206:
1202:
1198:
1193:
1185:
1181:
1177:
1173:
1168:
1143:
1139:
1135:
1132:
1127:
1123:
1119:
1114:
1110:
1093:
1090:
1089:
1088:
1077:
1074:
1071:
1066:
1063:
1058:
1055:
1052:
1049:
1046:
1041:
1038:
1035:
1031:
1025:
1022:
1019:
1016:
1013:
1010:
1007:
1004:
1001:
997:
993:
990:
954:
951:
913:
902:
901:
890:
887:
882:
879:
876:
873:
870:
866:
862:
857:
854:
842:
831:
828:
823:
820:
817:
814:
811:
807:
803:
800:
760:
730:
729:
717:
714:
711:
708:
703:
700:
697:
694:
691:
687:
681:
678:
675:
672:
669:
665:
661:
655:
650:
646:
643:
638:
634:
525:
522:
498:Riemann sphere
452:
449:
446:
443:
440:
436:
432:
427:
424:
420:
393:
390:
387:
384:
381:
377:
373:
370:
365:
362:
358:
329:
326:
284:at each point
277:
262:linear algebra
184:tangent bundle
171:
168:
165:
162:
159:
156:
153:
127:
124:
121:
116:
112:
63:
60:
58:in the 1940s.
15:
13:
10:
9:
6:
4:
3:
2:
3870:
3859:
3856:
3855:
3853:
3838:
3835:
3833:
3832:Supermanifold
3830:
3828:
3825:
3823:
3820:
3816:
3813:
3812:
3811:
3808:
3806:
3803:
3801:
3798:
3796:
3793:
3791:
3788:
3786:
3783:
3781:
3778:
3777:
3775:
3771:
3765:
3762:
3760:
3757:
3755:
3752:
3750:
3747:
3745:
3742:
3740:
3737:
3736:
3734:
3730:
3720:
3717:
3715:
3712:
3710:
3707:
3705:
3702:
3700:
3697:
3695:
3692:
3690:
3687:
3685:
3682:
3680:
3677:
3675:
3672:
3671:
3669:
3667:
3663:
3657:
3654:
3652:
3649:
3647:
3644:
3642:
3639:
3637:
3634:
3632:
3629:
3627:
3623:
3619:
3617:
3614:
3612:
3609:
3607:
3603:
3599:
3597:
3594:
3592:
3589:
3587:
3584:
3582:
3579:
3577:
3574:
3572:
3569:
3568:
3566:
3564:
3560:
3554:
3553:Wedge product
3551:
3549:
3546:
3542:
3539:
3538:
3537:
3534:
3532:
3529:
3525:
3522:
3521:
3520:
3517:
3515:
3512:
3510:
3507:
3505:
3502:
3498:
3497:Vector-valued
3495:
3494:
3493:
3490:
3488:
3485:
3481:
3478:
3477:
3476:
3473:
3471:
3468:
3466:
3463:
3462:
3460:
3456:
3450:
3447:
3445:
3442:
3440:
3437:
3433:
3430:
3429:
3428:
3427:Tangent space
3425:
3423:
3420:
3418:
3415:
3413:
3410:
3409:
3407:
3403:
3400:
3398:
3394:
3388:
3385:
3383:
3379:
3375:
3373:
3370:
3368:
3364:
3360:
3356:
3354:
3351:
3349:
3346:
3344:
3341:
3339:
3336:
3334:
3331:
3329:
3326:
3324:
3321:
3317:
3314:
3313:
3312:
3309:
3307:
3304:
3302:
3299:
3297:
3294:
3292:
3289:
3287:
3284:
3282:
3279:
3277:
3274:
3272:
3269:
3267:
3264:
3262:
3258:
3254:
3252:
3248:
3244:
3242:
3239:
3238:
3236:
3230:
3224:
3221:
3219:
3216:
3214:
3211:
3209:
3206:
3204:
3201:
3199:
3196:
3192:
3191:in Lie theory
3189:
3188:
3187:
3184:
3182:
3179:
3175:
3172:
3171:
3170:
3167:
3165:
3162:
3161:
3159:
3157:
3153:
3147:
3144:
3142:
3139:
3137:
3134:
3132:
3129:
3127:
3124:
3122:
3119:
3117:
3114:
3112:
3109:
3107:
3104:
3103:
3101:
3098:
3094:Main results
3092:
3086:
3083:
3081:
3078:
3076:
3075:Tangent space
3073:
3071:
3068:
3066:
3063:
3061:
3058:
3056:
3053:
3051:
3048:
3044:
3041:
3039:
3036:
3035:
3034:
3031:
3027:
3024:
3023:
3022:
3019:
3018:
3016:
3012:
3007:
3003:
2996:
2991:
2989:
2984:
2982:
2977:
2976:
2973:
2965:
2961:
2957:
2953:
2949:
2945:
2941:
2937:
2936:
2931:
2927:
2926:Borel, Armand
2923:
2919:
2913:
2909:
2904:
2899:
2897:0-387-90419-0
2893:
2889:
2885:
2881:
2876:
2874:3-540-42195-5
2870:
2866:
2862:
2858:
2854:
2850:
2846:
2842:
2838:
2834:
2830:
2826:
2822:
2818:
2817:
2812:
2807:
2806:
2798:
2794:
2790:
2786:
2781:
2776:
2772:
2768:
2767:
2762:
2756:
2753:
2748:
2744:
2739:
2734:
2729:
2724:
2720:
2716:
2712:
2708:
2707:
2702:
2695:
2692:
2685:
2680:
2677:
2675:
2672:
2669:
2666:
2663:
2660:
2658:
2655:
2652:
2649:
2643:
2640:
2639:
2635:
2633:
2631:
2627:
2623:
2619:
2615:
2611:
2606:
2604:
2600:
2596:
2592:
2588:
2584:
2581:
2577:
2573:
2569:
2565:
2561:
2557:
2556:Nigel Hitchin
2550:
2548:
2546:
2541:
2539:
2535:
2531:
2527:
2523:
2519:
2515:
2511:
2507:
2503:
2499:
2494:
2492:
2488:
2484:
2480:
2476:
2472:
2468:
2460:
2455:
2451:
2446:
2442:
2438:
2434:
2431:
2428:
2424:
2420:
2416:
2412:
2408:
2405:
2401:
2397:
2393:
2389:
2385:
2382:
2381:
2380:
2378:
2374:
2370:
2366:
2362:
2357:
2353:
2349:
2345:
2342: =
2341:
2337:
2333:
2329:
2324:
2320:
2317:
2312:
2308:
2304:
2300:
2299:nondegenerate
2296:
2292:
2288:
2284:
2281:
2277:
2274:
2270:
2262:
2260:
2258:
2254:
2250:
2246:
2243:
2242:real-analytic
2239:
2233:
2216:
2213:
2208:
2187:
2161:
2155:
2152:
2145:
2142:
2141:
2140:
2136:
2134:
2127:
2123:
2118:
2116:
2109:
2090:
2082:
2077:
2073:
2067:
2059:
2054:
2049:
2045:
2039:
2026:
2021:
2017:
2013:
2008:
2003:
1999:
1993:
1983:
1978:
1974:
1970:
1965:
1960:
1956:
1950:
1940:
1935:
1931:
1927:
1922:
1917:
1914:
1904:
1900:
1893:
1886:
1885:
1884:
1881:
1877:
1873:
1869:
1865:
1861:
1841:
1835:
1832:
1829:
1826:
1823:
1817:
1808:
1805:
1802:
1799:
1793:
1787:
1784:
1781:
1778:
1769:
1766:
1760:
1757:
1754:
1748:
1742:
1739:
1736:
1728:
1724:
1716:
1715:
1714:
1700:
1697:
1694:
1691:
1686:
1682:
1673:
1653:
1647:
1644:
1641:
1638:
1635:
1629:
1620:
1617:
1614:
1611:
1605:
1599:
1596:
1593:
1590:
1581:
1578:
1572:
1569:
1566:
1558:
1554:
1550:
1547:
1541:
1538:
1535:
1527:
1523:
1515:
1514:
1513:
1511:
1507:
1503:
1499:
1494:
1492:
1488:
1484:
1480:
1476:
1472:
1469:
1465:
1461:
1457:
1453:
1449:
1445:
1439:
1421:
1413:
1403:
1387:
1384:
1381:
1373:
1363:
1347:
1338:
1334:
1321:
1318:
1310:
1306:
1293:
1286:
1285:
1284:
1262:
1258:
1245:
1242:
1234:
1230:
1217:
1208:
1204:
1191:
1183:
1179:
1166:
1159:
1158:
1157:
1141:
1137:
1133:
1130:
1125:
1121:
1117:
1112:
1108:
1099:
1091:
1075:
1072:
1069:
1056:
1050:
1047:
1044:
1039:
1036:
1033:
1029:
1023:
1020:
1017:
1014:
1011:
1008:
1005:
1002:
999:
995:
991:
988:
981:
980:
979:
977:
972:
970:
939:
935:
931:
927:
888:
885:
880:
877:
874:
871:
868:
864:
860:
843:
829:
826:
821:
818:
815:
812:
809:
805:
801:
791:
790:
789:
787:
783:
780:which maps Ω(
779:
776:
772:
767:
763:
758:
753:
751:
748: +
747:
744: =
743:
739:
735:
715:
709:
698:
695:
692:
679:
676:
673:
670:
667:
663:
659:
644:
636:
624:
623:
622:
620:
616:
612:
608:
603:
601:
597:
596:
591:
587:
583:
579:
575:
571:
567:
563:
559:
555:
551:
547:
543:
539:
535:
531:
523:
521:
519:
515:
514:Hopf problem,
511:
507:
503:
499:
495:
491:
487:
483:
479:
475:
470:
468:
450:
447:
444:
441:
438:
434:
430:
425:
422:
418:
409:
391:
388:
385:
382:
379:
375:
371:
368:
363:
360:
356:
347:
343:
339:
335:
327:
325:
323:
317:
313:
305:
301:
295:
291:
287:
280:
276:
271:
268:-rank tensor
263:
258:
256:
252:
248:
243:
237:
231:
223:
217:
213:
209:
204:
200:
196:
191:
189:
185:
169:
166:
160:
157:
154:
151:
144:
141:
140:vector bundle
125:
122:
119:
114:
110:
98:
94:
91:
88:
84:
80:
76:
73:
69:
61:
59:
57:
53:
48:
46:
42:
38:
37:tangent space
34:
30:
26:
22:
3759:Moving frame
3754:Morse theory
3744:Gauge theory
3536:Tensor field
3465:Closed/Exact
3444:Vector field
3412:Distribution
3353:Hypercomplex
3348:Quaternionic
3246:
3085:Vector field
3043:Smooth atlas
2939:
2933:
2907:
2887:
2867:. Springer.
2864:
2820:
2814:
2770:
2764:
2755:
2710:
2704:
2694:
2625:
2607:
2582:
2563:
2554:
2542:
2533:
2529:
2525:
2521:
2520:). Also, if
2517:
2513:
2509:
2501:
2497:
2495:
2490:
2486:
2482:
2478:
2474:
2470:
2466:
2464:
2458:
2453:
2449:
2444:
2440:
2436:
2432:
2426:
2422:
2418:
2414:
2410:
2403:
2399:
2395:
2391:
2387:
2383:
2376:
2372:
2368:
2364:
2360:
2355:
2351:
2347:
2343:
2339:
2335:
2331:
2327:
2322:
2318:
2310:
2306:
2302:
2294:
2290:
2286:
2282:
2275:
2268:
2266:
2256:
2252:
2244:
2237:
2234:
2231:
2137:
2129:
2125:
2121:
2119:
2111:
2105:
1879:
1875:
1871:
1867:
1863:
1859:
1857:
1671:
1669:
1509:
1505:
1501:
1497:
1495:
1490:
1482:
1478:
1474:
1463:
1459:
1455:
1452:neighborhood
1447:
1443:
1442:given point
1440:
1436:
1282:
1095:
976:identity map
973:
937:
933:
929:
925:
903:
785:
781:
777:
770:
765:
761:
755:As with any
754:
749:
745:
741:
737:
733:
731:
618:
604:
599:
594:
589:
585:
582:vector field
580:is called a
577:
573:
569:
565:
561:
557:
553:
549:
541:
537:
533:
529:
527:
513:
501:
493:
489:
481:
477:
471:
466:
407:
345:
341:
337:
333:
331:
315:
311:
303:
299:
289:
285:
278:
274:
269:
259:
250:
246:
241:
235:
229:
221:
215:
211:
207:
202:
198:
194:
192:
187:
92:
90:tensor field
78:
74:
71:
67:
65:
49:
24:
18:
3704:Levi-Civita
3694:Generalized
3666:Connections
3616:Lie algebra
3548:Volume form
3449:Vector flow
3422:Pushforward
3417:Lie bracket
3316:Lie algebra
3281:G-structure
3070:Pushforward
3050:Submanifold
2651:Chern class
2622:pure spinor
2610:Lie bracket
1468:holomorphic
546:eigenspaces
143:isomorphism
21:mathematics
3827:Stratifold
3785:Diffeology
3581:Associated
3382:Symplectic
3367:Riemannian
3296:Hyperbolic
3223:Submersion
3131:Hopf–Rinow
3065:Submersion
3060:Smooth map
2780:1708.01068
2686:References
2591:direct sum
1674:such that
1487:integrable
757:direct sum
518:Heinz Hopf
255:orientable
102:such that
83:linear map
56:Heinz Hopf
3709:Principal
3684:Ehresmann
3641:Subbundle
3631:Principal
3606:Fibration
3586:Cotangent
3458:Covectors
3311:Lie group
3291:Hermitian
3234:manifolds
3203:Immersion
3198:Foliation
3136:Noether's
3121:Frobenius
3116:De Rham's
3111:Darboux's
3002:Manifolds
2837:0003-486X
2797:119297359
2599:subbundle
2587:isotropic
2375:) form a
2202:¯
2199:∂
2171:¯
2168:∂
2159:∂
2064:∂
2060:−
2036:∂
2014:−
1990:∂
1971:−
1947:∂
1894:−
1818:−
1695:−
1630:−
1551:−
1414:μ
1407:¯
1397:∂
1393:∂
1385:−
1374:μ
1367:¯
1357:∂
1353:∂
1339:μ
1331:∂
1327:∂
1311:μ
1303:∂
1299:∂
1263:μ
1255:∂
1251:∂
1246:−
1235:μ
1227:∂
1223:∂
1209:μ
1201:∂
1197:∂
1184:μ
1176:∂
1172:∂
1142:μ
1126:μ
1113:μ
1073:⋯
1065:¯
1062:∂
1054:∂
1045:∘
1030:π
996:∑
953:¯
950:∂
912:∂
886:∘
865:π
856:¯
853:∂
827:∘
806:π
799:∂
686:Ω
664:⨁
633:Ω
506:octonions
472:The only
465:for even
435:δ
389:−
376:δ
372:−
270:pointwise
257:as well.
234:. But if
164:→
155::
123:−
3852:Category
3805:Orbifold
3800:K-theory
3790:Diffiety
3514:Pullback
3328:Oriented
3306:Kenmotsu
3286:Hadamard
3232:Types of
3181:Geodesic
3006:Glossary
2886:(1980).
2863:(2001).
2747:16578639
2636:See also
2601:and its
2576:subspace
2303:TM → T*M
2289:. Since
2267:Suppose
1504:; i.e.,
940:)), and
904:so that
740:), with
406:for odd
328:Examples
232:) = (−1)
210: :
39:. Every
35:on each
3749:History
3732:Related
3646:Tangent
3624:)
3604:)
3571:Adjoint
3563:Bundles
3541:density
3439:Torsion
3405:Vectors
3397:Tensors
3380:)
3365:)
3361:,
3359:Pseudo−
3338:Poisson
3271:Finsler
3266:Fibered
3261:Contact
3259:)
3251:Complex
3249:)
3218:Section
2964:0058213
2956:2372495
2853:0088770
2845:1970051
2773:: 1–9.
2715:Bibcode
2536:) is a
928:,
784:) to Ω(
769:from Ω(
621:-forms
613:of the
609:out of
474:spheres
182:on the
3714:Vector
3699:Koszul
3679:Cartan
3674:Affine
3656:Vector
3651:Tensor
3636:Spinor
3626:Normal
3622:Stable
3576:Affine
3480:bundle
3432:bundle
3378:Almost
3301:Kähler
3257:Almost
3247:Almost
3241:Closed
3141:Sard's
3097:(list)
2962:
2954:
2914:
2894:
2871:
2851:
2843:
2835:
2795:
2745:
2738:224368
2735:
2504:is an
2251:; for
1489:'. If
1096:Every
516:after
266:(1, 1)
100:(1, 1)
97:degree
87:smooth
3822:Sheaf
3596:Fiber
3372:Rizza
3343:Prime
3174:Local
3164:Curve
3026:Atlas
2952:JSTOR
2841:JSTOR
2793:S2CID
2775:arXiv
2628:is a
2624:then
2439:) = (
1471:atlas
556:and −
544:(the
536:into
228:(det
226:then
27:is a
23:, an
3689:Form
3591:Dual
3524:flow
3387:Tame
3363:Sub−
3276:Flat
3156:Maps
2912:ISBN
2892:ISBN
2869:ISBN
2833:ISSN
2743:PMID
2570:and
2489:has
2477:(•,
2469:and
2417:) =
2394:) =
2297:are
2293:and
2278:, a
2120:The
1870:and
1862:and
1473:for
936:+1,
572:and
540:and
480:and
298:GL(2
282:= −1
240:det
224:= −1
66:Let
54:and
3611:Jet
2944:doi
2825:doi
2785:doi
2733:PMC
2723:doi
2461:)).
1672:A=J
1454:of
548:of
348:):
344:≤ 2
310:GL(
308:to
201:is
193:If
95:of
77:on
19:In
3854::
3602:Co
2960:MR
2958:.
2950:.
2940:75
2938:.
2928:;
2849:MR
2847:.
2839:.
2831:.
2821:65
2791:.
2783:.
2771:57
2769:.
2741:.
2731:.
2721:.
2711:55
2709:.
2703:.
2632:.
2583:TM
2547:.
2540:.
2532:,
2528:,
2518:Jv
2516:,
2448:)(
2425:,
2423:Ju
2413:,
2409:ω(
2404:Jv
2402:,
2390:,
2371:,
2367:,
2217:0.
1713:,
1481:.
971:.
752:.
586:TM
578:TM
574:TM
570:TM
566:TM
520:.
469:.
410:,
340:,
314:,
302:,
216:TM
214:→
212:TM
190:.
47:.
3620:(
3600:(
3376:(
3357:(
3255:(
3245:(
3008:)
3004:(
2994:e
2987:t
2980:v
2966:.
2946::
2920:.
2900:.
2877:.
2855:.
2827::
2799:.
2787::
2777::
2749:.
2725::
2717::
2626:M
2564:M
2534:J
2530:ω
2526:M
2522:J
2514:u
2512:(
2510:ω
2502:J
2498:ω
2487:ω
2483:M
2479:J
2475:ω
2471:J
2467:ω
2459:u
2457:(
2454:ω
2450:φ
2445:g
2441:φ
2437:u
2435:(
2433:J
2429:)
2427:v
2421:(
2419:g
2415:v
2411:u
2406:)
2400:u
2398:(
2396:ω
2392:v
2388:u
2386:(
2384:g
2373:J
2369:ω
2365:g
2361:g
2356:g
2352:φ
2348:u
2346:(
2344:ω
2340:ω
2336:u
2332:i
2328:u
2326:(
2323:ω
2319:φ
2311:ω
2307:φ
2295:g
2291:ω
2287:J
2283:g
2276:ω
2269:M
2257:J
2253:C
2245:J
2238:S
2214:=
2209:2
2162:+
2156:=
2153:d
2132:J
2130:N
2126:J
2114:A
2112:N
2091:.
2088:)
2083:m
2078:i
2074:A
2068:j
2055:m
2050:j
2046:A
2040:i
2032:(
2027:k
2022:m
2018:A
2009:k
2004:i
2000:A
1994:m
1984:m
1979:j
1975:A
1966:k
1961:j
1957:A
1951:m
1941:m
1936:i
1932:A
1928:=
1923:k
1918:j
1915:i
1911:)
1905:A
1901:N
1897:(
1880:A
1876:N
1872:Y
1868:X
1864:Y
1860:X
1842:.
1839:]
1836:Y
1833:J
1830:,
1827:X
1824:J
1821:[
1815:)
1812:]
1809:Y
1806:J
1803:,
1800:X
1797:[
1794:+
1791:]
1788:Y
1785:,
1782:X
1779:J
1776:[
1773:(
1770:J
1767:+
1764:]
1761:Y
1758:,
1755:X
1752:[
1749:=
1746:)
1743:Y
1740:,
1737:X
1734:(
1729:J
1725:N
1701:d
1698:I
1692:=
1687:2
1683:J
1654:.
1651:]
1648:Y
1645:A
1642:,
1639:X
1636:A
1633:[
1627:)
1624:]
1621:Y
1618:A
1615:,
1612:X
1609:[
1606:+
1603:]
1600:Y
1597:,
1594:X
1591:A
1588:[
1585:(
1582:A
1579:+
1576:]
1573:Y
1570:,
1567:X
1564:[
1559:2
1555:A
1548:=
1545:)
1542:Y
1539:,
1536:X
1533:(
1528:A
1524:N
1506:A
1502:M
1498:A
1491:J
1483:J
1479:J
1475:M
1464:J
1460:M
1456:p
1448:J
1444:p
1422:.
1404:z
1388:i
1382:=
1364:z
1348:J
1335:z
1322:i
1319:=
1307:z
1294:J
1259:x
1243:=
1231:y
1218:J
1205:y
1192:=
1180:x
1167:J
1138:y
1134:i
1131:+
1122:x
1118:=
1109:z
1076:.
1070:+
1057:+
1051:=
1048:d
1040:s
1037:,
1034:r
1024:1
1021:+
1018:q
1015:+
1012:p
1009:=
1006:s
1003:+
1000:r
992:=
989:d
938:q
934:p
930:q
926:p
889:d
881:1
878:+
875:q
872:,
869:p
861:=
830:d
822:q
819:,
816:1
813:+
810:p
802:=
786:M
782:M
778:d
771:M
766:q
764:,
762:p
750:q
746:p
742:r
738:M
734:M
716:.
713:)
710:M
707:(
702:)
699:q
696:,
693:p
690:(
680:r
677:=
674:q
671:+
668:p
660:=
654:C
649:)
645:M
642:(
637:r
619:r
600:i
595:i
590:J
562:M
558:i
554:i
550:J
542:V
538:V
534:V
530:V
502:S
494:S
490:S
484:(
482:S
478:S
467:i
451:1
448:+
445:j
442:,
439:i
431:=
426:j
423:i
419:J
408:i
392:1
386:j
383:,
380:i
369:=
364:j
361:i
357:J
346:n
342:j
338:i
334:R
318:)
316:C
312:n
306:)
304:R
300:n
290:M
286:p
279:p
275:J
251:M
247:n
242:J
236:M
230:J
222:J
208:J
203:n
199:M
195:M
170:M
167:T
161:M
158:T
152:J
126:1
120:=
115:2
111:J
93:J
79:M
75:J
68:M
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.