2570:
of a pendulum. The state of the pendulum is determined by its position (an angle) and its momentum (or equivalently, its velocity, since its mass is constant). The entire state space looks like a cylinder, which is the cotangent bundle of the circle. The above symplectic construction, along with an
1848:
1358:
340:
715:
1073:
1480:
2171:
633:
1661:
2362:. Proving that this form is, indeed, symplectic can be done by noting that being symplectic is a local property: since the cotangent bundle is locally trivial, this definition need only be checked on
2404:
2287:
811:
78:. In the smooth case, any Riemannian metric or symplectic form gives an isomorphism between the cotangent bundle and the tangent bundle, but they are not in general isomorphic in other categories.
907:
240:
1194:
533:
849:
1202:
757:
1127:
1528:
2491:
1615:
420:
570:
939:
463:
178:
2445:
2556:
1654:
1564:
3504:
2695:
2519:
260:
3499:
2786:
638:
2810:
3005:
947:
1366:
2082:
575:
2875:
3101:
3154:
2682:
3438:
2640:
2621:
1843:{\displaystyle T^{*}M={\bigl \{}(x,v^{*})\in T^{*}\mathbb {R} ^{n}\ :\ f(x)=0,\ v\in T_{x}\mathbb {R} ^{n},\ df_{x}(v)=0{\bigr \}}.}
3203:
2181:
470:
423:
2795:
3186:
3552:
2365:
2234:
766:
3398:
2605:
2044:
2334:
and applying ω to this projection. Note that the tautological one-form is not a pullback of a one-form on the base
858:
3383:
3106:
2880:
202:
3428:
1135:
3433:
3403:
3111:
3067:
3048:
2815:
2759:
2005:
This section can be constructed in several ways. The most elementary method uses local coordinates. Suppose that
2970:
2835:
476:
3355:
3220:
2912:
2754:
2196:
1353:{\displaystyle T^{*}M={\bigl \{}(x,v^{*})\in T^{*}\mathbb {R} ^{n}\ :\ f(x)=0,\ v^{*}\in T_{x}^{*}M{\bigr \}},}
819:
3547:
3052:
3022:
2946:
2936:
2892:
2722:
2675:
2592:
1927:
720:
109:
2820:
3393:
3012:
2907:
2727:
1979:
1130:
1081:
375:
242:
consists of equivalence classes of functions which vanish on the diagonal modulo higher order terms. The
63:
1485:
3042:
3037:
2651:
2576:
2450:
2355:
1947:
1935:
1919:
1883:
1569:
393:
247:
35:
2296:
in the tangent bundle of the cotangent bundle, the application of the tautological one-form θ to
1874:, it can be regarded as a manifold in its own right. Because at each point the tangent directions of
546:
3373:
3311:
3159:
2863:
2853:
2825:
2800:
2710:
2359:
1953:
912:
428:
75:
159:
3511:
3484:
3193:
3071:
3056:
2985:
2744:
2409:
2351:
1923:
1887:
346:
197:
181:
2528:
1620:
3453:
3408:
3305:
3176:
2980:
2805:
2668:
2609:
1533:
350:
185:
98:
2990:
3388:
3368:
3363:
3270:
3181:
2995:
2975:
2830:
2769:
2636:
2617:
121:
83:
71:
3557:
3526:
3320:
3198:
3027:
2960:
2955:
2950:
2940:
2732:
2715:
2522:
129:
94:
67:
3469:
3378:
3208:
3164:
2930:
2347:
1891:
372:
243:
90:
51:
43:
335:{\displaystyle \Gamma T^{*}M=\Delta ^{*}\left({\mathcal {I}}/{\mathcal {I}}^{2}\right).}
3335:
3260:
3230:
3128:
3121:
3061:
3032:
2902:
2897:
2858:
2504:
2076:
are coordinates in the fibre. The canonical one-form is given in these coordinates by
59:
3541:
3521:
3345:
3340:
3325:
3315:
3265:
3242:
3116:
3076:
3017:
2965:
2764:
2580:
1871:
358:
47:
3448:
3443:
3285:
3252:
3225:
3133:
2774:
852:
710:{\displaystyle T^{*}\mathbb {R} ^{n}=\mathbb {R} ^{n}\times (\mathbb {R} ^{n})^{*}}
17:
3291:
3280:
3237:
3138:
2739:
2567:
1931:
1915:
1895:
55:
31:
1068:{\displaystyle TM=\{(x,v)\in T\,\mathbb {R} ^{n}\ :\ f(x)=0,\ \,df_{x}(v)=0\},}
3516:
3474:
3300:
3213:
2845:
2749:
1907:
1475:{\displaystyle T_{x}^{*}M=\{v\in T_{x}\mathbb {R} ^{n}\ :\ df_{x}(v)=0\}^{*}.}
760:
3330:
3295:
3000:
2887:
2166:{\displaystyle \theta _{(x,p)}=\sum _{{\mathfrak {i}}=1}^{n}p_{i}\,dx^{i}.}
1952:
The cotangent bundle carries a canonical one-form θ also known as the
628:{\displaystyle T\,\mathbb {R} ^{n}=\mathbb {R} ^{n}\times \mathbb {R} ^{n}}
2176:
Intrinsically, the value of the canonical one-form in each fixed point of
3494:
3489:
3479:
2870:
2691:
379:
2660:
2575:
function, gives a complete determination of the physics of system. See
1926:, any real function on the cotangent bundle can be interpreted to be a
3086:
2583:
for an explicit construction of the
Hamiltonian equations of motion.
2572:
353:
of modules with respect to the sheaf of germs of smooth functions of
2447:, and the differential is the canonical symplectic form, the sum of
2013:. In terms of these base coordinates, there are fibre coordinates
89:
There are several equivalent ways to define the cotangent bundle.
54:
at every point in the manifold. It may be described also as the
2664:
27:
Vector bundle of cotangent spaces at every point in a manifold
2224:, and the tautological one-form θ assigns to the point (
1918:
can be defined on the cotangent bundle; these are called the
313:
300:
221:
208:
165:
2399:{\displaystyle \mathbb {R} ^{n}\times \mathbb {R} ^{n}}
2282:{\displaystyle \theta _{(x,\omega )}=\pi ^{*}\omega .}
1930:; thus the cotangent bundle can be understood to be a
1878:
can be paired with their dual covectors in the fiber,
1196:. By definition, the cotangent bundle in this case is
806:{\displaystyle v^{*}:\mathbb {R} ^{n}\to \mathbb {R} }
2531:
2507:
2453:
2412:
2368:
2237:
2085:
1978:
as a manifold in its own right, there is a canonical
1664:
1623:
1572:
1536:
1488:
1369:
1205:
1138:
1084:
950:
915:
861:
822:
769:
723:
641:
578:
549:
479:
431:
396:
263:
205:
162:
156:. The image of Δ is called the diagonal. Let
3462:
3421:
3354:
3251:
3147:
3094:
3085:
2921:
2844:
2783:
2703:
2653:
Symmetry in
Mechanics: A Gentle Modern Introduction
902:{\displaystyle f\in C^{\infty }(\mathbb {R} ^{n}),}
66:with more structure than smooth manifolds, such as
2550:
2513:
2485:
2439:
2398:
2281:
2165:
1914:is an orientable vector bundle). A special set of
1842:
1648:
1609:
1558:
1522:
1474:
1352:
1188:
1121:
1067:
933:
901:
843:
805:
751:
709:
627:
564:
527:
457:
414:
378:of the cotangent bundle are called (differential)
334:
234:
172:
2544:
2532:
1922:. Because cotangent bundles can be thought of as
1882:possesses a canonical one-form θ called the
1167:
855:represented by the vanishing locus of a function
235:{\displaystyle {\mathcal {I}}/{\mathcal {I}}^{2}}
2406:. But there the one form defined is the sum of
2521:represents the set of possible positions in a
1189:{\displaystyle df_{x}(v)=\nabla \!f(x)\cdot v}
2676:
2566:. For example, this is a way to describe the
2022: : a one-form at a particular point of
1832:
1683:
1342:
1224:
8:
1460:
1391:
1059:
960:
2009:are local coordinates on the base manifold
528:{\displaystyle \phi ^{*}(T^{*}N)\to T^{*}M}
3091:
2683:
2669:
2661:
2633:Riemannian Geometry and Geometric Analysis
2558:can be thought of as the set of possible
2538:
2533:
2530:
2506:
2477:
2461:
2452:
2431:
2423:
2417:
2411:
2390:
2386:
2385:
2375:
2371:
2370:
2367:
2267:
2242:
2236:
2154:
2146:
2140:
2130:
2117:
2116:
2115:
2090:
2084:
1831:
1830:
1809:
1790:
1786:
1785:
1778:
1729:
1725:
1724:
1717:
1701:
1682:
1681:
1669:
1663:
1634:
1622:
1577:
1571:
1547:
1535:
1511:
1506:
1493:
1487:
1463:
1438:
1416:
1412:
1411:
1404:
1379:
1374:
1368:
1341:
1340:
1331:
1326:
1313:
1270:
1266:
1265:
1258:
1242:
1223:
1222:
1210:
1204:
1146:
1137:
1110:
1105:
1092:
1083:
1038:
1030:
991:
987:
986:
984:
949:
914:
887:
883:
882:
872:
860:
844:{\displaystyle M\subset \mathbb {R} ^{n}}
835:
831:
830:
821:
799:
798:
789:
785:
784:
774:
768:
743:
733:
729:
728:
722:
701:
691:
687:
686:
673:
669:
668:
658:
654:
653:
646:
640:
619:
615:
614:
604:
600:
599:
589:
585:
584:
582:
577:
556:
552:
551:
548:
516:
497:
484:
478:
446:
436:
430:
395:
318:
312:
311:
305:
299:
298:
287:
271:
262:
226:
220:
219:
213:
207:
206:
204:
164:
163:
161:
196:which vanish on the diagonal. Then the
2072:'s are coordinates on the base and the
543:The tangent bundle of the vector space
752:{\displaystyle (\mathbb {R} ^{n})^{*}}
70:, or (in the form of cotangent sheaf)
2346:The cotangent bundle has a canonical
7:
2304:, ω) is computed by projecting
1970:-form. This means that if we regard
1122:{\displaystyle df_{x}\in T_{x}^{*}M}
2212:is the same as choosing of a point
2118:
1854:The cotangent bundle as phase space
1523:{\displaystyle v^{*}\in T_{x}^{*}M}
2486:{\displaystyle dy_{i}\land dx_{i}}
2055:itself carries local coordinates (
1610:{\displaystyle v^{*}(u)=v\cdot u,}
1164:
916:
873:
415:{\displaystyle \phi \colon M\to N}
284:
264:
25:
2199:of the bundle. Taking a point in
2650:Singer, Stephanie Frank (2001).
1894:, out of which a non-degenerate
565:{\displaystyle \mathbb {R} ^{n}}
1530:corresponds to a unique vector
934:{\displaystyle \nabla f\neq 0,}
763:of covectors, linear functions
458:{\displaystyle \phi ^{*}T^{*}N}
2723:Differentiable/Smooth manifold
2255:
2243:
2184:. Specifically, suppose that
2103:
2091:
1821:
1815:
1753:
1747:
1707:
1688:
1589:
1583:
1450:
1444:
1294:
1288:
1248:
1229:
1177:
1171:
1158:
1152:
1050:
1044:
1015:
1009:
975:
963:
893:
878:
795:
740:
724:
698:
682:
635:, and the cotangent bundle is
509:
506:
490:
406:
173:{\displaystyle {\mathcal {I}}}
1:
2616:. London: Benjamin-Cummings.
2525:, then the cotangent bundle
2440:{\displaystyle y_{i}\,dx_{i}}
2045:Einstein summation convention
1910:manifold (the tangent bundle
62:. This may be generalized to
2551:{\displaystyle \!\,T^{*}\!M}
1902:. For example, as a result
1649:{\displaystyle u\in T_{x}M,}
3429:Classification of manifolds
2635:. Berlin: Springer-Verlag.
2308:into the tangent bundle at
2047:implied). So the manifold
1858:Since the cotangent bundle
1559:{\displaystyle v\in T_{x}M}
3574:
1945:
3505:over commutative algebras
2220:and a one-form ω at
1942:The tautological one-form
386:Contravariance properties
3221:Riemann curvature tensor
2614:Foundations of Mechanics
909:with the condition that
816:Given a smooth manifold
422:of manifolds, induces a
2593:Legendre transformation
1886:, discussed below. The
188:of smooth functions on
3013:Manifold with boundary
2728:Differential structure
2552:
2515:
2487:
2441:
2400:
2292:That is, for a vector
2283:
2167:
2135:
1844:
1650:
1611:
1560:
1524:
1476:
1354:
1190:
1131:directional derivative
1123:
1069:
941:the tangent bundle is
935:
903:
845:
807:
753:
711:
629:
566:
529:
459:
416:
336:
236:
174:
82:Formal definition via
3553:Differential topology
2656:. Boston: BirkhÀuser.
2631:Jost, JĂŒrgen (2002).
2577:Hamiltonian mechanics
2553:
2516:
2488:
2442:
2401:
2356:tautological one-form
2284:
2168:
2111:
1982:of the vector bundle
1948:Tautological one-form
1936:Hamiltonian mechanics
1920:canonical coordinates
1884:tautological one-form
1845:
1651:
1612:
1561:
1525:
1482:Since every covector
1477:
1355:
1191:
1124:
1070:
936:
904:
846:
808:
754:
712:
630:
567:
530:
460:
417:
357:. Thus it defines a
337:
237:
175:
132:Δ sends a point
36:differential geometry
3160:Covariant derivative
2711:Topological manifold
2529:
2505:
2451:
2410:
2366:
2360:symplectic potential
2235:
2228:, ω) the value
2083:
1954:symplectic potential
1924:symplectic manifolds
1662:
1621:
1570:
1534:
1486:
1367:
1203:
1136:
1082:
948:
913:
859:
820:
767:
721:
639:
576:
547:
477:
429:
394:
261:
203:
160:
3194:Exterior derivative
2796:AtiyahâSinger index
2745:Riemannian manifold
2610:Marsden, Jerrold E.
2579:and the article on
2352:exterior derivative
1888:exterior derivative
1516:
1384:
1336:
1115:
72:algebraic varieties
3500:Secondary calculus
3454:Singularity theory
3409:Parallel transport
3177:De Rham cohomology
2816:Generalized Stokes
2548:
2511:
2483:
2437:
2396:
2279:
2163:
1840:
1646:
1607:
1556:
1520:
1502:
1472:
1370:
1350:
1322:
1186:
1119:
1101:
1065:
931:
899:
841:
803:
749:
707:
625:
562:
525:
473:of vector bundles
455:
412:
390:A smooth morphism
351:locally free sheaf
332:
246:is defined as the
232:
170:
128:with itself. The
18:Cotangent manifold
3535:
3534:
3417:
3416:
3182:Differential form
2836:Whitney embedding
2770:Differential form
2514:{\displaystyle M}
2348:symplectic 2-form
1898:can be built for
1892:symplectic 2-form
1801:
1767:
1743:
1737:
1617:for an arbitrary
1430:
1424:
1308:
1284:
1278:
1029:
1005:
999:
250:of this sheaf to
122:Cartesian product
84:diagonal morphism
68:complex manifolds
16:(Redirected from
3565:
3527:Stratified space
3485:Fréchet manifold
3199:Interior product
3092:
2789:
2685:
2678:
2671:
2662:
2657:
2646:
2627:
2557:
2555:
2554:
2549:
2543:
2542:
2523:dynamical system
2520:
2518:
2517:
2512:
2501:If the manifold
2492:
2490:
2489:
2484:
2482:
2481:
2466:
2465:
2446:
2444:
2443:
2438:
2436:
2435:
2422:
2421:
2405:
2403:
2402:
2397:
2395:
2394:
2389:
2380:
2379:
2374:
2333:
2288:
2286:
2285:
2280:
2272:
2271:
2259:
2258:
2194:
2172:
2170:
2169:
2164:
2159:
2158:
2145:
2144:
2134:
2129:
2122:
2121:
2107:
2106:
1849:
1847:
1846:
1841:
1836:
1835:
1814:
1813:
1799:
1795:
1794:
1789:
1783:
1782:
1765:
1741:
1735:
1734:
1733:
1728:
1722:
1721:
1706:
1705:
1687:
1686:
1674:
1673:
1655:
1653:
1652:
1647:
1639:
1638:
1616:
1614:
1613:
1608:
1582:
1581:
1565:
1563:
1562:
1557:
1552:
1551:
1529:
1527:
1526:
1521:
1515:
1510:
1498:
1497:
1481:
1479:
1478:
1473:
1468:
1467:
1443:
1442:
1428:
1422:
1421:
1420:
1415:
1409:
1408:
1383:
1378:
1359:
1357:
1356:
1351:
1346:
1345:
1335:
1330:
1318:
1317:
1306:
1282:
1276:
1275:
1274:
1269:
1263:
1262:
1247:
1246:
1228:
1227:
1215:
1214:
1195:
1193:
1192:
1187:
1151:
1150:
1128:
1126:
1125:
1120:
1114:
1109:
1097:
1096:
1074:
1072:
1071:
1066:
1043:
1042:
1027:
1003:
997:
996:
995:
990:
940:
938:
937:
932:
908:
906:
905:
900:
892:
891:
886:
877:
876:
850:
848:
847:
842:
840:
839:
834:
812:
810:
809:
804:
802:
794:
793:
788:
779:
778:
758:
756:
755:
750:
748:
747:
738:
737:
732:
716:
714:
713:
708:
706:
705:
696:
695:
690:
678:
677:
672:
663:
662:
657:
651:
650:
634:
632:
631:
626:
624:
623:
618:
609:
608:
603:
594:
593:
588:
571:
569:
568:
563:
561:
560:
555:
534:
532:
531:
526:
521:
520:
502:
501:
489:
488:
464:
462:
461:
456:
451:
450:
441:
440:
421:
419:
418:
413:
367:cotangent bundle
347:Taylor's theorem
341:
339:
338:
333:
328:
324:
323:
322:
317:
316:
309:
304:
303:
292:
291:
276:
275:
241:
239:
238:
233:
231:
230:
225:
224:
217:
212:
211:
179:
177:
176:
171:
169:
168:
130:diagonal mapping
95:diagonal mapping
52:cotangent spaces
40:cotangent bundle
21:
3573:
3572:
3568:
3567:
3566:
3564:
3563:
3562:
3538:
3537:
3536:
3531:
3470:Banach manifold
3463:Generalizations
3458:
3413:
3350:
3247:
3209:Ricci curvature
3165:Cotangent space
3143:
3081:
2923:
2917:
2876:Exponential map
2840:
2785:
2779:
2699:
2689:
2649:
2643:
2630:
2624:
2604:
2601:
2589:
2534:
2527:
2526:
2503:
2502:
2499:
2473:
2457:
2449:
2448:
2427:
2413:
2408:
2407:
2384:
2369:
2364:
2363:
2344:
2342:Symplectic form
2313:
2263:
2238:
2233:
2232:
2207:
2185:
2150:
2136:
2086:
2081:
2080:
2067:
2038:
2021:
1950:
1944:
1890:of θ is a
1856:
1805:
1784:
1774:
1723:
1713:
1697:
1665:
1660:
1659:
1630:
1619:
1618:
1573:
1568:
1567:
1543:
1532:
1531:
1489:
1484:
1483:
1459:
1434:
1410:
1400:
1365:
1364:
1309:
1264:
1254:
1238:
1206:
1201:
1200:
1142:
1134:
1133:
1088:
1080:
1079:
1034:
985:
946:
945:
911:
910:
881:
868:
857:
856:
829:
818:
817:
783:
770:
765:
764:
739:
727:
719:
718:
697:
685:
667:
652:
642:
637:
636:
613:
598:
583:
574:
573:
550:
545:
544:
541:
512:
493:
480:
475:
474:
442:
432:
427:
426:
392:
391:
388:
310:
297:
293:
283:
267:
259:
258:
244:cotangent sheaf
218:
201:
200:
158:
157:
110:smooth manifold
87:
44:smooth manifold
28:
23:
22:
15:
12:
11:
5:
3571:
3569:
3561:
3560:
3555:
3550:
3548:Vector bundles
3540:
3539:
3533:
3532:
3530:
3529:
3524:
3519:
3514:
3509:
3508:
3507:
3497:
3492:
3487:
3482:
3477:
3472:
3466:
3464:
3460:
3459:
3457:
3456:
3451:
3446:
3441:
3436:
3431:
3425:
3423:
3419:
3418:
3415:
3414:
3412:
3411:
3406:
3401:
3396:
3391:
3386:
3381:
3376:
3371:
3366:
3360:
3358:
3352:
3351:
3349:
3348:
3343:
3338:
3333:
3328:
3323:
3318:
3308:
3303:
3298:
3288:
3283:
3278:
3273:
3268:
3263:
3257:
3255:
3249:
3248:
3246:
3245:
3240:
3235:
3234:
3233:
3223:
3218:
3217:
3216:
3206:
3201:
3196:
3191:
3190:
3189:
3179:
3174:
3173:
3172:
3162:
3157:
3151:
3149:
3145:
3144:
3142:
3141:
3136:
3131:
3126:
3125:
3124:
3114:
3109:
3104:
3098:
3096:
3089:
3083:
3082:
3080:
3079:
3074:
3064:
3059:
3045:
3040:
3035:
3030:
3025:
3023:Parallelizable
3020:
3015:
3010:
3009:
3008:
2998:
2993:
2988:
2983:
2978:
2973:
2968:
2963:
2958:
2953:
2943:
2933:
2927:
2925:
2919:
2918:
2916:
2915:
2910:
2905:
2903:Lie derivative
2900:
2898:Integral curve
2895:
2890:
2885:
2884:
2883:
2873:
2868:
2867:
2866:
2859:Diffeomorphism
2856:
2850:
2848:
2842:
2841:
2839:
2838:
2833:
2828:
2823:
2818:
2813:
2808:
2803:
2798:
2792:
2790:
2781:
2780:
2778:
2777:
2772:
2767:
2762:
2757:
2752:
2747:
2742:
2737:
2736:
2735:
2730:
2720:
2719:
2718:
2707:
2705:
2704:Basic concepts
2701:
2700:
2690:
2688:
2687:
2680:
2673:
2665:
2659:
2658:
2647:
2641:
2628:
2622:
2606:Abraham, Ralph
2600:
2597:
2596:
2595:
2588:
2585:
2547:
2541:
2537:
2510:
2498:
2495:
2480:
2476:
2472:
2469:
2464:
2460:
2456:
2434:
2430:
2426:
2420:
2416:
2393:
2388:
2383:
2378:
2373:
2343:
2340:
2317:π :
2290:
2289:
2278:
2275:
2270:
2266:
2262:
2257:
2254:
2251:
2248:
2245:
2241:
2203:
2186:π :
2180:is given as a
2174:
2173:
2162:
2157:
2153:
2149:
2143:
2139:
2133:
2128:
2125:
2120:
2114:
2110:
2105:
2102:
2099:
2096:
2093:
2089:
2063:
2034:
2017:
1946:Main article:
1943:
1940:
1855:
1852:
1851:
1850:
1839:
1834:
1829:
1826:
1823:
1820:
1817:
1812:
1808:
1804:
1798:
1793:
1788:
1781:
1777:
1773:
1770:
1764:
1761:
1758:
1755:
1752:
1749:
1746:
1740:
1732:
1727:
1720:
1716:
1712:
1709:
1704:
1700:
1696:
1693:
1690:
1685:
1680:
1677:
1672:
1668:
1645:
1642:
1637:
1633:
1629:
1626:
1606:
1603:
1600:
1597:
1594:
1591:
1588:
1585:
1580:
1576:
1555:
1550:
1546:
1542:
1539:
1519:
1514:
1509:
1505:
1501:
1496:
1492:
1471:
1466:
1462:
1458:
1455:
1452:
1449:
1446:
1441:
1437:
1433:
1427:
1419:
1414:
1407:
1403:
1399:
1396:
1393:
1390:
1387:
1382:
1377:
1373:
1361:
1360:
1349:
1344:
1339:
1334:
1329:
1325:
1321:
1316:
1312:
1305:
1302:
1299:
1296:
1293:
1290:
1287:
1281:
1273:
1268:
1261:
1257:
1253:
1250:
1245:
1241:
1237:
1234:
1231:
1226:
1221:
1218:
1213:
1209:
1185:
1182:
1179:
1176:
1173:
1170:
1166:
1163:
1160:
1157:
1154:
1149:
1145:
1141:
1118:
1113:
1108:
1104:
1100:
1095:
1091:
1087:
1076:
1075:
1064:
1061:
1058:
1055:
1052:
1049:
1046:
1041:
1037:
1033:
1026:
1023:
1020:
1017:
1014:
1011:
1008:
1002:
994:
989:
983:
980:
977:
974:
971:
968:
965:
962:
959:
956:
953:
930:
927:
924:
921:
918:
898:
895:
890:
885:
880:
875:
871:
867:
864:
851:embedded as a
838:
833:
828:
825:
801:
797:
792:
787:
782:
777:
773:
746:
742:
736:
731:
726:
704:
700:
694:
689:
684:
681:
676:
671:
666:
661:
656:
649:
645:
622:
617:
612:
607:
602:
597:
592:
587:
581:
559:
554:
540:
537:
524:
519:
515:
511:
508:
505:
500:
496:
492:
487:
483:
469:. There is an
454:
449:
445:
439:
435:
424:pullback sheaf
411:
408:
405:
402:
399:
387:
384:
343:
342:
331:
327:
321:
315:
308:
302:
296:
290:
286:
282:
279:
274:
270:
266:
229:
223:
216:
210:
198:quotient sheaf
167:
140:to the point (
86:
80:
60:tangent bundle
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3570:
3559:
3556:
3554:
3551:
3549:
3546:
3545:
3543:
3528:
3525:
3523:
3522:Supermanifold
3520:
3518:
3515:
3513:
3510:
3506:
3503:
3502:
3501:
3498:
3496:
3493:
3491:
3488:
3486:
3483:
3481:
3478:
3476:
3473:
3471:
3468:
3467:
3465:
3461:
3455:
3452:
3450:
3447:
3445:
3442:
3440:
3437:
3435:
3432:
3430:
3427:
3426:
3424:
3420:
3410:
3407:
3405:
3402:
3400:
3397:
3395:
3392:
3390:
3387:
3385:
3382:
3380:
3377:
3375:
3372:
3370:
3367:
3365:
3362:
3361:
3359:
3357:
3353:
3347:
3344:
3342:
3339:
3337:
3334:
3332:
3329:
3327:
3324:
3322:
3319:
3317:
3313:
3309:
3307:
3304:
3302:
3299:
3297:
3293:
3289:
3287:
3284:
3282:
3279:
3277:
3274:
3272:
3269:
3267:
3264:
3262:
3259:
3258:
3256:
3254:
3250:
3244:
3243:Wedge product
3241:
3239:
3236:
3232:
3229:
3228:
3227:
3224:
3222:
3219:
3215:
3212:
3211:
3210:
3207:
3205:
3202:
3200:
3197:
3195:
3192:
3188:
3187:Vector-valued
3185:
3184:
3183:
3180:
3178:
3175:
3171:
3168:
3167:
3166:
3163:
3161:
3158:
3156:
3153:
3152:
3150:
3146:
3140:
3137:
3135:
3132:
3130:
3127:
3123:
3120:
3119:
3118:
3117:Tangent space
3115:
3113:
3110:
3108:
3105:
3103:
3100:
3099:
3097:
3093:
3090:
3088:
3084:
3078:
3075:
3073:
3069:
3065:
3063:
3060:
3058:
3054:
3050:
3046:
3044:
3041:
3039:
3036:
3034:
3031:
3029:
3026:
3024:
3021:
3019:
3016:
3014:
3011:
3007:
3004:
3003:
3002:
2999:
2997:
2994:
2992:
2989:
2987:
2984:
2982:
2979:
2977:
2974:
2972:
2969:
2967:
2964:
2962:
2959:
2957:
2954:
2952:
2948:
2944:
2942:
2938:
2934:
2932:
2929:
2928:
2926:
2920:
2914:
2911:
2909:
2906:
2904:
2901:
2899:
2896:
2894:
2891:
2889:
2886:
2882:
2881:in Lie theory
2879:
2878:
2877:
2874:
2872:
2869:
2865:
2862:
2861:
2860:
2857:
2855:
2852:
2851:
2849:
2847:
2843:
2837:
2834:
2832:
2829:
2827:
2824:
2822:
2819:
2817:
2814:
2812:
2809:
2807:
2804:
2802:
2799:
2797:
2794:
2793:
2791:
2788:
2784:Main results
2782:
2776:
2773:
2771:
2768:
2766:
2765:Tangent space
2763:
2761:
2758:
2756:
2753:
2751:
2748:
2746:
2743:
2741:
2738:
2734:
2731:
2729:
2726:
2725:
2724:
2721:
2717:
2714:
2713:
2712:
2709:
2708:
2706:
2702:
2697:
2693:
2686:
2681:
2679:
2674:
2672:
2667:
2666:
2663:
2655:
2654:
2648:
2644:
2642:3-540-63654-4
2638:
2634:
2629:
2625:
2623:0-8053-0102-X
2619:
2615:
2611:
2607:
2603:
2602:
2598:
2594:
2591:
2590:
2586:
2584:
2582:
2581:geodesic flow
2578:
2574:
2569:
2565:
2561:
2545:
2539:
2535:
2524:
2508:
2496:
2494:
2478:
2474:
2470:
2467:
2462:
2458:
2454:
2432:
2428:
2424:
2418:
2414:
2391:
2381:
2376:
2361:
2357:
2353:
2350:on it, as an
2349:
2341:
2339:
2337:
2332:
2328:
2324:
2320:
2316:
2311:
2307:
2303:
2299:
2295:
2276:
2273:
2268:
2264:
2260:
2252:
2249:
2246:
2239:
2231:
2230:
2229:
2227:
2223:
2219:
2215:
2211:
2206:
2202:
2198:
2193:
2189:
2183:
2179:
2160:
2155:
2151:
2147:
2141:
2137:
2131:
2126:
2123:
2112:
2108:
2100:
2097:
2094:
2087:
2079:
2078:
2077:
2075:
2071:
2066:
2062:
2058:
2054:
2050:
2046:
2042:
2037:
2033:
2030:has the form
2029:
2025:
2020:
2016:
2012:
2008:
2003:
2001:
1997:
1993:
1989:
1985:
1981:
1977:
1973:
1969:
1966:
1962:
1959:
1955:
1949:
1941:
1939:
1937:
1933:
1929:
1925:
1921:
1917:
1913:
1909:
1906:is always an
1905:
1901:
1897:
1893:
1889:
1885:
1881:
1877:
1873:
1872:vector bundle
1869:
1865:
1861:
1853:
1837:
1827:
1824:
1818:
1810:
1806:
1802:
1796:
1791:
1779:
1775:
1771:
1768:
1762:
1759:
1756:
1750:
1744:
1738:
1730:
1718:
1714:
1710:
1702:
1698:
1694:
1691:
1678:
1675:
1670:
1666:
1658:
1657:
1656:
1643:
1640:
1635:
1631:
1627:
1624:
1604:
1601:
1598:
1595:
1592:
1586:
1578:
1574:
1553:
1548:
1544:
1540:
1537:
1517:
1512:
1507:
1503:
1499:
1494:
1490:
1469:
1464:
1456:
1453:
1447:
1439:
1435:
1431:
1425:
1417:
1405:
1401:
1397:
1394:
1388:
1385:
1380:
1375:
1371:
1347:
1337:
1332:
1327:
1323:
1319:
1314:
1310:
1303:
1300:
1297:
1291:
1285:
1279:
1271:
1259:
1255:
1251:
1243:
1239:
1235:
1232:
1219:
1216:
1211:
1207:
1199:
1198:
1197:
1183:
1180:
1174:
1168:
1161:
1155:
1147:
1143:
1139:
1132:
1116:
1111:
1106:
1102:
1098:
1093:
1089:
1085:
1062:
1056:
1053:
1047:
1039:
1035:
1031:
1024:
1021:
1018:
1012:
1006:
1000:
992:
981:
978:
972:
969:
966:
957:
954:
951:
944:
943:
942:
928:
925:
922:
919:
896:
888:
869:
865:
862:
854:
836:
826:
823:
814:
790:
780:
775:
771:
762:
744:
734:
702:
692:
679:
674:
664:
659:
647:
643:
620:
610:
605:
595:
590:
579:
557:
538:
536:
522:
517:
513:
503:
498:
494:
485:
481:
472:
468:
452:
447:
443:
437:
433:
425:
409:
403:
400:
397:
385:
383:
381:
377:
374:
370:
368:
364:
360:
359:vector bundle
356:
352:
348:
329:
325:
319:
306:
294:
288:
280:
277:
272:
268:
257:
256:
255:
253:
249:
245:
227:
214:
199:
195:
191:
187:
183:
155:
151:
147:
143:
139:
135:
131:
127:
123:
119:
115:
111:
107:
102:
100:
96:
93:is through a
92:
85:
81:
79:
77:
73:
69:
65:
61:
57:
53:
49:
48:vector bundle
45:
41:
37:
34:, especially
33:
19:
3449:Moving frame
3444:Morse theory
3434:Gauge theory
3275:
3226:Tensor field
3169:
3155:Closed/Exact
3134:Vector field
3102:Distribution
3043:Hypercomplex
3038:Quaternionic
2775:Vector field
2733:Smooth atlas
2652:
2632:
2613:
2571:appropriate
2563:
2559:
2500:
2345:
2335:
2330:
2326:
2322:
2318:
2314:
2309:
2305:
2301:
2297:
2293:
2291:
2225:
2221:
2217:
2213:
2209:
2204:
2200:
2191:
2187:
2177:
2175:
2073:
2069:
2068:) where the
2064:
2060:
2056:
2052:
2048:
2040:
2035:
2031:
2027:
2023:
2018:
2014:
2010:
2006:
2004:
1999:
1995:
1991:
1987:
1983:
1975:
1971:
1967:
1964:
1960:
1957:
1951:
1911:
1903:
1899:
1879:
1875:
1867:
1863:
1859:
1857:
1362:
1077:
853:hypersurface
815:
759:denotes the
542:
466:
389:
371:
366:
362:
354:
349:, this is a
344:
251:
193:
189:
153:
149:
145:
141:
137:
133:
125:
117:
113:
105:
103:
97:Δ and
88:
39:
29:
3394:Levi-Civita
3384:Generalized
3356:Connections
3306:Lie algebra
3238:Volume form
3139:Vector flow
3112:Pushforward
3107:Lie bracket
3006:Lie algebra
2971:G-structure
2760:Pushforward
2740:Submanifold
2568:phase space
2497:Phase space
1938:plays out.
1932:phase space
1928:Hamiltonian
1916:coordinates
1896:volume form
471:induced map
56:dual bundle
50:of all the
32:mathematics
3542:Categories
3517:Stratifold
3475:Diffeology
3271:Associated
3072:Symplectic
3057:Riemannian
2986:Hyperbolic
2913:Submersion
2821:HopfâRinow
2755:Submersion
2750:Smooth map
2599:References
2329:) →
2197:projection
1963:-form, or
1908:orientable
1566:for which
761:dual space
64:categories
3399:Principal
3374:Ehresmann
3331:Subbundle
3321:Principal
3296:Fibration
3276:Cotangent
3148:Covectors
3001:Lie group
2981:Hermitian
2924:manifolds
2893:Immersion
2888:Foliation
2826:Noether's
2811:Frobenius
2806:De Rham's
2801:Darboux's
2692:Manifolds
2560:positions
2540:∗
2468:∧
2382:×
2274:ω
2269:∗
2265:π
2253:ω
2240:θ
2113:∑
2088:θ
1965:Liouville
1934:on which
1772:∈
1719:∗
1711:∈
1703:∗
1671:∗
1628:∈
1599:⋅
1579:∗
1541:∈
1513:∗
1500:∈
1495:∗
1465:∗
1398:∈
1381:∗
1333:∗
1320:∈
1315:∗
1260:∗
1252:∈
1244:∗
1212:∗
1181:⋅
1165:∇
1112:∗
1099:∈
979:∈
923:≠
917:∇
874:∞
866:∈
827:⊂
796:→
776:∗
745:∗
703:∗
680:×
648:∗
611:×
518:∗
510:→
499:∗
486:∗
482:ϕ
448:∗
438:∗
434:ϕ
407:→
401::
398:ϕ
380:one-forms
289:∗
285:Δ
273:∗
265:Γ
3495:Orbifold
3490:K-theory
3480:Diffiety
3204:Pullback
3018:Oriented
2996:Kenmotsu
2976:Hadamard
2922:Types of
2871:Geodesic
2696:Glossary
2612:(1978).
2587:See also
2190:→
2182:pullback
1958:Poincaré
717:, where
539:Examples
376:sections
248:pullback
112:and let
3558:Tensors
3439:History
3422:Related
3336:Tangent
3314:)
3294:)
3261:Adjoint
3253:Bundles
3231:density
3129:Torsion
3095:Vectors
3087:Tensors
3070:)
3055:)
3051:,
3049:Pseudoâ
3028:Poisson
2961:Finsler
2956:Fibered
2951:Contact
2949:)
2941:Complex
2939:)
2908:Section
2564:momenta
2354:of the
2195:is the
1994:) over
1980:section
1129:is the
365:: the
180:be the
120:be the
91:One way
76:schemes
58:to the
46:is the
3404:Vector
3389:Koszul
3369:Cartan
3364:Affine
3346:Vector
3341:Tensor
3326:Spinor
3316:Normal
3312:Stable
3266:Affine
3170:bundle
3122:bundle
3068:Almost
2991:KĂ€hler
2947:Almost
2937:Almost
2931:Closed
2831:Sard's
2787:(list)
2639:
2620:
2573:energy
2358:, the
2312:using
2039:
1800:
1766:
1742:
1736:
1429:
1423:
1363:where
1307:
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1078:where
1028:
1004:
998:
373:Smooth
192:×
152:×
116:×
38:, the
3512:Sheaf
3286:Fiber
3062:Rizza
3033:Prime
2864:Local
2854:Curve
2716:Atlas
1870:is a
186:germs
182:sheaf
148:) of
108:be a
99:germs
42:of a
3379:Form
3281:Dual
3214:flow
3077:Tame
3053:Subâ
2966:Flat
2846:Maps
2637:ISBN
2618:ISBN
2562:and
2300:at (
104:Let
3301:Jet
2216:in
2188:T*M
2178:T*M
2074:p's
572:is
465:on
361:on
345:By
184:of
136:in
124:of
74:or
30:In
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3292:Co
2608:;
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2338:.
2331:TM
2059:,
2041:dx
2002:.
1986:*(
1956:,
1912:TX
1862:=
813:.
535:.
382:.
369:.
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3066:(
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2471:d
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2459:y
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2327:M
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2015:p
2011:M
2007:x
2000:M
1998:*
1996:T
1992:M
1990:*
1988:T
1984:T
1976:M
1974:*
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1961:1
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