1443:
1289:
1452:
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
2112:
299:. 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
1448:
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.
2246:
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
1300:
2146: = 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.
2149:
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):
1173:
628:, 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
1097:
738:
1900:
910:
1438:{\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 }}}.}
2195:
851:
1165:
1676:
2238:
2476:
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,
976:
415:
474:
1284:{\displaystyle J{\frac {\partial }{\partial x^{\mu }}}={\frac {\partial }{\partial y^{\mu }}}\qquad J{\frac {\partial }{\partial y^{\mu }}}=-{\frac {\partial }{\partial x^{\mu }}}}
1864:
191:
1722:
933:
147:
995:
3825:
2716:
3016:
54:
is an almost complex manifold, but there are almost complex manifolds that are not complex manifolds. Almost complex structures have important applications in
3820:
2107:{\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}).}
638:
3107:
3131:
2259:
1497:
3326:
3196:
2926:
2251:
admits an integrable almost complex structure, despite a long history of ultimately unverified claims. Smoothness issues are important. For
3422:
2570:
2640:
3475:
3003:
275:
shows that any even dimensional vector space admits a linear complex structure. Therefore, an even dimensional manifold always admits a
2667:
2118:
3759:
2906:
2883:
3524:
857:
304:
3116:
2776:
2774:; Bazzoni, Giovanni; Goertsches, Oliver; Konstantis, Panagiotis; Rollenske, Sönke (2018). "On the history of the Hopf problem".
3507:
2945:
799:). Thus we may use the almost complex structure to refine the action of the exterior derivative to the forms of definite type
96:
which squares to −1) on each tangent space of the manifold, which varies smoothly on the manifold. In other words, we have a
3719:
2624:
3704:
3427:
3201:
2652:
2601:
767:
3749:
2159:
3754:
3724:
3432:
3388:
3369:
3136:
3080:
2619:
An almost complex structure integrates to a complex structure if the half-dimensional subspace is closed under the
805:
3868:
3291:
3156:
2516:
1114:
1529:
3676:
3541:
3233:
3075:
2201:
43:
954:
362:
3373:
3343:
3267:
3213:
3043:
2996:
1462:
424:
3141:
3714:
3333:
3228:
3048:
1730:
519:, inherits an almost complex structure from the octonion multiplication; the question of whether it has a
3363:
3358:
2826:
2504:: the complex structures on the tangent fibres compatible with the restriction to the symplectic forms.
208:
admits an almost complex structure, it must be even-dimensional. This can be seen as follows. Suppose
158:
3694:
3632:
3480:
3184:
3174:
3146:
3121:
3031:
2894:
2725:
2578:
1478:
978:
is a map which increases the antiholomorphic part of the type by one. These operators are called the
2582:
3832:
3805:
3514:
3392:
3377:
3306:
3065:
2871:
2689:
785:
265:
55:
3774:
3729:
3626:
3497:
3301:
3126:
2989:
2962:
2851:
2803:
2785:
2597:
979:
332:
3311:
2672:
2548:
1688:
1092:{\displaystyle d=\sum _{r+s=p+q+1}\pi _{r,s}\circ d=\partial +{\overline {\partial }}+\cdots .}
918:
3709:
3689:
3684:
3591:
3502:
3316:
3296:
3151:
3090:
2940:
2922:
2902:
2879:
2843:
2753:
2613:
2290:
2270:, analysis is required (with more difficult techniques as the regularity hypothesis weakens).
986:
617:
116:
62:
3847:
3641:
3596:
3519:
3490:
3348:
3281:
3276:
3271:
3261:
3053:
3036:
2954:
2835:
2795:
2743:
2733:
2678:
2605:
2326:
1481:
1108:
625:
609:
on the (1, 0)-vector fields of the complexified tangent bundle, and multiplication by −
347:
admits an almost complex structure. An example for such an almost complex structure is (1 ≤
51:
2974:
2863:
3790:
3699:
3529:
3485:
3251:
2970:
2859:
2821:
2628:
2586:
2309:
2283:
97:
39:
1869:
The individual expressions on the right depend on the choice of the smooth vector fields
2729:
2243:
Any of these conditions implies the existence of a unique compatible complex structure.
1469:. Such coordinates, if they exist, are called 'local holomorphic coordinates for J'. If
17:
3656:
3581:
3551:
3449:
3442:
3382:
3353:
3223:
3218:
3179:
2684:
2631:. If furthermore this half-dimensional space is the annihilator of a nowhere vanishing
2620:
2590:
935:
is a map which increases the holomorphic part of the type by one (takes forms of type (
621:
604:
508:
272:
194:
2748:
2711:
3862:
3842:
3666:
3661:
3646:
3636:
3586:
3563:
3437:
3397:
3338:
3286:
3085:
2807:
2771:
2566:
2555:
2252:
1504:
is induced by a complex structure, then it is induced by a unique complex structure.
733:{\displaystyle \Omega ^{r}(M)^{\mathbf {C} }=\bigoplus _{p+q=r}\Omega ^{(p,q)}(M).\,}
150:
47:
3769:
3764:
3606:
3573:
3546:
3454:
3095:
2936:
592:
100:
503:
cannot be given an almost complex structure (Ehresmann and Hopf). In the case of
2799:
2154:
The Lie bracket of any two (1, 0)-vector fields is again of type (1, 0)
3612:
3601:
3558:
3459:
3060:
2661:
2632:
153:
31:
579:(which is the vector bundle of complexified tangent spaces at each point) into
3837:
3795:
3621:
3534:
3166:
3070:
556:
528:
93:
66:
2847:
507:, the almost complex structure comes from an honest complex structure on the
3651:
3616:
3321:
3208:
2609:
2757:
2738:
2596:. A generalized almost complex structure is a choice of a half-dimensional
2121:, which generalizes the Lie bracket of vector fields, the Nijenhuis tensor
3815:
3810:
3800:
3191:
3012:
2585:. An ordinary almost complex structure is a choice of a half-dimensional
516:
283:(which is just a linear transformation on each tangent space) such that
2981:
2966:
2890:
Information on compatible triples, Kähler and
Hermitian manifolds, etc.
2855:
1111:
is itself an almost complex manifold. In local holomorphic coordinates
2712:"On the Chern numbers of certain complex and almost complex manifolds"
1457:. In general, however, it is not possible to find coordinates so that
3407:
2577:, which was elaborated in the doctoral dissertations of his students
1877:, but the left side actually depends only on the pointwise values of
484:
107:
2958:
2839:
2824:(1957). "Complex analytic coordinates in almost complex manifolds".
197:. A manifold equipped with an almost complex structure is called an
2790:
2390:
when each structure can be specified by the two others as follows:
2675: – Manifold with Riemannian, complex and symplectic structure
2985:
2496:
whose sections are the almost complex structures compatible to
264:
has an almost complex structure. One can show that it must be
2943:(1953). "Groupes de Lie et puissances réduites de Steenrod".
1894:
is a tensor. This is also clear from the component formula
905:{\displaystyle {\overline {\partial }}=\pi _{p,q+1}\circ d}
2511:, one can show that a compatible almost complex structure
575:
allows a decomposition of the complexified tangent bundle
2623:. A generalized almost complex structure integrates to a
520:
2913:
Short section which introduces standard basic material.
2681: – Mathematical structure in differential geometry
2657:
Pages displaying short descriptions of redirect targets
2608:. In both cases one demands that the direct sum of the
1477:
around every point then these patch together to form a
1681:
or, for the usual case of an almost complex structure
1488:
giving it a complex structure, which moreover induces
989:, we note that the exterior derivative can be written
2204:
2162:
1903:
1733:
1691:
1532:
1303:
1176:
1117:
998:
957:
921:
860:
808:
641:
427:
365:
161:
119:
515:, when considered as the set of unit norm imaginary
3783:
3742:
3675:
3572:
3468:
3415:
3406:
3242:
3165:
3104:
3024:
2507:Using elementary properties of the symplectic form
2258:, the Newlander–Nirenberg theorem follows from the
571:, respectively), so an almost complex structure on
2232:
2189:
2106:
1858:
1716:
1670:
1437:
1294:(just like a counterclockwise rotation of π/2) or
1283:
1159:
1091:
970:
927:
904:
845:
732:
468:
409:
185:
141:
2692: – Type of manifold in differential geometry
985:Since the sum of all the projections must be the
535:Differential topology of almost complex manifolds
2664: – Characteristic classes of vector bundles
1519:is a tensor field of rank (1, 1), then the
2717:Proceedings of the National Academy of Sciences
2374:. With this understood, the three structures (
2190:{\displaystyle d=\partial +{\bar {\partial }}}
539:Just as a complex structure on a vector space
2997:
846:{\displaystyle \partial =\pi _{p+1,q}\circ d}
8:
1160:{\displaystyle z^{\mu }=x^{\mu }+iy^{\mu }}
3412:
3004:
2990:
2982:
2899:Differential Analysis on Complex Manifolds
2777:Differential Geometry and Its Applications
2370:, is given by the analogous operation for
1671:{\displaystyle N_{A}(X,Y)=-A^{2}+A(+)-.\,}
496:
487:which admit almost complex structures are
331:. The existence question is then a purely
92:is a linear complex structure (that is, a
2789:
2747:
2737:
2492:•) is a Riemannian metric. The bundle on
2233:{\displaystyle {\bar {\partial }}^{2}=0.}
2218:
2207:
2206:
2203:
2176:
2175:
2161:
2092:
2087:
2077:
2064:
2059:
2049:
2036:
2031:
2018:
2013:
2003:
1993:
1988:
1975:
1970:
1960:
1950:
1945:
1932:
1924:
1914:
1902:
1855:
1738:
1732:
1696:
1690:
1667:
1568:
1537:
1531:
1523:is a tensor field of rank (1,2) given by
1473:admits local holomorphic coordinates for
1423:
1412:
1411:
1401:
1383:
1372:
1371:
1361:
1348:
1335:
1320:
1307:
1302:
1272:
1259:
1244:
1231:
1218:
1205:
1193:
1180:
1175:
1151:
1135:
1122:
1116:
1070:
1043:
1009:
997:
958:
956:
920:
878:
861:
859:
819:
807:
747:) admits a decomposition into a sum of Ω(
729:
699:
677:
663:
662:
646:
640:
448:
432:
426:
389:
370:
364:
160:
124:
118:
2919:Algebraic Geometry, a concise dictionary
2556:2 out of 3 property of the unitary group
2135:states that an almost complex structure
2702:
971:{\displaystyle {\overline {\partial }}}
599:is a vector field of type (0, 1). Thus
410:{\displaystyle J_{ij}=-\delta _{i,j-1}}
1461:takes the canonical form on an entire
469:{\displaystyle J_{ij}=\delta _{i,j+1}}
7:
2921:. Berlin/Boston: Walter De Gruyter.
2627:if the subspace is closed under the
2571:generalized almost complex structure
2562:Generalized almost complex structure
2312:, each induces a bundle isomorphism
1859:{\displaystyle N_{J}(X,Y)=+J(+)-.\,}
1103:Integrable almost complex structures
343:For every integer n, the flat space
770:, there is a canonical projection π
595:of type (1, 0), while a section of
335:one and is fairly well understood.
230:be an almost complex structure. If
2589:of each fiber of the complexified
2296:, and an almost complex structure
2209:
2178:
2169:
2074:
2046:
2000:
1957:
1407:
1403:
1367:
1363:
1341:
1337:
1313:
1309:
1265:
1261:
1237:
1233:
1211:
1207:
1186:
1182:
1072:
1064:
960:
922:
863:
809:
696:
643:
613:on the (0, 1)-vector fields.
25:
2554:These triples are related to the
2361:, •) and the other, denoted
603:corresponds to multiplication by
2604:of the complexified tangent and
664:
305:reduction of the structure group
186:{\displaystyle J\colon TM\to TM}
2946:American Journal of Mathematics
2876:Lectures on Symplectic Geometry
2641:generalized Calabi–Yau manifold
2316:, where the first map, denoted
1357:
1227:
3044:Differentiable/Smooth manifold
2600:subspace of each fiber of the
2484:are compatible if and only if
2212:
2181:
2098:
2042:
1921:
1907:
1849:
1831:
1825:
1822:
1807:
1801:
1786:
1783:
1774:
1762:
1756:
1744:
1661:
1643:
1637:
1634:
1619:
1613:
1598:
1595:
1586:
1574:
1555:
1543:
1417:
1377:
723:
717:
712:
700:
659:
652:
174:
1:
2901:. New York: Springer-Verlag.
2625:generalized complex structure
2139:is integrable if and only if
2800:10.1016/j.difgeo.2017.10.014
2710:Van de Ven, A. (June 1966).
2653:Almost quaternionic manifold
1075:
963:
866:
3750:Classification of manifolds
2668:Frölicher–Nijenhuis bracket
2655: – Concept in geometry
2616:yield the original bundle.
2569:introduced the notion of a
2133:Newlander–Nirenberg theorem
2119:Frölicher–Nijenhuis bracket
307:of the tangent bundle from
3885:
2519:for the Riemannian metric
543:allows a decomposition of
3826:over commutative algebras
1717:{\displaystyle J^{2}=-Id}
1511:on each tangent space of
928:{\displaystyle \partial }
784:) to Ω. We also have the
249:is a real manifold, then
81:be a smooth manifold. An
3542:Riemann curvature tensor
1167:one can define the maps
497:Borel & Serre (1953)
256:is a real number – thus
142:{\displaystyle J^{2}=-1}
83:almost complex structure
44:linear complex structure
18:Almost-complex structure
2517:almost Kähler structure
743:In other words, each Ω(
199:almost complex manifold
42:equipped with a smooth
36:almost complex manifold
3334:Manifold with boundary
3049:Differential structure
2739:10.1073/pnas.55.6.1624
2234:
2191:
2128:is just one-half of .
2108:
1860:
1718:
1672:
1439:
1285:
1161:
1093:
972:
929:
906:
847:
734:
470:
411:
216:-dimensional, and let
187:
143:
61:The concept is due to
2917:Rubei, Elena (2014).
2827:Annals of Mathematics
2535:is integrable, then (
2235:
2192:
2109:
1861:
1719:
1673:
1507:Given any linear map
1440:
1286:
1162:
1094:
973:
930:
907:
848:
735:
471:
412:
333:algebraic topological
188:
144:
3481:Covariant derivative
3032:Topological manifold
2872:Cannas da Silva, Ana
2202:
2160:
1901:
1731:
1689:
1530:
1496:is then said to be '
1301:
1174:
1115:
996:
955:
943:) to forms of type (
919:
858:
806:
639:
425:
363:
271:An easy exercise in
159:
117:
3515:Exterior derivative
3117:Atiyah–Singer index
3066:Riemannian manifold
2820:Newlander, August;
2730:1966PNAS...55.1624V
2690:Symplectic manifold
2502:contractible fibres
2282:is equipped with a
2097:
2069:
2041:
2023:
1998:
1980:
1955:
1937:
980:Dolbeault operators
786:exterior derivative
303:is equivalent to a
149:when regarded as a
56:symplectic geometry
3821:Secondary calculus
3775:Singularity theory
3730:Parallel transport
3498:De Rham cohomology
3137:Generalized Stokes
2941:Serre, Jean-Pierre
2325:, is given by the
2274:Compatible triples
2266:(and less smooth)
2230:
2187:
2104:
2083:
2055:
2027:
2009:
1984:
1966:
1941:
1920:
1856:
1714:
1668:
1435:
1281:
1157:
1089:
1038:
968:
925:
902:
843:
730:
694:
618:differential forms
563:corresponding to +
499:). In particular,
466:
407:
183:
139:
3856:
3855:
3738:
3737:
3503:Differential form
3157:Whitney embedding
3091:Differential form
2928:978-3-11-031622-3
2895:Wells, Raymond O.
2830:. Second Series.
2614:complex conjugate
2606:cotangent bundles
2388:compatible triple
2291:Riemannian metric
2260:Frobenius theorem
2215:
2184:
1430:
1420:
1390:
1380:
1355:
1327:
1279:
1251:
1225:
1200:
1078:
1005:
966:
869:
673:
616:Just as we build
521:complex structure
73:Formal definition
63:Charles Ehresmann
16:(Redirected from
3876:
3869:Smooth manifolds
3848:Stratified space
3806:Fréchet manifold
3520:Interior product
3413:
3110:
3006:
2999:
2992:
2983:
2978:
2932:
2912:
2889:
2867:
2822:Nirenberg, Louis
2812:
2811:
2793:
2768:
2762:
2761:
2751:
2741:
2724:(6): 1624–1627.
2707:
2679:Poisson manifold
2658:
2573:on the manifold
2327:interior product
2239:
2237:
2236:
2231:
2223:
2222:
2217:
2216:
2208:
2196:
2194:
2193:
2188:
2186:
2185:
2177:
2117:In terms of the
2113:
2111:
2110:
2105:
2096:
2091:
2082:
2081:
2068:
2063:
2054:
2053:
2040:
2035:
2022:
2017:
2008:
2007:
1997:
1992:
1979:
1974:
1965:
1964:
1954:
1949:
1936:
1931:
1919:
1918:
1865:
1863:
1862:
1857:
1743:
1742:
1723:
1721:
1720:
1715:
1701:
1700:
1677:
1675:
1674:
1669:
1573:
1572:
1542:
1541:
1521:Nijenhuis tensor
1444:
1442:
1441:
1436:
1431:
1429:
1428:
1427:
1422:
1421:
1413:
1402:
1391:
1389:
1388:
1387:
1382:
1381:
1373:
1362:
1356:
1354:
1353:
1352:
1336:
1328:
1326:
1325:
1324:
1308:
1290:
1288:
1287:
1282:
1280:
1278:
1277:
1276:
1260:
1252:
1250:
1249:
1248:
1232:
1226:
1224:
1223:
1222:
1206:
1201:
1199:
1198:
1197:
1181:
1166:
1164:
1163:
1158:
1156:
1155:
1140:
1139:
1127:
1126:
1109:complex manifold
1098:
1096:
1095:
1090:
1079:
1071:
1054:
1053:
1037:
977:
975:
974:
969:
967:
959:
934:
932:
931:
926:
911:
909:
908:
903:
895:
894:
870:
862:
852:
850:
849:
844:
836:
835:
739:
737:
736:
731:
716:
715:
693:
669:
668:
667:
651:
650:
626:cotangent bundle
523:is known as the
511:. The 6-sphere,
475:
473:
472:
467:
465:
464:
440:
439:
416:
414:
413:
408:
406:
405:
378:
377:
330:
318:
294:
278:
260:must be even if
255:
244:
236:
229:
192:
190:
189:
184:
148:
146:
145:
140:
129:
128:
112:
52:complex manifold
21:
3884:
3883:
3879:
3878:
3877:
3875:
3874:
3873:
3859:
3858:
3857:
3852:
3791:Banach manifold
3784:Generalizations
3779:
3734:
3671:
3568:
3530:Ricci curvature
3486:Cotangent space
3464:
3402:
3244:
3238:
3197:Exponential map
3161:
3106:
3100:
3020:
3010:
2959:10.2307/2372495
2935:
2929:
2916:
2909:
2893:
2886:
2870:
2840:10.2307/1970051
2819:
2816:
2815:
2770:
2769:
2765:
2709:
2708:
2704:
2699:
2673:Kähler manifold
2656:
2649:
2629:Courant bracket
2579:Marco Gualtieri
2564:
2549:Kähler manifold
2467:
2458:
2369:
2349:
2336:
2324:
2284:symplectic form
2276:
2205:
2200:
2199:
2158:
2157:
2144:
2126:
2073:
2045:
1999:
1956:
1910:
1899:
1898:
1893:
1885:, which is why
1734:
1729:
1728:
1692:
1687:
1686:
1564:
1533:
1528:
1527:
1410:
1406:
1370:
1366:
1344:
1340:
1316:
1312:
1299:
1298:
1268:
1264:
1240:
1236:
1214:
1210:
1189:
1185:
1172:
1171:
1147:
1131:
1118:
1113:
1112:
1105:
1039:
994:
993:
953:
952:
917:
916:
874:
856:
855:
815:
804:
803:
779:
695:
658:
642:
637:
636:
622:exterior powers
587:. A section of
537:
444:
428:
423:
422:
385:
366:
361:
360:
341:
320:
308:
292:
284:
276:
250:
238:
231:
217:
157:
156:
120:
115:
114:
110:
75:
40:smooth manifold
28:
27:Smooth manifold
23:
22:
15:
12:
11:
5:
3882:
3880:
3872:
3871:
3861:
3860:
3854:
3853:
3851:
3850:
3845:
3840:
3835:
3830:
3829:
3828:
3818:
3813:
3808:
3803:
3798:
3793:
3787:
3785:
3781:
3780:
3778:
3777:
3772:
3767:
3762:
3757:
3752:
3746:
3744:
3740:
3739:
3736:
3735:
3733:
3732:
3727:
3722:
3717:
3712:
3707:
3702:
3697:
3692:
3687:
3681:
3679:
3673:
3672:
3670:
3669:
3664:
3659:
3654:
3649:
3644:
3639:
3629:
3624:
3619:
3609:
3604:
3599:
3594:
3589:
3584:
3578:
3576:
3570:
3569:
3567:
3566:
3561:
3556:
3555:
3554:
3544:
3539:
3538:
3537:
3527:
3522:
3517:
3512:
3511:
3510:
3500:
3495:
3494:
3493:
3483:
3478:
3472:
3470:
3466:
3465:
3463:
3462:
3457:
3452:
3447:
3446:
3445:
3435:
3430:
3425:
3419:
3417:
3410:
3404:
3403:
3401:
3400:
3395:
3385:
3380:
3366:
3361:
3356:
3351:
3346:
3344:Parallelizable
3341:
3336:
3331:
3330:
3329:
3319:
3314:
3309:
3304:
3299:
3294:
3289:
3284:
3279:
3274:
3264:
3254:
3248:
3246:
3240:
3239:
3237:
3236:
3231:
3226:
3224:Lie derivative
3221:
3219:Integral curve
3216:
3211:
3206:
3205:
3204:
3194:
3189:
3188:
3187:
3180:Diffeomorphism
3177:
3171:
3169:
3163:
3162:
3160:
3159:
3154:
3149:
3144:
3139:
3134:
3129:
3124:
3119:
3113:
3111:
3102:
3101:
3099:
3098:
3093:
3088:
3083:
3078:
3073:
3068:
3063:
3058:
3057:
3056:
3051:
3041:
3040:
3039:
3028:
3026:
3025:Basic concepts
3022:
3021:
3011:
3009:
3008:
3001:
2994:
2986:
2980:
2979:
2953:(3): 409–448.
2933:
2927:
2914:
2907:
2891:
2884:
2868:
2834:(3): 391–404.
2814:
2813:
2772:Agricola, Ilka
2763:
2701:
2700:
2698:
2695:
2694:
2693:
2687:
2685:Rizza manifold
2682:
2676:
2670:
2665:
2659:
2648:
2645:
2591:tangent bundle
2583:Gil Cavalcanti
2563:
2560:
2474:
2473:
2463:
2454:
2441:
2418:
2365:
2345:
2341:) =
2332:
2320:
2275:
2272:
2241:
2240:
2229:
2226:
2221:
2214:
2211:
2197:
2183:
2180:
2174:
2171:
2168:
2165:
2155:
2142:
2124:
2115:
2114:
2103:
2100:
2095:
2090:
2086:
2080:
2076:
2072:
2067:
2062:
2058:
2052:
2048:
2044:
2039:
2034:
2030:
2026:
2021:
2016:
2012:
2006:
2002:
1996:
1991:
1987:
1983:
1978:
1973:
1969:
1963:
1959:
1953:
1948:
1944:
1940:
1935:
1930:
1927:
1923:
1917:
1913:
1909:
1906:
1889:
1867:
1866:
1854:
1851:
1848:
1845:
1842:
1839:
1836:
1833:
1830:
1827:
1824:
1821:
1818:
1815:
1812:
1809:
1806:
1803:
1800:
1797:
1794:
1791:
1788:
1785:
1782:
1779:
1776:
1773:
1770:
1767:
1764:
1761:
1758:
1755:
1752:
1749:
1746:
1741:
1737:
1713:
1710:
1707:
1704:
1699:
1695:
1679:
1678:
1666:
1663:
1660:
1657:
1654:
1651:
1648:
1645:
1642:
1639:
1636:
1633:
1630:
1627:
1624:
1621:
1618:
1615:
1612:
1609:
1606:
1603:
1600:
1597:
1594:
1591:
1588:
1585:
1582:
1579:
1576:
1571:
1567:
1563:
1560:
1557:
1554:
1551:
1548:
1545:
1540:
1536:
1446:
1445:
1434:
1426:
1419:
1416:
1409:
1405:
1400:
1397:
1394:
1386:
1379:
1376:
1369:
1365:
1360:
1351:
1347:
1343:
1339:
1334:
1331:
1323:
1319:
1315:
1311:
1306:
1292:
1291:
1275:
1271:
1267:
1263:
1258:
1255:
1247:
1243:
1239:
1235:
1230:
1221:
1217:
1213:
1209:
1204:
1196:
1192:
1188:
1184:
1179:
1154:
1150:
1146:
1143:
1138:
1134:
1130:
1125:
1121:
1104:
1101:
1100:
1099:
1088:
1085:
1082:
1077:
1074:
1069:
1066:
1063:
1060:
1057:
1052:
1049:
1046:
1042:
1036:
1033:
1030:
1027:
1024:
1021:
1018:
1015:
1012:
1008:
1004:
1001:
965:
962:
924:
913:
912:
901:
898:
893:
890:
887:
884:
881:
877:
873:
868:
865:
853:
842:
839:
834:
831:
828:
825:
822:
818:
814:
811:
771:
741:
740:
728:
725:
722:
719:
714:
711:
708:
705:
702:
698:
692:
689:
686:
683:
680:
676:
672:
666:
661:
657:
654:
649:
645:
536:
533:
509:Riemann sphere
463:
460:
457:
454:
451:
447:
443:
438:
435:
431:
404:
401:
398:
395:
392:
388:
384:
381:
376:
373:
369:
340:
337:
295:at each point
288:
273:linear algebra
195:tangent bundle
182:
179:
176:
173:
170:
167:
164:
138:
135:
132:
127:
123:
74:
71:
69:in the 1940s.
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3881:
3870:
3867:
3866:
3864:
3849:
3846:
3844:
3843:Supermanifold
3841:
3839:
3836:
3834:
3831:
3827:
3824:
3823:
3822:
3819:
3817:
3814:
3812:
3809:
3807:
3804:
3802:
3799:
3797:
3794:
3792:
3789:
3788:
3786:
3782:
3776:
3773:
3771:
3768:
3766:
3763:
3761:
3758:
3756:
3753:
3751:
3748:
3747:
3745:
3741:
3731:
3728:
3726:
3723:
3721:
3718:
3716:
3713:
3711:
3708:
3706:
3703:
3701:
3698:
3696:
3693:
3691:
3688:
3686:
3683:
3682:
3680:
3678:
3674:
3668:
3665:
3663:
3660:
3658:
3655:
3653:
3650:
3648:
3645:
3643:
3640:
3638:
3634:
3630:
3628:
3625:
3623:
3620:
3618:
3614:
3610:
3608:
3605:
3603:
3600:
3598:
3595:
3593:
3590:
3588:
3585:
3583:
3580:
3579:
3577:
3575:
3571:
3565:
3564:Wedge product
3562:
3560:
3557:
3553:
3550:
3549:
3548:
3545:
3543:
3540:
3536:
3533:
3532:
3531:
3528:
3526:
3523:
3521:
3518:
3516:
3513:
3509:
3508:Vector-valued
3506:
3505:
3504:
3501:
3499:
3496:
3492:
3489:
3488:
3487:
3484:
3482:
3479:
3477:
3474:
3473:
3471:
3467:
3461:
3458:
3456:
3453:
3451:
3448:
3444:
3441:
3440:
3439:
3438:Tangent space
3436:
3434:
3431:
3429:
3426:
3424:
3421:
3420:
3418:
3414:
3411:
3409:
3405:
3399:
3396:
3394:
3390:
3386:
3384:
3381:
3379:
3375:
3371:
3367:
3365:
3362:
3360:
3357:
3355:
3352:
3350:
3347:
3345:
3342:
3340:
3337:
3335:
3332:
3328:
3325:
3324:
3323:
3320:
3318:
3315:
3313:
3310:
3308:
3305:
3303:
3300:
3298:
3295:
3293:
3290:
3288:
3285:
3283:
3280:
3278:
3275:
3273:
3269:
3265:
3263:
3259:
3255:
3253:
3250:
3249:
3247:
3241:
3235:
3232:
3230:
3227:
3225:
3222:
3220:
3217:
3215:
3212:
3210:
3207:
3203:
3202:in Lie theory
3200:
3199:
3198:
3195:
3193:
3190:
3186:
3183:
3182:
3181:
3178:
3176:
3173:
3172:
3170:
3168:
3164:
3158:
3155:
3153:
3150:
3148:
3145:
3143:
3140:
3138:
3135:
3133:
3130:
3128:
3125:
3123:
3120:
3118:
3115:
3114:
3112:
3109:
3105:Main results
3103:
3097:
3094:
3092:
3089:
3087:
3086:Tangent space
3084:
3082:
3079:
3077:
3074:
3072:
3069:
3067:
3064:
3062:
3059:
3055:
3052:
3050:
3047:
3046:
3045:
3042:
3038:
3035:
3034:
3033:
3030:
3029:
3027:
3023:
3018:
3014:
3007:
3002:
3000:
2995:
2993:
2988:
2987:
2984:
2976:
2972:
2968:
2964:
2960:
2956:
2952:
2948:
2947:
2942:
2938:
2937:Borel, Armand
2934:
2930:
2924:
2920:
2915:
2910:
2908:0-387-90419-0
2904:
2900:
2896:
2892:
2887:
2885:3-540-42195-5
2881:
2877:
2873:
2869:
2865:
2861:
2857:
2853:
2849:
2845:
2841:
2837:
2833:
2829:
2828:
2823:
2818:
2817:
2809:
2805:
2801:
2797:
2792:
2787:
2783:
2779:
2778:
2773:
2767:
2764:
2759:
2755:
2750:
2745:
2740:
2735:
2731:
2727:
2723:
2719:
2718:
2713:
2706:
2703:
2696:
2691:
2688:
2686:
2683:
2680:
2677:
2674:
2671:
2669:
2666:
2663:
2660:
2654:
2651:
2650:
2646:
2644:
2642:
2638:
2634:
2630:
2626:
2622:
2617:
2615:
2611:
2607:
2603:
2599:
2595:
2592:
2588:
2584:
2580:
2576:
2572:
2568:
2567:Nigel Hitchin
2561:
2559:
2557:
2552:
2550:
2546:
2542:
2538:
2534:
2530:
2526:
2522:
2518:
2514:
2510:
2505:
2503:
2499:
2495:
2491:
2487:
2483:
2479:
2471:
2466:
2462:
2457:
2453:
2449:
2445:
2442:
2439:
2435:
2431:
2427:
2423:
2419:
2416:
2412:
2408:
2404:
2400:
2396:
2393:
2392:
2391:
2389:
2385:
2381:
2377:
2373:
2368:
2364:
2360:
2356:
2353: =
2352:
2348:
2344:
2340:
2335:
2331:
2328:
2323:
2319:
2315:
2311:
2310:nondegenerate
2307:
2303:
2299:
2295:
2292:
2288:
2285:
2281:
2273:
2271:
2269:
2265:
2261:
2257:
2254:
2253:real-analytic
2250:
2244:
2227:
2224:
2219:
2198:
2172:
2166:
2163:
2156:
2153:
2152:
2151:
2147:
2145:
2138:
2134:
2129:
2127:
2120:
2101:
2093:
2088:
2084:
2078:
2070:
2065:
2060:
2056:
2050:
2037:
2032:
2028:
2024:
2019:
2014:
2010:
2004:
1994:
1989:
1985:
1981:
1976:
1971:
1967:
1961:
1951:
1946:
1942:
1938:
1933:
1928:
1925:
1915:
1911:
1904:
1897:
1896:
1895:
1892:
1888:
1884:
1880:
1876:
1872:
1852:
1846:
1843:
1840:
1837:
1834:
1828:
1819:
1816:
1813:
1810:
1804:
1798:
1795:
1792:
1789:
1780:
1777:
1771:
1768:
1765:
1759:
1753:
1750:
1747:
1739:
1735:
1727:
1726:
1725:
1711:
1708:
1705:
1702:
1697:
1693:
1684:
1664:
1658:
1655:
1652:
1649:
1646:
1640:
1631:
1628:
1625:
1622:
1616:
1610:
1607:
1604:
1601:
1592:
1589:
1583:
1580:
1577:
1569:
1565:
1561:
1558:
1552:
1549:
1546:
1538:
1534:
1526:
1525:
1524:
1522:
1518:
1514:
1510:
1505:
1503:
1499:
1495:
1491:
1487:
1483:
1480:
1476:
1472:
1468:
1464:
1460:
1456:
1450:
1432:
1424:
1414:
1398:
1395:
1392:
1384:
1374:
1358:
1349:
1345:
1332:
1329:
1321:
1317:
1304:
1297:
1296:
1295:
1273:
1269:
1256:
1253:
1245:
1241:
1228:
1219:
1215:
1202:
1194:
1190:
1177:
1170:
1169:
1168:
1152:
1148:
1144:
1141:
1136:
1132:
1128:
1123:
1119:
1110:
1102:
1086:
1083:
1080:
1067:
1061:
1058:
1055:
1050:
1047:
1044:
1040:
1034:
1031:
1028:
1025:
1022:
1019:
1016:
1013:
1010:
1006:
1002:
999:
992:
991:
990:
988:
983:
981:
950:
946:
942:
938:
899:
896:
891:
888:
885:
882:
879:
875:
871:
854:
840:
837:
832:
829:
826:
823:
820:
816:
812:
802:
801:
800:
798:
794:
791:which maps Ω(
790:
787:
783:
778:
774:
769:
764:
762:
759: +
758:
755: =
754:
750:
746:
726:
720:
709:
706:
703:
690:
687:
684:
681:
678:
674:
670:
655:
647:
635:
634:
633:
631:
627:
623:
619:
614:
612:
608:
607:
602:
598:
594:
590:
586:
582:
578:
574:
570:
566:
562:
558:
554:
550:
546:
542:
534:
532:
530:
526:
525:Hopf problem,
522:
518:
514:
510:
506:
502:
498:
494:
490:
486:
481:
479:
461:
458:
455:
452:
449:
445:
441:
436:
433:
429:
420:
402:
399:
396:
393:
390:
386:
382:
379:
374:
371:
367:
358:
354:
350:
346:
338:
336:
334:
328:
324:
316:
312:
306:
302:
298:
291:
287:
282:
279:-rank tensor
274:
269:
267:
263:
259:
254:
248:
242:
234:
228:
224:
220:
215:
211:
207:
202:
200:
196:
180:
177:
171:
168:
165:
162:
155:
152:
151:vector bundle
136:
133:
130:
125:
121:
109:
105:
102:
99:
95:
91:
87:
84:
80:
72:
70:
68:
64:
59:
57:
53:
49:
48:tangent space
45:
41:
37:
33:
19:
3770:Moving frame
3765:Morse theory
3755:Gauge theory
3547:Tensor field
3476:Closed/Exact
3455:Vector field
3423:Distribution
3364:Hypercomplex
3359:Quaternionic
3257:
3096:Vector field
3054:Smooth atlas
2950:
2944:
2918:
2898:
2878:. Springer.
2875:
2831:
2825:
2781:
2775:
2766:
2721:
2715:
2705:
2636:
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1470:
1466:
1463:neighborhood
1458:
1454:
1453:given point
1451:
1447:
1293:
1106:
987:identity map
984:
948:
944:
940:
936:
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792:
788:
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776:
772:
766:As with any
765:
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752:
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600:
596:
593:vector field
591:is called a
588:
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564:
560:
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544:
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203:
198:
103:
101:tensor field
89:
85:
82:
78:
76:
60:
35:
29:
3715:Levi-Civita
3705:Generalized
3677:Connections
3627:Lie algebra
3559:Volume form
3460:Vector flow
3433:Pushforward
3428:Lie bracket
3327:Lie algebra
3292:G-structure
3081:Pushforward
3061:Submanifold
2662:Chern class
2633:pure spinor
2621:Lie bracket
1479:holomorphic
557:eigenspaces
154:isomorphism
32:mathematics
3838:Stratifold
3796:Diffeology
3592:Associated
3393:Symplectic
3378:Riemannian
3307:Hyperbolic
3234:Submersion
3142:Hopf–Rinow
3076:Submersion
3071:Smooth map
2791:1708.01068
2697:References
2602:direct sum
1685:such that
1498:integrable
768:direct sum
529:Heinz Hopf
266:orientable
113:such that
94:linear map
67:Heinz Hopf
3720:Principal
3695:Ehresmann
3652:Subbundle
3642:Principal
3617:Fibration
3597:Cotangent
3469:Covectors
3322:Lie group
3302:Hermitian
3245:manifolds
3214:Immersion
3209:Foliation
3147:Noether's
3132:Frobenius
3127:De Rham's
3122:Darboux's
3013:Manifolds
2848:0003-486X
2808:119297359
2610:subbundle
2598:isotropic
2386:) form a
2213:¯
2210:∂
2182:¯
2179:∂
2170:∂
2075:∂
2071:−
2047:∂
2025:−
2001:∂
1982:−
1958:∂
1905:−
1829:−
1706:−
1641:−
1562:−
1425:μ
1418:¯
1408:∂
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1396:−
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1378:¯
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1364:∂
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1342:∂
1338:∂
1322:μ
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1274:μ
1266:∂
1262:∂
1257:−
1246:μ
1238:∂
1234:∂
1220:μ
1212:∂
1208:∂
1195:μ
1187:∂
1183:∂
1153:μ
1137:μ
1124:μ
1084:⋯
1076:¯
1073:∂
1065:∂
1056:∘
1041:π
1007:∑
964:¯
961:∂
923:∂
897:∘
876:π
867:¯
864:∂
838:∘
817:π
810:∂
697:Ω
675:⨁
644:Ω
517:octonions
483:The only
476:for even
446:δ
400:−
387:δ
383:−
281:pointwise
268:as well.
245:. But if
175:→
166::
134:−
3863:Category
3816:Orbifold
3811:K-theory
3801:Diffiety
3525:Pullback
3339:Oriented
3317:Kenmotsu
3297:Hadamard
3243:Types of
3192:Geodesic
3017:Glossary
2897:(1980).
2874:(2001).
2758:16578639
2647:See also
2612:and its
2587:subspace
2314:TM → T*M
2300:. Since
2278:Suppose
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951:)), and
915:so that
751:), with
417:for odd
339:Examples
243:) = (−1)
221: :
50:. Every
46:on each
3760:History
3743:Related
3657:Tangent
3635:)
3615:)
3582:Adjoint
3574:Bundles
3552:density
3450:Torsion
3416:Vectors
3408:Tensors
3391:)
3376:)
3372:,
3370:Pseudo−
3349:Poisson
3282:Finsler
3277:Fibered
3272:Contact
3270:)
3262:Complex
3260:)
3229:Section
2975:0058213
2967:2372495
2864:0088770
2856:1970051
2784:: 1–9.
2726:Bibcode
2547:) is a
939:,
795:) to Ω(
780:from Ω(
632:-forms
624:of the
620:out of
485:spheres
193:on the
3725:Vector
3710:Koszul
3690:Cartan
3685:Affine
3667:Vector
3662:Tensor
3647:Spinor
3637:Normal
3633:Stable
3587:Affine
3491:bundle
3443:bundle
3389:Almost
3312:Kähler
3268:Almost
3258:Almost
3252:Closed
3152:Sard's
3108:(list)
2973:
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1500:'. If
1107:Every
527:after
277:(1, 1)
111:(1, 1)
108:degree
98:smooth
3833:Sheaf
3607:Fiber
3383:Rizza
3354:Prime
3185:Local
3175:Curve
3037:Atlas
2963:JSTOR
2852:JSTOR
2804:S2CID
2786:arXiv
2639:is a
2635:then
2450:) = (
1482:atlas
567:and −
555:(the
547:into
239:(det
237:then
38:is a
34:, an
3700:Form
3602:Dual
3535:flow
3398:Tame
3374:Sub−
3287:Flat
3167:Maps
2923:ISBN
2903:ISBN
2880:ISBN
2844:ISSN
2754:PMID
2581:and
2500:has
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2480:and
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2308:are
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2131:The
1881:and
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583:and
551:and
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