1515:
1150:
1091:
1510:{\displaystyle _{x}\ =\ \left.{\tfrac {1}{2}}{\tfrac {\mathrm {d} ^{2}}{\mathrm {d} t^{2}}}\right|_{t=0}(\Phi _{-t}^{Y}\circ \Phi _{-t}^{X}\circ \Phi _{t}^{Y}\circ \Phi _{t}^{X})(x)\ =\ \left.{\tfrac {\mathrm {d} }{\mathrm {d} t}}\right|_{t=0}(\Phi _{\!-{\sqrt {t}}}^{Y}\circ \Phi _{\!-{\sqrt {t}}}^{X}\circ \Phi _{\!{\sqrt {t}}}^{Y}\circ \Phi _{\!{\sqrt {t}}}^{X})(x).}
2055:
796:
1086:{\displaystyle _{x}\ =\ ({\mathcal {L}}_{X}Y)_{x}\ :=\ \lim _{t\to 0}{\frac {(\mathrm {D} \Phi _{-t}^{X})Y_{\Phi _{t}^{X}(x)}\,-\,Y_{x}}{t}}\ =\ \left.{\tfrac {\mathrm {d} }{\mathrm {d} t}}\right|_{t=0}(\mathrm {D} \Phi _{-t}^{X})Y_{\Phi _{t}^{X}(x)}.}
1812:
3144:
714:
3315:
513:
3456:
1618:
1750:
1684:
3374:
1804:
3526:
3581:
3677:
2935:
2156:
2115:
2209:
2050:{\displaystyle :=\sum _{i=1}^{n}\left(X(Y^{i})-Y(X^{i})\right)\partial _{i}=\sum _{i=1}^{n}\sum _{j=1}^{n}\left(X^{j}\partial _{j}Y^{i}-Y^{j}\partial _{j}X^{i}\right)\partial _{i}.}
101:
2420:
2383:
225:
3605:
3211:
2276:
2550:
344:
2481:
754:
567:
540:
1564:
1138:
2731:
2514:
3476:
3241:
587:
288:
2333:
2306:
3025:
373:
262:
2772:
2616:
2792:
393:
308:
3038:
595:
164:
refers to this as the "fisherman derivative", as one can imagine being a fisherman, holding a fishing rod, sitting in a boat. Both the boat and the
3863:
4752:
3943:
3254:
452:
3379:
589:
denotes composition of operators. This can be used to define the Lie bracket as the vector field corresponding to the commutator derivation:
4747:
1569:
4034:
4058:
151:
1689:
1623:
4253:
3323:
4123:
3881:
3749:
3251:. The Lie bracket of two left invariant vector fields is also left invariant, which defines the JacobiâLie bracket operation
4349:
4402:
3930:
1526:
4686:
3696:
3900:
3848:
3828:
4451:
1755:
4434:
4043:
3692:
3485:
3168:
4805:
4800:
3744:. Applied mathematical sciences (Corr. 2. printing ed.). New York Berlin Heidelberg: Springer. p. 6.
3531:
4646:
4053:
3774:
3617:
2852:
2120:
2079:
420:
2161:
4810:
4631:
4128:
4676:
3769:
72:
2388:
2351:
4681:
4651:
4359:
4315:
4296:
4063:
4007:
765:
195:
4218:
4083:
3586:
3192:
2217:
2523:
313:
176:. The Lie bracket is the amount of dragging on the fishing float relative to the surrounding water.
4603:
4468:
4160:
4002:
2448:
1620:
for the associated local basis of the tangent bundle, so that general vector fields can be written
727:
155:
3691:
can be seen as a generalization of the Lie bracket. Another generalization of the Lie bracket (to
4300:
4270:
4194:
4184:
4140:
3970:
3923:
3764:
545:
518:
4795:
4641:
4260:
4155:
4068:
3975:
3727:
1536:
1533:), in practice one often wants to compute the bracket in terms of a specific coordinate system
1099:
396:
2626:
184:
There are three conceptually different but equivalent approaches to defining the Lie bracket:
4290:
4285:
2490:
232:
147:
143:
20:
3461:
3216:
572:
267:
4621:
4559:
4407:
4111:
4101:
4073:
4048:
3958:
2311:
2284:
2991:
349:
238:
172:, and the fisherman lengthens/shrinks and turns the fishing rod according to vector field
8:
4759:
4441:
4319:
4304:
4233:
3992:
3719:
4732:
2739:
2568:
4701:
4656:
4553:
4424:
4228:
3916:
3799:
2777:
757:
378:
293:
4238:
4636:
4616:
4611:
4518:
4429:
4243:
4223:
4078:
4017:
3896:
3877:
3857:
3844:
3824:
3745:
124:
3803:
4774:
4568:
4523:
4446:
4417:
4275:
4208:
4203:
4198:
4188:
3980:
3963:
3814:
3810:
3791:
3608:
4717:
4626:
4456:
4412:
4178:
3139:{\displaystyle (\Phi _{t}^{Y}\Phi _{s}^{X})(x)=(\Phi _{s}^{X}\,\Phi _{t}^{Y})(x)}
2971:
means that following the flows in these directions defines a surface embedded in
2956:
with the vector field . This turns the vector fields with the Lie bracket into a
2620:
2345:
161:
47:
3868:
Extensive discussion of Lie brackets, and the general theory of Lie derivatives.
4583:
4508:
4478:
4376:
4369:
4309:
4280:
4150:
4145:
4106:
3838:
3688:
3244:
787:
105:
709:{\displaystyle (f)=X(Y(f))-Y(X(f))\;\;{\text{ for all }}f\in C^{\infty }(M).}
4789:
4769:
4593:
4588:
4573:
4563:
4513:
4490:
4364:
4324:
4265:
4213:
4012:
3820:
2957:
165:
3795:
4696:
4691:
4533:
4500:
4473:
4381:
4022:
2798:
2560:
109:
36:
4539:
4528:
4485:
4386:
3987:
3310:{\displaystyle :{\mathfrak {g}}\times {\mathfrak {g}}\to {\mathfrak {g}}}
3187:
2517:
508:{\displaystyle \delta _{1}\circ \delta _{2}-\delta _{2}\circ \delta _{1}}
136:
61:
1096:
This also measures the failure of the flow in the successive directions
4764:
4722:
4548:
4461:
4093:
3997:
3908:
3451:{\displaystyle T_{g}G=g\cdot T_{I}G\subset M_{n\times n}(\mathbb {R} )}
447:
128:
4578:
4543:
4248:
4135:
3836:
3180:
3482:
is the identity matrix. The invariant vector field corresponding to
4742:
4737:
4727:
4118:
3939:
1613:{\displaystyle \partial _{i}={\tfrac {\partial }{\partial x^{i}}}}
4334:
2445:
The Lie bracket of vector fields equips the real vector space
1745:{\displaystyle \textstyle Y=\sum _{i=1}^{n}Y^{i}\partial _{i}}
1679:{\displaystyle \textstyle X=\sum _{i=1}^{n}X^{i}\partial _{i}}
2801:" for Lie brackets. Given a smooth (scalar-valued) function
1355:
1187:
981:
1529:(independent of the choice of coordinates on the manifold
3369:{\displaystyle g\in G\subset M_{n\times n}(\mathbb {R} )}
104:("Lie derivative of Y along X"). This generalizes to the
2736:
An immediate consequence of the second property is that
3893:
Foundations of differentiable manifolds and Lie groups
1693:
1627:
1587:
1358:
1203:
1191:
984:
3620:
3589:
3534:
3488:
3464:
3382:
3376:, each tangent space can be represented as matrices:
3326:
3257:
3219:
3195:
3041:
2994:
2855:
2780:
2742:
2629:
2571:
2526:
2493:
2451:
2391:
2354:
2314:
2287:
2220:
2164:
2123:
2082:
1815:
1758:
1692:
1626:
1572:
1539:
1153:
1102:
799:
730:
598:
575:
548:
521:
455:
381:
352:
316:
296:
270:
241:
198:
154:, and is also fundamental in the geometric theory of
75:
3320:
For a matrix Lie group, whose elements are matrices
56:
Conceptually, the Lie bracket is the derivative of
3243:, which can be identified with the vector space of
2422:respectively using index notation) multiplying the
3671:
3599:
3575:
3520:
3470:
3450:
3368:
3309:
3235:
3205:
3138:
3019:
2929:
2786:
2766:
2725:
2610:
2544:
2508:
2475:
2414:
2377:
2327:
2300:
2270:
2203:
2150:
2109:
2049:
1798:
1744:
1678:
1612:
1558:
1509:
1132:
1085:
748:
708:
581:
561:
534:
507:
387:
367:
338:
302:
282:
256:
219:
95:
3843:, Berlin, Heidelberg, New York: Springer-Verlag,
2886:
1477:
1454:
1428:
1402:
4787:
3782:Isaiah, Pantelis (2009), "Controlled parking ",
1525:Though the above definitions of Lie bracket are
871:
187:
3837:KolĂĄĆ, I., Michor, P., and SlovĂĄk, J. (1993),
375:to be another function whose value at a point
3924:
3739:
3583:, and a computation shows the Lie bracket on
2487:(i.e., smooth sections of the tangent bundle
1799:{\displaystyle X^{i},Y^{i}:M\to \mathbb {R} }
3862:: CS1 maint: multiple names: authors list (
1806:. Then the Lie bracket can be computed as:
1553:
1540:
3887:For generalizations to infinite dimensions.
3840:Natural operations in differential geometry
3521:{\displaystyle X\in {\mathfrak {g}}=T_{I}G}
439:) arises from a unique smooth vector field
142:The Lie bracket plays an important role in
3931:
3917:
2076:can be written as smooth maps of the form
672:
671:
3740:ArnolÊčd, V. I.; Khesin, Boris A. (1999).
3441:
3359:
3273:
3269:
3265:
3261:
3108:
3016:
2908:
2904:
2900:
2896:
2191:
2138:
2097:
1792:
953:
949:
415:). In this way, each smooth vector field
35:, is an operator that assigns to any two
3938:
3871:
3819:(3rd ed.), Upper Saddle River, NJ:
3576:{\displaystyle X_{g}=g\cdot X\in T_{g}G}
3672:{\displaystyle \ =\ X\cdot Y-Y\cdot X.}
2930:{\displaystyle \ =\ X\!(f)\,Y\,+\,f\,,}
2151:{\displaystyle Y:M\to \mathbb {R} ^{n}}
2110:{\displaystyle X:M\to \mathbb {R} ^{n}}
4788:
3890:
3809:
3781:
3723:
3715:
2940:where we multiply the scalar function
2204:{\displaystyle :M\to \mathbb {R} ^{n}}
168:are flowing according to vector field
3912:
3874:Differential and Riemannian manifolds
3213:is the tangent space at the identity
3895:, New York-Berlin: Springer-Verlag,
3742:Topological methods in hydrodynamics
3592:
3497:
3302:
3292:
3282:
3198:
719:
127:operation and turns the set of all
96:{\displaystyle {\mathcal {L}}_{X}Y}
13:
3682:
3110:
3094:
3061:
3046:
2458:
2415:{\displaystyle \partial _{j}X^{i}}
2393:
2378:{\displaystyle \partial _{j}Y^{i}}
2356:
2035:
2010:
1977:
1907:
1732:
1666:
1593:
1589:
1574:
1473:
1450:
1424:
1398:
1367:
1361:
1318:
1300:
1279:
1258:
1219:
1207:
1055:
1029:
1024:
993:
987:
924:
898:
893:
837:
732:
689:
431:). Furthermore, any derivation on
322:
79:
14:
4822:
1520:
760:associated with the vector field
220:{\displaystyle X:M\rightarrow TM}
16:Operator in differential topology
3693:vector-valued differential forms
3478:means matrix multiplication and
2963:Vanishing of the Lie bracket of
3600:{\displaystyle {\mathfrak {g}}}
3206:{\displaystyle {\mathfrak {g}}}
3169:Frobenius integrability theorem
2271:{\displaystyle :=J_{Y}X-J_{X}Y}
766:tangent map derivative operator
152:Frobenius integrability theorem
135:into an (infinite-dimensional)
53:a third vector field denoted .
3971:Differentiable/Smooth manifold
3733:
3709:
3633:
3621:
3445:
3437:
3363:
3355:
3297:
3274:
3258:
3167:This is a special case of the
3133:
3127:
3124:
3090:
3084:
3078:
3075:
3042:
3007:
2995:
2921:
2909:
2893:
2887:
2871:
2856:
2755:
2743:
2714:
2711:
2699:
2690:
2684:
2681:
2669:
2660:
2654:
2651:
2639:
2630:
2605:
2593:
2584:
2572:
2545:{\displaystyle V\times V\to V}
2536:
2500:
2470:
2461:
2233:
2221:
2186:
2177:
2165:
2133:
2092:
1898:
1885:
1876:
1863:
1828:
1816:
1788:
1501:
1495:
1492:
1394:
1341:
1335:
1332:
1254:
1167:
1154:
1075:
1069:
1046:
1020:
944:
938:
915:
889:
878:
852:
831:
813:
800:
700:
694:
668:
665:
659:
653:
644:
641:
635:
629:
620:
614:
611:
599:
362:
356:
339:{\displaystyle C^{\infty }(M)}
333:
327:
251:
245:
208:
179:
131:vector fields on the manifold
1:
3784:IEEE Control Systems Magazine
3702:
2983:as coordinate vector fields:
2476:{\displaystyle V=\Gamma (TM)}
2440:
749:{\displaystyle \Phi _{t}^{X}}
569:is again a derivation, where
19:In the mathematical field of
2817:, we get a new vector field
235:acting on smooth functions
188:Vector fields as derivations
112:along the flow generated by
25:Lie bracket of vector fields
7:
4677:Classification of manifolds
3770:Encyclopedia of Mathematics
3697:FrölicherâNijenhuis bracket
3174:
562:{\displaystyle \delta _{2}}
535:{\displaystyle \delta _{1}}
68:, and is sometimes denoted
33:commutator of vector fields
10:
4827:
2952:, and the scalar function
2821:by multiplying the vector
2516:) with the structure of a
768:. Then the Lie bracket of
4753:over commutative algebras
4710:
4669:
4602:
4499:
4395:
4342:
4333:
4169:
4092:
4031:
3951:
3607:corresponds to the usual
3035:commute locally, meaning
2797:Furthermore, there is a "
2520:, which means is a map
2068:, then the vector fields
1559:{\displaystyle \{x^{i}\}}
1133:{\displaystyle X,Y,-X,-Y}
192:Each smooth vector field
156:nonlinear control systems
4469:Riemann curvature tensor
3687:As mentioned above, the
2948:) with the vector field
2726:{\displaystyle ]+]+]=0.}
2483:of all vector fields on
29:Jacobi–Lie bracket
3891:Warner, Frank (1983) ,
3796:10.1109/MCS.2009.932394
3156:and sufficiently small
2509:{\displaystyle TM\to M}
2064:is (an open subset of)
1140:to return to the point
764:, and let D denote the
515:of any two derivations
4261:Manifold with boundary
3976:Differential structure
3728:feedback linearization
3673:
3601:
3577:
3522:
3472:
3471:{\displaystyle \cdot }
3452:
3370:
3311:
3237:
3236:{\displaystyle T_{e}G}
3207:
3140:
3021:
2931:
2788:
2768:
2727:
2612:
2546:
2510:
2477:
2416:
2379:
2329:
2302:
2272:
2205:
2158:, and the Lie bracket
2152:
2111:
2051:
1960:
1939:
1854:
1800:
1746:
1720:
1680:
1654:
1614:
1560:
1511:
1134:
1087:
786:can be defined as the
750:
710:
583:
582:{\displaystyle \circ }
563:
536:
509:
397:directional derivative
389:
369:
340:
304:
284:
283:{\displaystyle p\in M}
258:
221:
150:, for instance in the
119:The Lie bracket is an
97:
4806:Differential topology
4801:Differential geometry
3674:
3602:
3578:
3523:
3473:
3453:
3371:
3312:
3238:
3208:
3141:
3022:
2932:
2789:
2769:
2728:
2613:
2547:
2511:
2478:
2417:
2380:
2330:
2328:{\displaystyle J_{X}}
2303:
2301:{\displaystyle J_{Y}}
2273:
2206:
2153:
2112:
2052:
1940:
1919:
1834:
1801:
1752:for smooth functions
1747:
1700:
1681:
1634:
1615:
1561:
1512:
1135:
1088:
751:
711:
584:
564:
537:
510:
390:
370:
341:
305:
285:
259:
233:differential operator
231:may be regarded as a
222:
148:differential topology
144:differential geometry
98:
21:differential topology
4408:Covariant derivative
3959:Topological manifold
3726:, pp. 523â530,
3720:nonholonomic systems
3618:
3587:
3532:
3486:
3462:
3380:
3324:
3255:
3217:
3193:
3186:, the corresponding
3039:
3020:{\displaystyle =0\,}
2992:
2853:
2778:
2740:
2627:
2569:
2524:
2491:
2449:
2389:
2352:
2312:
2285:
2218:
2162:
2121:
2080:
1813:
1756:
1690:
1624:
1570:
1537:
1151:
1100:
797:
728:
596:
573:
546:
519:
453:
379:
368:{\displaystyle X(f)}
350:
314:
294:
268:
257:{\displaystyle f(p)}
239:
196:
73:
27:, also known as the
4811:Riemannian geometry
4442:Exterior derivative
4044:AtiyahâSinger index
3993:Riemannian manifold
3876:, Springer-Verlag,
3123:
3107:
3074:
3059:
2809:and a vector field
1491:
1468:
1445:
1419:
1331:
1313:
1295:
1274:
1068:
1045:
937:
914:
745:
675: for all
4748:Secondary calculus
4702:Singularity theory
4657:Parallel transport
4425:De Rham cohomology
4064:Generalized Stokes
3718:, pp. 20â21,
3669:
3597:
3573:
3518:
3468:
3448:
3366:
3307:
3233:
3203:
3136:
3109:
3093:
3060:
3045:
3017:
2927:
2784:
2767:{\displaystyle =0}
2764:
2723:
2611:{\displaystyle =-}
2608:
2542:
2506:
2473:
2412:
2375:
2325:
2298:
2268:
2201:
2148:
2107:
2047:
1796:
1742:
1741:
1676:
1675:
1610:
1608:
1556:
1507:
1472:
1449:
1423:
1397:
1376:
1317:
1299:
1278:
1257:
1235:
1200:
1130:
1083:
1054:
1028:
1002:
923:
897:
885:
746:
731:
706:
579:
559:
532:
505:
385:
365:
336:
300:
280:
254:
217:
93:
4783:
4782:
4665:
4664:
4430:Differential form
4084:Whitney embedding
4018:Differential form
3883:978-0-387-94338-1
3872:Lang, S. (1995),
3816:Nonlinear Systems
3790:(3): 17â21, 132,
3751:978-0-387-94947-5
3644:
3638:
3247:vector fields on
3027:iff the flows of
2882:
2876:
2787:{\displaystyle X}
2346:Jacobian matrices
1607:
1483:
1460:
1437:
1411:
1375:
1352:
1346:
1234:
1199:
1184:
1178:
1001:
978:
972:
968:
870:
869:
863:
830:
824:
676:
407:in the direction
388:{\displaystyle p}
346:) when we define
303:{\displaystyle f}
4818:
4775:Stratified space
4733:Fréchet manifold
4447:Interior product
4340:
4339:
4037:
3933:
3926:
3919:
3910:
3909:
3905:
3886:
3867:
3861:
3853:
3833:
3806:
3778:
3756:
3755:
3737:
3731:
3713:
3678:
3676:
3675:
3670:
3642:
3636:
3606:
3604:
3603:
3598:
3596:
3595:
3582:
3580:
3579:
3574:
3569:
3568:
3544:
3543:
3527:
3525:
3524:
3519:
3514:
3513:
3501:
3500:
3477:
3475:
3474:
3469:
3457:
3455:
3454:
3449:
3444:
3436:
3435:
3414:
3413:
3392:
3391:
3375:
3373:
3372:
3367:
3362:
3354:
3353:
3316:
3314:
3313:
3308:
3306:
3305:
3296:
3295:
3286:
3285:
3242:
3240:
3239:
3234:
3229:
3228:
3212:
3210:
3209:
3204:
3202:
3201:
3155:
3145:
3143:
3142:
3137:
3122:
3117:
3106:
3101:
3073:
3068:
3058:
3053:
3026:
3024:
3023:
3018:
2936:
2934:
2933:
2928:
2880:
2874:
2845:
2836:) at each point
2793:
2791:
2790:
2785:
2773:
2771:
2770:
2765:
2732:
2730:
2729:
2724:
2617:
2615:
2614:
2609:
2551:
2549:
2548:
2543:
2515:
2513:
2512:
2507:
2482:
2480:
2479:
2474:
2428:
2421:
2419:
2418:
2413:
2411:
2410:
2401:
2400:
2384:
2382:
2381:
2376:
2374:
2373:
2364:
2363:
2344:
2334:
2332:
2331:
2326:
2324:
2323:
2307:
2305:
2304:
2299:
2297:
2296:
2277:
2275:
2274:
2269:
2264:
2263:
2248:
2247:
2210:
2208:
2207:
2202:
2200:
2199:
2194:
2157:
2155:
2154:
2149:
2147:
2146:
2141:
2116:
2114:
2113:
2108:
2106:
2105:
2100:
2056:
2054:
2053:
2048:
2043:
2042:
2033:
2029:
2028:
2027:
2018:
2017:
2008:
2007:
1995:
1994:
1985:
1984:
1975:
1974:
1959:
1954:
1938:
1933:
1915:
1914:
1905:
1901:
1897:
1896:
1875:
1874:
1853:
1848:
1805:
1803:
1802:
1797:
1795:
1781:
1780:
1768:
1767:
1751:
1749:
1748:
1743:
1740:
1739:
1730:
1729:
1719:
1714:
1685:
1683:
1682:
1677:
1674:
1673:
1664:
1663:
1653:
1648:
1619:
1617:
1616:
1611:
1609:
1606:
1605:
1604:
1588:
1582:
1581:
1565:
1563:
1562:
1557:
1552:
1551:
1516:
1514:
1513:
1508:
1490:
1485:
1484:
1479:
1467:
1462:
1461:
1456:
1444:
1439:
1438:
1433:
1418:
1413:
1412:
1407:
1393:
1392:
1381:
1377:
1374:
1370:
1364:
1359:
1350:
1344:
1330:
1325:
1312:
1307:
1294:
1289:
1273:
1268:
1253:
1252:
1241:
1237:
1236:
1233:
1232:
1231:
1222:
1216:
1215:
1210:
1204:
1201:
1192:
1182:
1176:
1175:
1174:
1139:
1137:
1136:
1131:
1092:
1090:
1089:
1084:
1079:
1078:
1067:
1062:
1044:
1039:
1027:
1019:
1018:
1007:
1003:
1000:
996:
990:
985:
976:
970:
969:
964:
963:
962:
948:
947:
936:
931:
913:
908:
896:
887:
884:
867:
861:
860:
859:
847:
846:
841:
840:
828:
822:
821:
820:
785:
755:
753:
752:
747:
744:
739:
720:Flows and limits
715:
713:
712:
707:
693:
692:
677:
674:
588:
586:
585:
580:
568:
566:
565:
560:
558:
557:
541:
539:
538:
533:
531:
530:
514:
512:
511:
506:
504:
503:
491:
490:
478:
477:
465:
464:
446:In general, the
394:
392:
391:
386:
374:
372:
371:
366:
345:
343:
342:
337:
326:
325:
309:
307:
306:
301:
289:
287:
286:
281:
263:
261:
260:
255:
226:
224:
223:
218:
102:
100:
99:
94:
89:
88:
83:
82:
4826:
4825:
4821:
4820:
4819:
4817:
4816:
4815:
4786:
4785:
4784:
4779:
4718:Banach manifold
4711:Generalizations
4706:
4661:
4598:
4495:
4457:Ricci curvature
4413:Cotangent space
4391:
4329:
4171:
4165:
4124:Exponential map
4088:
4033:
4027:
3947:
3937:
3903:
3884:
3855:
3854:
3851:
3831:
3763:
3760:
3759:
3752:
3738:
3734:
3714:
3710:
3705:
3685:
3683:Generalizations
3619:
3616:
3615:
3591:
3590:
3588:
3585:
3584:
3564:
3560:
3539:
3535:
3533:
3530:
3529:
3509:
3505:
3496:
3495:
3487:
3484:
3483:
3463:
3460:
3459:
3440:
3425:
3421:
3409:
3405:
3387:
3383:
3381:
3378:
3377:
3358:
3343:
3339:
3325:
3322:
3321:
3301:
3300:
3291:
3290:
3281:
3280:
3256:
3253:
3252:
3224:
3220:
3218:
3215:
3214:
3197:
3196:
3194:
3191:
3190:
3177:
3147:
3118:
3113:
3102:
3097:
3069:
3064:
3054:
3049:
3040:
3037:
3036:
2993:
2990:
2989:
2854:
2851:
2850:
2837:
2826:
2779:
2776:
2775:
2741:
2738:
2737:
2628:
2625:
2624:
2621:Jacobi identity
2570:
2567:
2566:
2565:Anti-symmetry,
2525:
2522:
2521:
2492:
2489:
2488:
2450:
2447:
2446:
2443:
2429:column vectors
2423:
2406:
2402:
2396:
2392:
2390:
2387:
2386:
2369:
2365:
2359:
2355:
2353:
2350:
2349:
2336:
2319:
2315:
2313:
2310:
2309:
2292:
2288:
2286:
2283:
2282:
2259:
2255:
2243:
2239:
2219:
2216:
2215:
2195:
2190:
2189:
2163:
2160:
2159:
2142:
2137:
2136:
2122:
2119:
2118:
2101:
2096:
2095:
2081:
2078:
2077:
2038:
2034:
2023:
2019:
2013:
2009:
2003:
1999:
1990:
1986:
1980:
1976:
1970:
1966:
1965:
1961:
1955:
1944:
1934:
1923:
1910:
1906:
1892:
1888:
1870:
1866:
1859:
1855:
1849:
1838:
1814:
1811:
1810:
1791:
1776:
1772:
1763:
1759:
1757:
1754:
1753:
1735:
1731:
1725:
1721:
1715:
1704:
1691:
1688:
1687:
1669:
1665:
1659:
1655:
1649:
1638:
1625:
1622:
1621:
1600:
1596:
1592:
1586:
1577:
1573:
1571:
1568:
1567:
1547:
1543:
1538:
1535:
1534:
1523:
1486:
1478:
1476:
1463:
1455:
1453:
1440:
1432:
1427:
1414:
1406:
1401:
1382:
1366:
1365:
1360:
1357:
1354:
1353:
1326:
1321:
1308:
1303:
1290:
1282:
1269:
1261:
1242:
1227:
1223:
1218:
1217:
1211:
1206:
1205:
1202:
1190:
1189:
1186:
1185:
1170:
1166:
1152:
1149:
1148:
1101:
1098:
1097:
1063:
1058:
1053:
1049:
1040:
1032:
1023:
1008:
992:
991:
986:
983:
980:
979:
958:
954:
932:
927:
922:
918:
909:
901:
892:
888:
886:
874:
855:
851:
842:
836:
835:
834:
816:
812:
798:
795:
794:
777:
740:
735:
729:
726:
725:
722:
688:
684:
673:
597:
594:
593:
574:
571:
570:
553:
549:
547:
544:
543:
526:
522:
520:
517:
516:
499:
495:
486:
482:
473:
469:
460:
456:
454:
451:
450:
380:
377:
376:
351:
348:
347:
321:
317:
315:
312:
311:
295:
292:
291:
269:
266:
265:
240:
237:
236:
197:
194:
193:
190:
182:
84:
78:
77:
76:
74:
71:
70:
48:smooth manifold
17:
12:
11:
5:
4824:
4814:
4813:
4808:
4803:
4798:
4781:
4780:
4778:
4777:
4772:
4767:
4762:
4757:
4756:
4755:
4745:
4740:
4735:
4730:
4725:
4720:
4714:
4712:
4708:
4707:
4705:
4704:
4699:
4694:
4689:
4684:
4679:
4673:
4671:
4667:
4666:
4663:
4662:
4660:
4659:
4654:
4649:
4644:
4639:
4634:
4629:
4624:
4619:
4614:
4608:
4606:
4600:
4599:
4597:
4596:
4591:
4586:
4581:
4576:
4571:
4566:
4556:
4551:
4546:
4536:
4531:
4526:
4521:
4516:
4511:
4505:
4503:
4497:
4496:
4494:
4493:
4488:
4483:
4482:
4481:
4471:
4466:
4465:
4464:
4454:
4449:
4444:
4439:
4438:
4437:
4427:
4422:
4421:
4420:
4410:
4405:
4399:
4397:
4393:
4392:
4390:
4389:
4384:
4379:
4374:
4373:
4372:
4362:
4357:
4352:
4346:
4344:
4337:
4331:
4330:
4328:
4327:
4322:
4312:
4307:
4293:
4288:
4283:
4278:
4273:
4271:Parallelizable
4268:
4263:
4258:
4257:
4256:
4246:
4241:
4236:
4231:
4226:
4221:
4216:
4211:
4206:
4201:
4191:
4181:
4175:
4173:
4167:
4166:
4164:
4163:
4158:
4153:
4151:Lie derivative
4148:
4146:Integral curve
4143:
4138:
4133:
4132:
4131:
4121:
4116:
4115:
4114:
4107:Diffeomorphism
4104:
4098:
4096:
4090:
4089:
4087:
4086:
4081:
4076:
4071:
4066:
4061:
4056:
4051:
4046:
4040:
4038:
4029:
4028:
4026:
4025:
4020:
4015:
4010:
4005:
4000:
3995:
3990:
3985:
3984:
3983:
3978:
3968:
3967:
3966:
3955:
3953:
3952:Basic concepts
3949:
3948:
3936:
3935:
3928:
3921:
3913:
3907:
3906:
3901:
3888:
3882:
3869:
3849:
3834:
3829:
3807:
3779:
3758:
3757:
3750:
3732:
3707:
3706:
3704:
3701:
3689:Lie derivative
3684:
3681:
3680:
3679:
3668:
3665:
3662:
3659:
3656:
3653:
3650:
3647:
3641:
3635:
3632:
3629:
3626:
3623:
3594:
3572:
3567:
3563:
3559:
3556:
3553:
3550:
3547:
3542:
3538:
3517:
3512:
3508:
3504:
3499:
3494:
3491:
3467:
3447:
3443:
3439:
3434:
3431:
3428:
3424:
3420:
3417:
3412:
3408:
3404:
3401:
3398:
3395:
3390:
3386:
3365:
3361:
3357:
3352:
3349:
3346:
3342:
3338:
3335:
3332:
3329:
3304:
3299:
3294:
3289:
3284:
3279:
3276:
3272:
3268:
3264:
3260:
3245:left invariant
3232:
3227:
3223:
3200:
3176:
3173:
3135:
3132:
3129:
3126:
3121:
3116:
3112:
3105:
3100:
3096:
3092:
3089:
3086:
3083:
3080:
3077:
3072:
3067:
3063:
3057:
3052:
3048:
3044:
3015:
3012:
3009:
3006:
3003:
3000:
2997:
2938:
2937:
2926:
2923:
2920:
2917:
2914:
2911:
2907:
2903:
2899:
2895:
2892:
2889:
2885:
2879:
2873:
2870:
2867:
2864:
2861:
2858:
2828:by the scalar
2824:
2783:
2763:
2760:
2757:
2754:
2751:
2748:
2745:
2734:
2733:
2722:
2719:
2716:
2713:
2710:
2707:
2704:
2701:
2698:
2695:
2692:
2689:
2686:
2683:
2680:
2677:
2674:
2671:
2668:
2665:
2662:
2659:
2656:
2653:
2650:
2647:
2644:
2641:
2638:
2635:
2632:
2618:
2607:
2604:
2601:
2598:
2595:
2592:
2589:
2586:
2583:
2580:
2577:
2574:
2563:
2541:
2538:
2535:
2532:
2529:
2505:
2502:
2499:
2496:
2472:
2469:
2466:
2463:
2460:
2457:
2454:
2442:
2439:
2409:
2405:
2399:
2395:
2372:
2368:
2362:
2358:
2322:
2318:
2295:
2291:
2279:
2278:
2267:
2262:
2258:
2254:
2251:
2246:
2242:
2238:
2235:
2232:
2229:
2226:
2223:
2198:
2193:
2188:
2185:
2182:
2179:
2176:
2173:
2170:
2167:
2145:
2140:
2135:
2132:
2129:
2126:
2104:
2099:
2094:
2091:
2088:
2085:
2058:
2057:
2046:
2041:
2037:
2032:
2026:
2022:
2016:
2012:
2006:
2002:
1998:
1993:
1989:
1983:
1979:
1973:
1969:
1964:
1958:
1953:
1950:
1947:
1943:
1937:
1932:
1929:
1926:
1922:
1918:
1913:
1909:
1904:
1900:
1895:
1891:
1887:
1884:
1881:
1878:
1873:
1869:
1865:
1862:
1858:
1852:
1847:
1844:
1841:
1837:
1833:
1830:
1827:
1824:
1821:
1818:
1794:
1790:
1787:
1784:
1779:
1775:
1771:
1766:
1762:
1738:
1734:
1728:
1724:
1718:
1713:
1710:
1707:
1703:
1699:
1696:
1672:
1668:
1662:
1658:
1652:
1647:
1644:
1641:
1637:
1633:
1630:
1603:
1599:
1595:
1591:
1585:
1580:
1576:
1555:
1550:
1546:
1542:
1522:
1521:In coordinates
1519:
1518:
1517:
1506:
1503:
1500:
1497:
1494:
1489:
1482:
1475:
1471:
1466:
1459:
1452:
1448:
1443:
1436:
1431:
1426:
1422:
1417:
1410:
1405:
1400:
1396:
1391:
1388:
1385:
1380:
1373:
1369:
1363:
1356:
1349:
1343:
1340:
1337:
1334:
1329:
1324:
1320:
1316:
1311:
1306:
1302:
1298:
1293:
1288:
1285:
1281:
1277:
1272:
1267:
1264:
1260:
1256:
1251:
1248:
1245:
1240:
1230:
1226:
1221:
1214:
1209:
1198:
1195:
1188:
1181:
1173:
1169:
1165:
1162:
1159:
1156:
1129:
1126:
1123:
1120:
1117:
1114:
1111:
1108:
1105:
1094:
1093:
1082:
1077:
1074:
1071:
1066:
1061:
1057:
1052:
1048:
1043:
1038:
1035:
1031:
1026:
1022:
1017:
1014:
1011:
1006:
999:
995:
989:
982:
975:
967:
961:
957:
952:
946:
943:
940:
935:
930:
926:
921:
917:
912:
907:
904:
900:
895:
891:
883:
880:
877:
873:
866:
858:
854:
850:
845:
839:
833:
827:
819:
815:
811:
808:
805:
802:
788:Lie derivative
743:
738:
734:
721:
718:
717:
716:
705:
702:
699:
696:
691:
687:
683:
680:
670:
667:
664:
661:
658:
655:
652:
649:
646:
643:
640:
637:
634:
631:
628:
625:
622:
619:
616:
613:
610:
607:
604:
601:
578:
556:
552:
529:
525:
502:
498:
494:
489:
485:
481:
476:
472:
468:
463:
459:
384:
364:
361:
358:
355:
335:
332:
329:
324:
320:
299:
279:
276:
273:
253:
250:
247:
244:
227:on a manifold
216:
213:
210:
207:
204:
201:
189:
186:
181:
178:
106:Lie derivative
92:
87:
81:
15:
9:
6:
4:
3:
2:
4823:
4812:
4809:
4807:
4804:
4802:
4799:
4797:
4796:Bilinear maps
4794:
4793:
4791:
4776:
4773:
4771:
4770:Supermanifold
4768:
4766:
4763:
4761:
4758:
4754:
4751:
4750:
4749:
4746:
4744:
4741:
4739:
4736:
4734:
4731:
4729:
4726:
4724:
4721:
4719:
4716:
4715:
4713:
4709:
4703:
4700:
4698:
4695:
4693:
4690:
4688:
4685:
4683:
4680:
4678:
4675:
4674:
4672:
4668:
4658:
4655:
4653:
4650:
4648:
4645:
4643:
4640:
4638:
4635:
4633:
4630:
4628:
4625:
4623:
4620:
4618:
4615:
4613:
4610:
4609:
4607:
4605:
4601:
4595:
4592:
4590:
4587:
4585:
4582:
4580:
4577:
4575:
4572:
4570:
4567:
4565:
4561:
4557:
4555:
4552:
4550:
4547:
4545:
4541:
4537:
4535:
4532:
4530:
4527:
4525:
4522:
4520:
4517:
4515:
4512:
4510:
4507:
4506:
4504:
4502:
4498:
4492:
4491:Wedge product
4489:
4487:
4484:
4480:
4477:
4476:
4475:
4472:
4470:
4467:
4463:
4460:
4459:
4458:
4455:
4453:
4450:
4448:
4445:
4443:
4440:
4436:
4435:Vector-valued
4433:
4432:
4431:
4428:
4426:
4423:
4419:
4416:
4415:
4414:
4411:
4409:
4406:
4404:
4401:
4400:
4398:
4394:
4388:
4385:
4383:
4380:
4378:
4375:
4371:
4368:
4367:
4366:
4365:Tangent space
4363:
4361:
4358:
4356:
4353:
4351:
4348:
4347:
4345:
4341:
4338:
4336:
4332:
4326:
4323:
4321:
4317:
4313:
4311:
4308:
4306:
4302:
4298:
4294:
4292:
4289:
4287:
4284:
4282:
4279:
4277:
4274:
4272:
4269:
4267:
4264:
4262:
4259:
4255:
4252:
4251:
4250:
4247:
4245:
4242:
4240:
4237:
4235:
4232:
4230:
4227:
4225:
4222:
4220:
4217:
4215:
4212:
4210:
4207:
4205:
4202:
4200:
4196:
4192:
4190:
4186:
4182:
4180:
4177:
4176:
4174:
4168:
4162:
4159:
4157:
4154:
4152:
4149:
4147:
4144:
4142:
4139:
4137:
4134:
4130:
4129:in Lie theory
4127:
4126:
4125:
4122:
4120:
4117:
4113:
4110:
4109:
4108:
4105:
4103:
4100:
4099:
4097:
4095:
4091:
4085:
4082:
4080:
4077:
4075:
4072:
4070:
4067:
4065:
4062:
4060:
4057:
4055:
4052:
4050:
4047:
4045:
4042:
4041:
4039:
4036:
4032:Main results
4030:
4024:
4021:
4019:
4016:
4014:
4013:Tangent space
4011:
4009:
4006:
4004:
4001:
3999:
3996:
3994:
3991:
3989:
3986:
3982:
3979:
3977:
3974:
3973:
3972:
3969:
3965:
3962:
3961:
3960:
3957:
3956:
3954:
3950:
3945:
3941:
3934:
3929:
3927:
3922:
3920:
3915:
3914:
3911:
3904:
3902:0-387-90894-3
3898:
3894:
3889:
3885:
3879:
3875:
3870:
3865:
3859:
3852:
3850:3-540-56235-4
3846:
3842:
3841:
3835:
3832:
3830:0-13-067389-7
3826:
3822:
3821:Prentice Hall
3818:
3817:
3812:
3808:
3805:
3801:
3797:
3793:
3789:
3785:
3780:
3776:
3772:
3771:
3766:
3765:"Lie bracket"
3762:
3761:
3753:
3747:
3743:
3736:
3729:
3725:
3721:
3717:
3712:
3708:
3700:
3698:
3694:
3690:
3666:
3663:
3660:
3657:
3654:
3651:
3648:
3645:
3639:
3630:
3627:
3624:
3614:
3613:
3612:
3611:of matrices:
3610:
3570:
3565:
3561:
3557:
3554:
3551:
3548:
3545:
3540:
3536:
3515:
3510:
3506:
3502:
3492:
3489:
3481:
3465:
3432:
3429:
3426:
3422:
3418:
3415:
3410:
3406:
3402:
3399:
3396:
3393:
3388:
3384:
3350:
3347:
3344:
3340:
3336:
3333:
3330:
3327:
3318:
3287:
3277:
3270:
3266:
3262:
3250:
3246:
3230:
3225:
3221:
3189:
3185:
3182:
3172:
3170:
3165:
3163:
3159:
3154:
3150:
3130:
3119:
3114:
3103:
3098:
3087:
3081:
3070:
3065:
3055:
3050:
3034:
3030:
3013:
3010:
3004:
3001:
2998:
2988:
2984:
2982:
2978:
2974:
2970:
2966:
2961:
2959:
2958:Lie algebroid
2955:
2951:
2947:
2943:
2924:
2918:
2915:
2912:
2905:
2901:
2897:
2890:
2883:
2877:
2868:
2865:
2862:
2859:
2849:
2848:
2847:
2844:
2840:
2835:
2831:
2827:
2820:
2816:
2812:
2808:
2804:
2800:
2795:
2781:
2761:
2758:
2752:
2749:
2746:
2720:
2717:
2708:
2705:
2702:
2696:
2693:
2687:
2678:
2675:
2672:
2666:
2663:
2657:
2648:
2645:
2642:
2636:
2633:
2622:
2619:
2602:
2599:
2596:
2590:
2587:
2581:
2578:
2575:
2564:
2562:
2558:
2555:
2554:
2553:
2539:
2533:
2530:
2527:
2519:
2503:
2497:
2494:
2486:
2467:
2464:
2455:
2452:
2438:
2436:
2432:
2426:
2407:
2403:
2397:
2370:
2366:
2360:
2347:
2343:
2339:
2320:
2316:
2293:
2289:
2265:
2260:
2256:
2252:
2249:
2244:
2240:
2236:
2230:
2227:
2224:
2214:
2213:
2212:
2211:is given by:
2196:
2183:
2180:
2174:
2171:
2168:
2143:
2130:
2127:
2124:
2102:
2089:
2086:
2083:
2075:
2071:
2067:
2063:
2044:
2039:
2030:
2024:
2020:
2014:
2004:
2000:
1996:
1991:
1987:
1981:
1971:
1967:
1962:
1956:
1951:
1948:
1945:
1941:
1935:
1930:
1927:
1924:
1920:
1916:
1911:
1902:
1893:
1889:
1882:
1879:
1871:
1867:
1860:
1856:
1850:
1845:
1842:
1839:
1835:
1831:
1825:
1822:
1819:
1809:
1808:
1807:
1785:
1782:
1777:
1773:
1769:
1764:
1760:
1736:
1726:
1722:
1716:
1711:
1708:
1705:
1701:
1697:
1694:
1670:
1660:
1656:
1650:
1645:
1642:
1639:
1635:
1631:
1628:
1601:
1597:
1583:
1578:
1548:
1544:
1532:
1528:
1504:
1498:
1487:
1480:
1469:
1464:
1457:
1446:
1441:
1434:
1429:
1420:
1415:
1408:
1403:
1389:
1386:
1383:
1378:
1371:
1347:
1338:
1327:
1322:
1314:
1309:
1304:
1296:
1291:
1286:
1283:
1275:
1270:
1265:
1262:
1249:
1246:
1243:
1238:
1228:
1224:
1212:
1196:
1193:
1179:
1171:
1163:
1160:
1157:
1147:
1146:
1145:
1143:
1127:
1124:
1121:
1118:
1115:
1112:
1109:
1106:
1103:
1080:
1072:
1064:
1059:
1050:
1041:
1036:
1033:
1015:
1012:
1009:
1004:
997:
973:
965:
959:
955:
950:
941:
933:
928:
919:
910:
905:
902:
881:
875:
864:
856:
848:
843:
825:
817:
809:
806:
803:
793:
792:
791:
789:
784:
780:
776:at the point
775:
771:
767:
763:
759:
741:
736:
703:
697:
685:
681:
678:
662:
656:
650:
647:
638:
632:
626:
623:
617:
608:
605:
602:
592:
591:
590:
576:
554:
550:
527:
523:
500:
496:
492:
487:
483:
479:
474:
470:
466:
461:
457:
449:
444:
442:
438:
434:
430:
426:
422:
418:
414:
410:
406:
402:
398:
382:
359:
353:
330:
318:
297:
277:
274:
271:
248:
242:
234:
230:
214:
211:
205:
202:
199:
185:
177:
175:
171:
167:
163:
159:
157:
153:
149:
145:
140:
138:
134:
130:
126:
122:
117:
115:
111:
107:
103:
90:
85:
67:
64:generated by
63:
59:
54:
52:
49:
45:
41:
38:
37:vector fields
34:
30:
26:
22:
4697:Moving frame
4692:Morse theory
4682:Gauge theory
4474:Tensor field
4403:Closed/Exact
4382:Vector field
4354:
4350:Distribution
4291:Hypercomplex
4286:Quaternionic
4023:Vector field
3981:Smooth atlas
3892:
3873:
3839:
3815:
3811:Khalil, H.K.
3787:
3783:
3768:
3741:
3735:
3711:
3686:
3528:is given by
3479:
3319:
3248:
3183:
3178:
3166:
3161:
3157:
3152:
3148:
3032:
3028:
2986:
2985:
2980:
2976:
2972:
2968:
2964:
2962:
2953:
2949:
2945:
2941:
2939:
2842:
2838:
2833:
2829:
2822:
2818:
2814:
2810:
2806:
2802:
2799:product rule
2796:
2735:
2556:
2484:
2444:
2434:
2430:
2424:
2341:
2337:
2280:
2073:
2069:
2065:
2061:
2059:
1530:
1524:
1141:
1095:
782:
778:
773:
769:
761:
723:
445:
440:
436:
432:
428:
424:
416:
412:
408:
404:
400:
228:
191:
183:
173:
169:
162:V. I. Arnold
160:
141:
132:
120:
118:
113:
110:tensor field
69:
65:
57:
55:
50:
43:
39:
32:
28:
24:
18:
4642:Levi-Civita
4632:Generalized
4604:Connections
4554:Lie algebra
4486:Volume form
4387:Vector flow
4360:Pushforward
4355:Lie bracket
4254:Lie algebra
4219:G-structure
4008:Pushforward
3988:Submanifold
3724:Khalil 2002
3716:Isaiah 2009
3188:Lie algebra
2561:bilinearity
2518:Lie algebra
1566:. We write
180:Definitions
137:Lie algebra
4790:Categories
4765:Stratifold
4723:Diffeology
4519:Associated
4320:Symplectic
4305:Riemannian
4234:Hyperbolic
4161:Submersion
4069:HopfâRinow
4003:Submersion
3998:Smooth map
3703:References
3609:commutator
2441:Properties
448:commutator
421:derivation
419:becomes a
60:along the
4647:Principal
4622:Ehresmann
4579:Subbundle
4569:Principal
4544:Fibration
4524:Cotangent
4396:Covectors
4249:Lie group
4229:Hermitian
4172:manifolds
4141:Immersion
4136:Foliation
4074:Noether's
4059:Frobenius
4054:De Rham's
4049:Darboux's
3940:Manifolds
3775:EMS Press
3695:) is the
3661:⋅
3655:−
3649:⋅
3558:∈
3552:⋅
3493:∈
3466:⋅
3430:×
3419:⊂
3403:⋅
3348:×
3337:⊂
3331:∈
3298:→
3288:×
3271:⋅
3263:⋅
3181:Lie group
3111:Φ
3095:Φ
3062:Φ
3047:Φ
2591:−
2537:→
2531:×
2501:→
2459:Γ
2394:∂
2357:∂
2253:−
2187:→
2134:→
2093:→
2036:∂
2011:∂
1997:−
1978:∂
1942:∑
1921:∑
1908:∂
1880:−
1836:∑
1789:→
1733:∂
1702:∑
1667:∂
1636:∑
1594:∂
1590:∂
1575:∂
1527:intrinsic
1474:Φ
1470:∘
1451:Φ
1447:∘
1430:−
1425:Φ
1421:∘
1404:−
1399:Φ
1319:Φ
1315:∘
1301:Φ
1297:∘
1284:−
1280:Φ
1276:∘
1263:−
1259:Φ
1125:−
1116:−
1056:Φ
1034:−
1030:Φ
951:−
925:Φ
903:−
899:Φ
879:→
733:Φ
690:∞
682:∈
648:−
577:∘
551:δ
524:δ
497:δ
493:∘
484:δ
480:−
471:δ
467:∘
458:δ
323:∞
310:of class
275:∈
209:→
4743:Orbifold
4738:K-theory
4728:Diffiety
4452:Pullback
4266:Oriented
4244:Kenmotsu
4224:Hadamard
4170:Types of
4119:Geodesic
3944:Glossary
3858:citation
3813:(2002),
3804:42908664
3458:, where
3175:Examples
3151:∈
3146:for all
2987:Theorem:
2846:. Then:
2841:∈
2774:for any
781:∈
125:bilinear
4687:History
4670:Related
4584:Tangent
4562:)
4542:)
4509:Adjoint
4501:Bundles
4479:density
4377:Torsion
4343:Vectors
4335:Tensors
4318:)
4303:)
4299:,
4297:Pseudoâ
4276:Poisson
4209:Finsler
4204:Fibered
4199:Contact
4197:)
4189:Complex
4187:)
4156:Section
3777:, 2001
2975:, with
756:be the
395:is the
264:(where
108:of any
31:or the
4652:Vector
4637:Koszul
4617:Cartan
4612:Affine
4594:Vector
4589:Tensor
4574:Spinor
4564:Normal
4560:Stable
4514:Affine
4418:bundle
4370:bundle
4316:Almost
4239:KĂ€hler
4195:Almost
4185:Almost
4179:Closed
4079:Sard's
4035:(list)
3899:
3880:
3847:
3827:
3802:
3748:
3643:
3637:
3179:For a
2881:
2875:
2552:with:
2281:where
1351:
1345:
1183:
1177:
977:
971:
868:
862:
829:
823:
129:smooth
23:, the
4760:Sheaf
4534:Fiber
4310:Rizza
4281:Prime
4112:Local
4102:Curve
3964:Atlas
3800:S2CID
166:float
46:on a
4627:Form
4529:Dual
4462:flow
4325:Tame
4301:Subâ
4214:Flat
4094:Maps
3897:ISBN
3878:ISBN
3864:link
3845:ISBN
3825:ISBN
3746:ISBN
3031:and
2979:and
2967:and
2433:and
2385:and
2335:are
2308:and
2117:and
2072:and
1686:and
772:and
758:flow
724:Let
542:and
290:and
146:and
62:flow
42:and
4549:Jet
3792:doi
2813:on
2805:on
2427:Ă 1
2060:If
872:lim
423:on
403:at
399:of
4792::
4540:Co
3860:}}
3856:{{
3823:,
3798:,
3788:29
3786:,
3773:,
3767:,
3722:;
3699:.
3317:.
3171:.
3164:.
3160:,
2960:.
2819:fY
2794:.
2721:0.
2623:,
2437:.
2340:Ă
2237::=
1832::=
1144::
865::=
790::
443:.
158:.
139:.
116:.
4558:(
4538:(
4314:(
4295:(
4193:(
4183:(
3946:)
3942:(
3932:e
3925:t
3918:v
3866:)
3794::
3754:.
3730:.
3667:.
3664:X
3658:Y
3652:Y
3646:X
3640:=
3634:]
3631:Y
3628:,
3625:X
3622:[
3593:g
3571:G
3566:g
3562:T
3555:X
3549:g
3546:=
3541:g
3537:X
3516:G
3511:I
3507:T
3503:=
3498:g
3490:X
3480:I
3446:)
3442:R
3438:(
3433:n
3427:n
3423:M
3416:G
3411:I
3407:T
3400:g
3397:=
3394:G
3389:g
3385:T
3364:)
3360:R
3356:(
3351:n
3345:n
3341:M
3334:G
3328:g
3303:g
3293:g
3283:g
3278::
3275:]
3267:,
3259:[
3249:G
3231:G
3226:e
3222:T
3199:g
3184:G
3162:t
3158:s
3153:M
3149:x
3134:)
3131:x
3128:(
3125:)
3120:Y
3115:t
3104:X
3099:s
3091:(
3088:=
3085:)
3082:x
3079:(
3076:)
3071:X
3066:s
3056:Y
3051:t
3043:(
3033:Y
3029:X
3014:0
3011:=
3008:]
3005:Y
3002:,
2999:X
2996:[
2981:Y
2977:X
2973:M
2969:Y
2965:X
2954:f
2950:Y
2946:f
2944:(
2942:X
2925:,
2922:]
2919:Y
2916:,
2913:X
2910:[
2906:f
2902:+
2898:Y
2894:)
2891:f
2888:(
2884:X
2878:=
2872:]
2869:Y
2866:f
2863:,
2860:X
2857:[
2843:M
2839:x
2834:x
2832:(
2830:f
2825:x
2823:Y
2815:M
2811:Y
2807:M
2803:f
2782:X
2762:0
2759:=
2756:]
2753:X
2750:,
2747:X
2744:[
2718:=
2715:]
2712:]
2709:X
2706:,
2703:Z
2700:[
2697:,
2694:Y
2691:[
2688:+
2685:]
2682:]
2679:Y
2676:,
2673:X
2670:[
2667:,
2664:Z
2661:[
2658:+
2655:]
2652:]
2649:Z
2646:,
2643:Y
2640:[
2637:,
2634:X
2631:[
2606:]
2603:X
2600:,
2597:Y
2594:[
2588:=
2585:]
2582:Y
2579:,
2576:X
2573:[
2559:-
2557:R
2540:V
2534:V
2528:V
2504:M
2498:M
2495:T
2485:M
2471:)
2468:M
2465:T
2462:(
2456:=
2453:V
2435:Y
2431:X
2425:n
2408:i
2404:X
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40:X
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