891:
806:
267:
509:
131:
5602:
is a closed submanifold and whose
Hessian is non-degenerate in the normal direction. (Equivalently, the kernel of the Hessian at a critical point equals the tangent space to the critical submanifold.) A Morse function is the special case where the critical manifolds are zero-dimensional (so the
4851:
5740:
plus the dimension of the critical manifold. If the MorseâBott function is perturbed by a small function on the critical locus, the index of all critical points of the perturbed function on a critical manifold of the unperturbed function will lie between
5800:
MorseâBott functions are useful because generic Morse functions are difficult to work with; the functions one can visualize, and with which one can easily calculate, typically have symmetries. They often lead to positive-dimensional critical manifolds.
4322:
Using the fact that the alternating sum of the ranks of the homology groups of a topological space is equal to the alternating sum of the ranks of the chain groups from which the homology is computed, then by using the cellular chain groups (see
3212:
5570:. The basic theorem is that the resulting homology is an invariant of the manifold (that is, independent of the function and metric) and isomorphic to the singular homology of the manifold; this implies that the Morse and singular
4672:
5877: â smooth dynamical system whose non-wandering set consists of finitely many hyperbolic equilibrium points and hyperbolic periodic orbits and satisfying a transversality condition on the stable and unstable manifolds
368:
3341:
if it has no degenerate critical points. A basic result of Morse theory says that almost all functions are Morse functions. Technically, the Morse functions form an open, dense subset of all smooth functions
259:. Contour lines may also have points of higher order (triple points, etc.), but these are unstable and may be removed by a slight deformation of the landscape. Double points in contour lines occur at
4422:
5825:. Frederic Bourgeois sketched an approach in the course of his work on a MorseâBott version of symplectic field theory, but this work was never published due to substantial analytic difficulties.
2973:
4912:
4964:
2442:
5657:
3463:
4150:
Using the two previous results and the fact that there exists a Morse function on any differentiable manifold, one can prove that any differentiable manifold is a CW complex with an
3083:
5235:
2078:
5000:
4257:
2175:
5545:
4121:
4084:
1983:
1509:
1386:
3368:
969:
884:
211:
2713:
2625:
1255:
4614:
2130:
2104:
1788:
2489:
2464:
237:
4449:
1833:
3058:
2901:
4472:
4357:
3933:
5399:
4634:
4575:
4277:
4223:
4141:
3971:
3561:
3255:
2563:
1726:
1679:
1141:
5796:
5488:
5128:
4028:
3773:
3713:
3605:
1063:
1008:
7184:
5766:
5738:
5711:
5684:
3803:
3740:
3679:
3395:
1903:
1706:
1639:
1609:
1536:
1413:
1282:
1168:
919:
834:
780:
499:
5351:
2537:
683:
6375:
6310:
5154:
5070:
2651:
4657:
4555:
4320:
4191:
4051:
3652:
3533:
3235:
2201:
1943:
1876:
1749:
1728:
is the index of the point. This does not address what happens when two critical points are at the same height, which can be resolved by a slight perturbation of
1583:
1456:
1333:
1211:
706:
6088:
5508:
5462:
5442:
5418:
5370:
5322:
5302:
5282:
5262:
5098:
5044:
5024:
4532:
4512:
4492:
4297:
4168:
3910:
3890:
3870:
3850:
3823:
3629:
3510:
3483:
3335:
3295:
3275:
3078:
3016:
2996:
2866:
2834:
2810:
2790:
2766:
2746:
2671:
2583:
2508:
2347:
2317:
2293:
2269:
2249:
2225:
2027:
2007:
1923:
1853:
1659:
1560:
1433:
1310:
1188:
1103:
1083:
1028:
800:
753:
729:
654:
630:
610:
582:
562:
542:
456:
436:
408:
388:
292:
180:
156:
6050:
7179:
6466:
4193:
To do this, one needs the technical fact that one can arrange to have a single critical point on each critical level, which is usually proven by using
6490:
6685:
4846:{\displaystyle C^{\gamma }-C^{\gamma -1}\pm \cdots +(-1)^{\gamma }C^{0}\geq b_{\gamma }(M)-b_{\gamma -1}(M)\pm \cdots +(-1)^{\gamma }b_{0}(M).}
4362:
6555:
6211:
6173:
297:
6781:
5865:
6834:
6362:
4858:
2836:
decreases. The degeneracy and index of a critical point are independent of the choice of the local coordinate system used, as shown by
7118:
4917:
2365:
6331:
6270:
6021:
5835:
6883:
5928:
6475:
5590:
The notion of a Morse function can be generalized to consider functions that have nondegenerate manifolds of critical points. A
6866:
5609:
2272:
5994:
5965:
248:
4205:
Morse theory can be used to prove some strong results on the homology of manifolds. The number of critical points of index
2909:
65:, a typical differentiable function on a manifold will reflect the topology quite directly. Morse theory allows one to find
7078:
6199:
2837:
7063:
6786:
6560:
2724:
2228:
517:
411:
97:
5868: â integer-valued homotopy invariant of spaces; the size of the minimal open cover consisting of contractible sets
119:
7108:
520:âbasins, passes, and peaks (i.e. minima, saddles, and maxima)âone associates a number called the index, the number of
755:
again taking a point to its height above the plane. One can again analyze how the topology of the underwater surface
7113:
7083:
6791:
6747:
6728:
6495:
6439:
5574:
agree and gives an immediate proof of the Morse inequalities. An infinite dimensional analog of Morse homology in
4194:
3302:
890:
5161:
4968:
This gives a powerful tool to study manifold topology. Suppose on a closed manifold there exists a Morse function
390:
or below. Consider how the topology of this surface changes as the water rises. It appears unchanged except when
7237:
6650:
6515:
6165:
256:
7035:
6900:
6592:
6434:
5806:
2813:
585:
112:
104:
3407:
6732:
6702:
6626:
6616:
6572:
6402:
6355:
6157:
5901:
5856:
2976:
2204:
2178:
58:
6500:
5874:
5175:
3306:
2032:
7073:
6692:
6587:
6407:
6120:
5821:
can also be formulated for MorseâBott functions; the differential in MorseâBott homology is computed by a
5559:
183:
74:
4971:
4228:
2146:
7227:
6722:
6717:
5883:
5846:
5515:
5510:
4093:
4056:
1461:
1338:
471:
101:
42:
3345:
2009:
vanishes and the critical point is degenerate. This situation is unstable, since by slightly deforming
924:
839:
188:
2676:
2588:
7053:
6991:
6839:
6543:
6533:
6505:
6480:
6390:
5840:
4328:
1216:
70:
4583:
2109:
2083:
1765:
805:
7191:
7164:
6873:
6751:
6736:
6665:
6424:
6203:
5575:
5077:
4087:
3742:
2469:
1762:
One must take care to make the critical points non-degenerate. To see what can pose a problem, let
1539:
1289:
521:
85:
5599:
2447:
220:
7133:
7088:
6985:
6856:
6660:
6485:
6348:
6304:
6134:
6067:
4427:
1793:
1415:
is a cylinder, and is homotopy equivalent to a disk with a 1-cell attached (image at left). Once
1285:
6670:
5815:
are examples of MorseâBott functions, where the critical sets are (disjoint unions of) circles.
3021:
501:
does not change except when the water either (1) starts filling a basin, (2) covers a saddle (a
3207:{\displaystyle f(x)=f(p)-x_{1}^{2}-\cdots -x_{\gamma }^{2}+x_{\gamma +1}^{2}+\cdots +x_{n}^{2}}
2871:
2816:. This corresponds to the intuitive notion that the index is the number of directions in which
263:, or passes, where the surrounding landscape curves up in one direction and down in the other.
7068:
7048:
7043:
6950:
6861:
6675:
6655:
6510:
6449:
6327:
6266:
6217:
6207:
6179:
6169:
6017:
5990:
5961:
5822:
5567:
4454:
4333:
4324:
3915:
1948:
50:
5886: â mathematical theorem about a sufficient condition for the existence of a saddle point
5547:
In particular two closed 2-manifolds are homeomorphic if and only if they are diffeomorphic.
5375:
4619:
4560:
4262:
4208:
4126:
3938:
3537:
3240:
2542:
1711:
1664:
1108:
7206:
7000:
6955:
6878:
6849:
6707:
6640:
6635:
6630:
6620:
6412:
6395:
6239:
6097:
6059:
5936:
5771:
5467:
5103:
3976:
3748:
3688:
3565:
3398:
2904:
1033:
978:
6282:
5744:
5716:
5689:
5662:
3781:
3718:
3657:
3373:
1881:
1684:
1617:
1587:
1514:
1391:
1260:
1146:
897:
812:
758:
477:
7232:
7149:
7058:
6888:
6844:
6610:
5603:
Hessian at critical points is non-degenerate in every direction, that is, has no kernel).
5563:
5327:
5169:
2513:
2141:
1756:
659:
459:
31:
6112:
5989:. Monographs and Textbooks in Pure and Applied Mathematics. Vol. 72. Marcel Dekker.
5892:
5686:
is the dimension of the unstable manifold at a given point of the critical manifold, and
5133:
5049:
2630:
27:
Analyzes the topology of a manifold by studying differentiable functions on that manifold
4639:
4537:
4302:
4173:
4033:
3634:
3515:
3217:
2183:
1928:
1858:
1731:
1565:
1438:
1315:
1193:
688:
7015:
6940:
6910:
6808:
6801:
6741:
6712:
6582:
6577:
6538:
5983:
5818:
5812:
5579:
5555:
5493:
5447:
5427:
5403:
5355:
5307:
5287:
5267:
5247:
5083:
5029:
5009:
4517:
4497:
4477:
4282:
4153:
3895:
3875:
3855:
3835:
3808:
3614:
3495:
3468:
3320:
3298:
3280:
3260:
3063:
3001:
2981:
2851:
2819:
2795:
2775:
2751:
2731:
2656:
2568:
2493:
2332:
2302:
2296:
2278:
2254:
2234:
2210:
2012:
1992:
1986:
1908:
1838:
1644:
1545:
1418:
1295:
1173:
1088:
1068:
1013:
785:
738:
714:
639:
633:
615:
595:
567:
547:
544:
decreases from the point. More precisely, the index of a non-degenerate critical point
527:
441:
421:
393:
373:
277:
165:
141:
6045:
5960:. Pitman Research Notes in Mathematics Series. Vol. 395 (2nd ed.). Longman.
3297:. As a corollary of the Morse lemma, one sees that non-degenerate critical points are
1759:, this perturbation might simply be tilting slightly, rotating the coordinate system.
7221:
7201:
7025:
7020:
7005:
6995:
6945:
6922:
6796:
6756:
6697:
6645:
6444:
5421:
5165:
3682:
3608:
2769:
2716:
589:
502:
467:
260:
214:
81:
62:
6320:
6102:
6083:
6071:
7128:
6965:
6932:
6905:
6813:
6454:
5571:
5244:
Morse theory has been used to classify closed 2-manifolds up to diffeomorphism. If
5073:
4578:
252:
240:
4147:
These results generalize and formalize the 'rule' stated in the previous section.
6011:
6971:
6960:
6917:
6818:
6419:
6258:
3485:
varies. Half of the answer to this question is given by the following theorem.
38:
17:
6299:. American Mathematical Society Colloquium Publication. Vol. 18. New York.
3404:
As indicated before, we are interested in the question of when the topology of
7196:
7154:
6980:
6893:
6525:
6429:
6191:
6079:
6041:
5851:
5802:
3397:
topology. This is sometimes expressed as "a typical function is Morse" or "a
2715:). A less trivial example of a degenerate critical point is the origin of the
463:
266:
108:
89:
66:
6183:
5941:
5923:
1292:
to a point (a 0-cell) which has been "attached" to the empty set. Next, when
274:
Imagine flooding this landscape with water. When the water reaches elevation
7010:
6975:
6680:
6567:
6221:
5859: â The space of lagrangian subspaces of a fixed symplectic vector space
5168:
developed an analytic approach to the Morse inequalities by considering the
1752:
244:
5843: â Digital adaptation of continuum Morse theory for scalar volume data
508:
130:
7174:
7169:
7159:
6550:
6371:
1611:
is a torus, i.e. a torus with a disk (a 2-cell) removed and re-attached.
971:, is homotopy equivalent to a 1-cell attached to a cylinder (lower left).
415:
159:
93:
54:
6340:
6063:
3825:
passes a critical point. The following theorem answers that question.
1085:
corresponding to the basin, two saddles, and peak, respectively. When
363:{\displaystyle M^{a}\,{\stackrel {\text{def}}{=}}\,f^{-1}(-\infty ,a]}
6766:
6277:
A classic advanced reference in mathematics and mathematical physics.
6139:
886:, is homotopy equivalent to a 1-cell attached to a disk (lower left).
6168:(Reprint of 1993 ed.). Mineola, New York: Dover Publications.
656:
is negative definite. The indices of basins, passes, and peaks are
4616:, is less than or equal to the number of critical points of index
1538:
is a torus with a disk removed, which is homotopy equivalent to a
889:
804:
732:
507:
265:
129:
88:
had developed some of the ideas of Morse theory in the context of
30:"Morse function" redirects here. For anharmonic oscillators, see
5006:
critical points. In what way does the existence of the function
6344:
73:
on manifolds and to obtain substantial information about their
5566:. It is defined using a generic choice of Morse function and
5156:
is possible only in a small number of low dimensions, and
4417:{\displaystyle \sum (-1)^{\gamma }C^{\gamma }\,=\chi (M)}
2106:) or breaks up into two non-degenerate critical points (
1661:
passes the height of a critical point; at this point, a
1542:
with a 1-cell attached (image at right). Finally, when
138:
To illustrate, consider a mountainous landscape surface
5897:
Pages displaying short descriptions of redirect targets
2968:{\displaystyle \left(x_{1},x_{2},\ldots ,x_{n}\right)}
118:
The analogue of Morse theory for complex manifolds is
107:
on the space of paths). These techniques were used in
5774:
5747:
5719:
5692:
5665:
5612:
5518:
5496:
5470:
5450:
5430:
5406:
5378:
5358:
5330:
5310:
5290:
5270:
5250:
5178:
5136:
5106:
5086:
5052:
5032:
5012:
4974:
4920:
4861:
4675:
4642:
4622:
4586:
4563:
4540:
4520:
4500:
4480:
4457:
4430:
4365:
4336:
4305:
4285:
4265:
4231:
4211:
4176:
4156:
4129:
4096:
4059:
4036:
3979:
3941:
3918:
3898:
3878:
3858:
3838:
3811:
3784:
3751:
3721:
3691:
3660:
3637:
3617:
3568:
3540:
3518:
3498:
3471:
3410:
3376:
3348:
3323:
3301:. (Regarding an extension to the complex domain see
3283:
3263:
3243:
3220:
3086:
3066:
3024:
3004:
2984:
2912:
2874:
2854:
2822:
2798:
2778:
2754:
2734:
2679:
2659:
2633:
2591:
2571:
2545:
2516:
2496:
2472:
2450:
2368:
2335:
2305:
2281:
2257:
2237:
2213:
2186:
2149:
2112:
2086:
2035:
2015:
1995:
1951:
1931:
1911:
1884:
1861:
1841:
1796:
1768:
1734:
1714:
1687:
1667:
1647:
1620:
1614:
This illustrates the following rule: the topology of
1590:
1568:
1548:
1517:
1464:
1441:
1421:
1394:
1341:
1318:
1298:
1263:
1219:
1196:
1176:
1149:
1111:
1091:
1071:
1036:
1016:
981:
927:
900:
894:
A torus with a disk removed (upper right), formed by
842:
815:
788:
761:
741:
717:
691:
662:
642:
618:
598:
570:
550:
530:
480:
444:
424:
396:
376:
300:
280:
223:
191:
168:
144:
61:
on that manifold. According to the basic insights of
5888:
Pages displaying wikidata descriptions as a fallback
5879:
Pages displaying wikidata descriptions as a fallback
5870:
Pages displaying wikidata descriptions as a fallback
5861:
Pages displaying wikidata descriptions as a fallback
5805:
used MorseâBott theory in his original proof of the
7142:
7101:
7034:
6931:
6827:
6774:
6765:
6601:
6524:
6463:
6383:
6133:Guest, Martin (2001). "Morse Theory in the 1990s".
5240:
Application to classification of closed 2-manifolds
4030:is compact and contains no critical points besides
3778:It is also of interest to know how the topology of
2080:, the degenerate critical point is either removed (
6319:
5982:
5958:Elliptic Operators, Topology and Asymptotic Method
5790:
5760:
5732:
5705:
5678:
5651:
5539:
5502:
5482:
5456:
5436:
5412:
5393:
5364:
5345:
5316:
5296:
5276:
5256:
5229:
5148:
5122:
5092:
5064:
5038:
5018:
4994:
4958:
4907:{\displaystyle \gamma \in \{0,\ldots ,n=\dim M\},}
4906:
4845:
4651:
4628:
4608:
4569:
4549:
4526:
4506:
4486:
4466:
4443:
4416:
4351:
4314:
4291:
4271:
4251:
4217:
4185:
4162:
4135:
4115:
4078:
4045:
4022:
3965:
3927:
3904:
3884:
3864:
3844:
3817:
3797:
3767:
3734:
3707:
3673:
3646:
3623:
3599:
3555:
3527:
3504:
3477:
3457:
3389:
3362:
3329:
3289:
3269:
3249:
3229:
3206:
3072:
3052:
3010:
2990:
2967:
2895:
2860:
2828:
2804:
2784:
2760:
2740:
2707:
2665:
2645:
2619:
2577:
2557:
2531:
2502:
2483:
2458:
2436:
2341:
2311:
2287:
2263:
2243:
2219:
2195:
2169:
2124:
2098:
2072:
2021:
2001:
1977:
1937:
1917:
1897:
1870:
1847:
1827:
1782:
1743:
1720:
1700:
1673:
1653:
1633:
1603:
1577:
1554:
1530:
1503:
1450:
1427:
1407:
1380:
1327:
1304:
1276:
1249:
1205:
1182:
1162:
1135:
1097:
1077:
1057:
1022:
1002:
963:
913:
878:
828:
794:
774:
747:
723:
700:
677:
648:
624:
604:
576:
556:
536:
493:
450:
430:
402:
382:
362:
286:
231:
205:
174:
150:
5606:The index is most naturally thought of as a pair
2768:is the dimension of the largest subspace of the
4959:{\displaystyle C^{\gamma }\geq b_{\gamma }(M).}
2437:{\displaystyle f(x)=a+bx+cx^{2}+dx^{3}+\cdots }
5464:is unorientable, it is classified by a number
4659:These facts can be strengthened to obtain the
2295:the matrix of second partial derivatives (the
6356:
6089:Bulletin of the American Mathematical Society
5558:is a particularly easy way to understand the
5490:and is diffeomorphic to the connected sum of
1945:The problem is that the second derivative is
213:giving the elevation of each point, then the
8:
4898:
4868:
3317:A smooth real-valued function on a manifold
4474:Also by cellular homology, the rank of the
3611:, and there are no critical values between
975:Starting from the bottom of the torus, let
6771:
6363:
6349:
6341:
6309:: CS1 maint: location missing publisher (
4451:is the number of critical points of index
6138:
6101:
5940:
5779:
5773:
5752:
5746:
5724:
5718:
5697:
5691:
5670:
5664:
5652:{\displaystyle \left(i_{-},i_{+}\right),}
5635:
5622:
5611:
5598:is a smooth function on a manifold whose
5528:
5520:
5517:
5495:
5469:
5449:
5429:
5405:
5377:
5372:is diffeomorphic to the 2-sphere; and if
5357:
5329:
5309:
5289:
5269:
5249:
5215:
5196:
5183:
5177:
5135:
5111:
5105:
5085:
5051:
5031:
5011:
4988:
4987:
4973:
4938:
4925:
4919:
4860:
4825:
4815:
4772:
4750:
4737:
4727:
4693:
4680:
4674:
4641:
4621:
4591:
4585:
4562:
4539:
4519:
4499:
4479:
4456:
4435:
4429:
4398:
4392:
4382:
4364:
4335:
4304:
4284:
4264:
4245:
4244:
4230:
4210:
4175:
4155:
4128:
4101:
4095:
4064:
4058:
4035:
3984:
3978:
3940:
3917:
3897:
3877:
3857:
3837:
3810:
3789:
3783:
3756:
3750:
3726:
3720:
3696:
3690:
3665:
3659:
3636:
3616:
3573:
3567:
3539:
3517:
3497:
3470:
3428:
3415:
3409:
3381:
3375:
3356:
3355:
3347:
3322:
3282:
3262:
3242:
3219:
3198:
3193:
3174:
3163:
3150:
3145:
3126:
3121:
3085:
3065:
3029:
3023:
3003:
2983:
2954:
2935:
2922:
2911:
2873:
2853:
2821:
2797:
2777:
2753:
2733:
2693:
2678:
2658:
2632:
2605:
2590:
2570:
2544:
2515:
2495:
2474:
2473:
2471:
2452:
2451:
2449:
2422:
2406:
2367:
2334:
2304:
2280:
2256:
2236:
2212:
2185:
2163:
2162:
2148:
2111:
2085:
2055:
2034:
2014:
1994:
1950:
1930:
1910:
1889:
1883:
1860:
1840:
1816:
1795:
1776:
1775:
1767:
1751:In the case of a landscape or a manifold
1733:
1713:
1692:
1686:
1666:
1646:
1625:
1619:
1595:
1589:
1567:
1547:
1522:
1516:
1463:
1440:
1420:
1399:
1393:
1340:
1317:
1297:
1268:
1262:
1218:
1195:
1175:
1154:
1148:
1110:
1090:
1070:
1035:
1015:
980:
926:
905:
899:
841:
820:
814:
787:
766:
760:
740:
716:
690:
661:
641:
617:
597:
569:
549:
529:
485:
479:
443:
423:
395:
375:
333:
328:
320:
315:
313:
312:
311:
305:
299:
279:
225:
224:
222:
199:
198:
190:
167:
143:
92:. Morse originally applied his theory to
5895: â Theorem in mathematical analysis
3458:{\displaystyle M^{a}=f^{-1}(-\infty ,a]}
711:Considering a more general surface, let
6297:The Calculus of Variations in the Large
6084:"Lectures on Morse theory, old and new"
5914:
4514:is less than or equal to the number of
4170:-cell for each critical point of index
251:of a contour line is either a point, a
6302:
5985:Differential Topology: an Introduction
5304:and is diffeomorphic to a sphere with
3892:is a non-degenerate critical point of
2868:be a non-degenerate critical point of
2510:has a critical point at the origin if
1562:is greater than the critical level of
6196:Fundamentals of Differential Geometry
5230:{\displaystyle d_{t}=e^{-tf}de^{tf}.}
2073:{\displaystyle f(x)=x^{3}+\epsilon x}
1030:be the four critical points of index
7:
6051:Publications MathĂ©matiques de l'IHĂS
3852:is a smooth real-valued function on
3512:is a smooth real-valued function on
809:A cylinder (upper right), formed by
6284:Lectures on the h-cobordism theorem
4995:{\displaystyle f:M\to \mathbb {R} }
4252:{\displaystyle f:M\to \mathbb {R} }
2728:of a non-degenerate critical point
2170:{\displaystyle f:M\to \mathbb {R} }
474:). In other words, the topology of
270:Contour lines around a saddle point
5540:{\displaystyle \mathbf {RP} ^{2}.}
4197:to rearrange the critical points.
4116:{\displaystyle M^{q-\varepsilon }}
4079:{\displaystyle M^{q+\varepsilon }}
3443:
2329:; if the Hessian is singular then
1504:{\displaystyle f(r)<a<f(s),}
1381:{\displaystyle f(q)<a<f(r),}
348:
25:
6013:Elements of Differential Topology
3363:{\displaystyle M\to \mathbb {R} }
964:{\displaystyle f(r)<a<f(s)}
879:{\displaystyle f(q)<a<f(r)}
735:oriented as in the picture, with
206:{\displaystyle M\to \mathbb {R} }
5924:"Supersymmetry and Morse theory"
5524:
5521:
2708:{\displaystyle a+dx^{3}+\cdots }
2620:{\displaystyle a+cx^{2}+\cdots }
6231:An Introduction to Morse Theory
6103:10.1090/s0273-0979-1982-15038-8
5866:LusternikâSchnirelmann category
4494:homology group of a CW complex
1250:{\displaystyle 0<a<f(q),}
588:of the largest subspace of the
6403:Differentiable/Smooth manifold
6265:. Princeton University Press.
4984:
4950:
4944:
4837:
4831:
4812:
4802:
4790:
4784:
4762:
4756:
4724:
4714:
4609:{\displaystyle b_{\gamma }(M)}
4603:
4597:
4411:
4405:
4379:
4369:
4346:
4340:
4241:
4017:
3993:
3951:
3945:
3594:
3582:
3452:
3437:
3352:
3111:
3105:
3096:
3090:
3041:
3035:
2884:
2378:
2372:
2159:
2125:{\displaystyle \epsilon <0}
2099:{\displaystyle \epsilon >0}
2045:
2039:
1966:
1960:
1806:
1800:
1783:{\displaystyle M=\mathbb {R} }
1495:
1489:
1474:
1468:
1372:
1366:
1351:
1345:
1241:
1235:
1121:
1115:
958:
952:
937:
931:
873:
867:
852:
846:
357:
342:
195:
1:
6238:Maxwell, James Clerk (1870).
6200:Graduate Texts in Mathematics
4577:homology group, that is, the
4279:cells in the CW structure on
2484:{\displaystyle \mathbb {R} ,}
2324:non-degenerate critical point
6113:"On Contour and Slope Lines"
5836:AlmgrenâPitts min-max theory
5172:for the perturbed operator
5100:is homeomorphic to a sphere
3305:. For a generalization, see
2459:{\displaystyle \mathbb {R} }
1641:does not change except when
505:), or (3) submerges a peak.
370:, the points with elevation
294:, the underwater surface is
232:{\displaystyle \mathbb {R} }
7109:Classification of manifolds
6202:. Vol. 191. New York:
5284:is classified by its genus
4557:Therefore, the rank of the
4444:{\displaystyle C^{\gamma }}
4195:gradient-like vector fields
2539:which is non-degenerate if
1828:{\displaystyle f(x)=x^{3}.}
782:changes as the water level
255:, or a closed curve with a
49:enables one to analyze the
7254:
6318:Schwarz, Matthias (1993).
6247:The Philosophical Magazine
6121:The Philosophical Magazine
6046:"Morse Theory Indomitable"
6010:Shastri, Anant R. (2011).
4259:is equal to the number of
3053:{\displaystyle x_{i}(p)=0}
2275:. If at a critical point
29:
7185:over commutative algebras
6229:Matsumoto, Yukio (2002).
6166:Dover Book on Mathematics
6152:(2nd ed.). Springer.
4299:obtained from "climbing"
3257:is equal to the index of
2896:{\displaystyle f:M\to R.}
2354:degenerate critical point
1170:is the empty set. After
6901:Riemann curvature tensor
5981:Gauld, David B. (1982).
5807:Bott periodicity theorem
5420:is diffeomorphic to the
4467:{\displaystyle \gamma .}
4352:{\displaystyle \chi (M)}
3928:{\displaystyle \gamma ,}
2812:on which the Hessian is
2299:) is non-singular, then
1978:{\displaystyle f''(0)=0}
516:To these three types of
59:differentiable functions
6295:Morse, Marston (1934).
6111:Cayley, Arthur (1859).
5922:Witten, Edward (1982).
5902:Stratified Morse theory
5857:Lagrangian Grassmannian
5394:{\displaystyle g>0,}
4855:In particular, for any
4636:of a Morse function on
4629:{\displaystyle \gamma }
4570:{\displaystyle \gamma }
4327:) it is clear that the
4272:{\displaystyle \gamma }
4218:{\displaystyle \gamma }
4136:{\displaystyle \gamma }
3966:{\displaystyle f(p)=q.}
3556:{\displaystyle a<b,}
3250:{\displaystyle \gamma }
2558:{\displaystyle c\neq 0}
2251:and their images under
2179:differentiable manifold
1855:is a critical point of
1721:{\displaystyle \gamma }
1674:{\displaystyle \gamma }
1136:{\displaystyle f(p)=0,}
410:passes the height of a
120:PicardâLefschetz theory
6693:Manifold with boundary
6408:Differential structure
6162:Differential Manifolds
5942:10.4310/jdg/1214437492
5792:
5791:{\displaystyle i_{+}.}
5762:
5734:
5707:
5680:
5653:
5541:
5511:real projective spaces
5504:
5484:
5483:{\displaystyle g>0}
5458:
5438:
5414:
5395:
5366:
5347:
5318:
5298:
5278:
5258:
5231:
5160:is homeomorphic to an
5150:
5124:
5123:{\displaystyle S^{n}.}
5094:
5066:
5040:
5020:
4996:
4960:
4908:
4847:
4653:
4630:
4610:
4571:
4551:
4528:
4508:
4488:
4468:
4445:
4418:
4353:
4316:
4293:
4273:
4253:
4219:
4187:
4164:
4137:
4117:
4080:
4047:
4024:
4023:{\displaystyle f^{-1}}
3967:
3929:
3906:
3886:
3866:
3846:
3819:
3799:
3769:
3768:{\displaystyle M^{a}.}
3736:
3709:
3708:{\displaystyle M^{b},}
3675:
3648:
3625:
3601:
3600:{\displaystyle f^{-1}}
3557:
3529:
3506:
3479:
3459:
3391:
3364:
3331:
3291:
3271:
3251:
3231:
3208:
3074:
3054:
3012:
2992:
2969:
2897:
2862:
2830:
2806:
2786:
2762:
2742:
2709:
2667:
2647:
2621:
2579:
2559:
2533:
2504:
2485:
2460:
2438:
2343:
2313:
2289:
2265:
2245:
2221:
2197:
2171:
2126:
2100:
2074:
2023:
2003:
1979:
1939:
1919:
1899:
1872:
1849:
1829:
1784:
1745:
1722:
1702:
1675:
1655:
1635:
1605:
1579:
1556:
1532:
1505:
1452:
1429:
1409:
1382:
1329:
1306:
1278:
1251:
1207:
1184:
1164:
1137:
1099:
1079:
1059:
1058:{\displaystyle 0,1,1,}
1024:
1004:
1003:{\displaystyle p,q,r,}
972:
965:
915:
887:
880:
830:
796:
776:
749:
725:
702:
679:
650:
626:
606:
578:
558:
538:
513:
495:
470:does not have maximal
452:
432:
404:
384:
364:
288:
271:
233:
207:
176:
152:
135:
6281:Milnor, John (1965).
6150:Differential Topology
5929:J. Differential Geom.
5884:Mountain pass theorem
5847:Discrete Morse theory
5793:
5763:
5761:{\displaystyle i_{-}}
5735:
5733:{\displaystyle i_{-}}
5708:
5706:{\displaystyle i_{+}}
5681:
5679:{\displaystyle i_{-}}
5654:
5542:
5505:
5485:
5459:
5439:
5415:
5396:
5367:
5348:
5319:
5299:
5279:
5259:
5232:
5162:EellsâKuiper manifold
5151:
5125:
5095:
5067:
5041:
5021:
4997:
4961:
4909:
4848:
4654:
4631:
4611:
4572:
4552:
4529:
4509:
4489:
4469:
4446:
4419:
4354:
4317:
4294:
4274:
4254:
4220:
4188:
4165:
4138:
4118:
4081:
4048:
4025:
3968:
3930:
3907:
3887:
3867:
3847:
3820:
3800:
3798:{\displaystyle M^{a}}
3770:
3737:
3735:{\displaystyle M^{b}}
3710:
3676:
3674:{\displaystyle M^{a}}
3649:
3626:
3602:
3558:
3530:
3507:
3480:
3460:
3392:
3390:{\displaystyle C^{2}}
3365:
3332:
3292:
3272:
3252:
3232:
3209:
3075:
3055:
3013:
2993:
2970:
2898:
2863:
2831:
2807:
2787:
2763:
2743:
2710:
2668:
2648:
2622:
2580:
2560:
2534:
2505:
2486:
2461:
2439:
2344:
2314:
2290:
2266:
2246:
2222:
2203:the points where the
2198:
2172:
2127:
2101:
2075:
2024:
2004:
1980:
1940:
1920:
1905:does not change when
1900:
1898:{\displaystyle M^{a}}
1873:
1850:
1830:
1785:
1746:
1723:
1703:
1701:{\displaystyle M^{a}}
1681:-cell is attached to
1676:
1656:
1636:
1634:{\displaystyle M^{a}}
1606:
1604:{\displaystyle M^{a}}
1580:
1557:
1533:
1531:{\displaystyle M^{a}}
1506:
1453:
1430:
1410:
1408:{\displaystyle M^{a}}
1383:
1330:
1312:exceeds the level of
1307:
1279:
1277:{\displaystyle M^{a}}
1252:
1208:
1185:
1165:
1163:{\displaystyle M^{a}}
1138:
1100:
1080:
1060:
1025:
1005:
966:
916:
914:{\displaystyle M^{a}}
893:
881:
831:
829:{\displaystyle M^{a}}
808:
797:
777:
775:{\displaystyle M^{a}}
750:
726:
703:
680:
651:
627:
607:
579:
559:
539:
511:
496:
494:{\displaystyle M^{a}}
458:(more generally, the
453:
433:
405:
385:
365:
289:
269:
234:
208:
177:
153:
133:
71:handle decompositions
43:differential topology
6840:Covariant derivative
6391:Topological manifold
6240:"On Hills and Dales"
5841:Digital Morse theory
5772:
5745:
5717:
5690:
5663:
5610:
5516:
5494:
5468:
5448:
5428:
5404:
5376:
5356:
5346:{\displaystyle g=0,}
5328:
5308:
5288:
5268:
5248:
5176:
5134:
5104:
5084:
5050:
5030:
5010:
4972:
4918:
4859:
4673:
4640:
4620:
4584:
4561:
4538:
4518:
4498:
4478:
4455:
4428:
4363:
4359:is equal to the sum
4334:
4329:Euler characteristic
4303:
4283:
4263:
4229:
4209:
4174:
4154:
4127:
4094:
4057:
4034:
3977:
3939:
3916:
3896:
3876:
3856:
3836:
3809:
3782:
3749:
3743:deformation retracts
3719:
3689:
3658:
3635:
3615:
3566:
3538:
3516:
3496:
3469:
3408:
3401:function is Morse".
3374:
3346:
3321:
3313:Fundamental theorems
3281:
3261:
3241:
3218:
3084:
3064:
3022:
3002:
2982:
2910:
2903:Then there exists a
2872:
2852:
2820:
2796:
2776:
2752:
2732:
2677:
2657:
2631:
2627:) and degenerate if
2589:
2569:
2543:
2532:{\displaystyle b=0,}
2514:
2494:
2470:
2448:
2366:
2333:
2303:
2279:
2255:
2235:
2227:vanishes are called
2211:
2184:
2147:
2110:
2084:
2033:
2013:
1993:
1949:
1929:
1909:
1882:
1878:but the topology of
1859:
1839:
1794:
1766:
1732:
1712:
1685:
1665:
1645:
1618:
1588:
1566:
1546:
1515:
1462:
1439:
1435:passes the level of
1419:
1392:
1339:
1316:
1296:
1261:
1217:
1194:
1190:passes the level of
1174:
1147:
1109:
1089:
1069:
1034:
1014:
979:
925:
898:
840:
813:
786:
759:
739:
715:
689:
678:{\displaystyle 0,1,}
660:
640:
616:
596:
568:
548:
528:
524:directions in which
478:
442:
422:
394:
374:
298:
278:
221:
189:
166:
142:
6874:Exterior derivative
6476:AtiyahâSinger index
6425:Riemannian manifold
6160:(19 October 2007).
6158:Kosinski, Antoni A.
6148:Hirsch, M. (1994).
5594:MorseâBott function
5576:symplectic geometry
5149:{\displaystyle k=3}
5078:Reeb sphere theorem
5065:{\displaystyle k=2}
4088:homotopy equivalent
3303:Complex Morse Lemma
3203:
3179:
3155:
3131:
2646:{\displaystyle c=0}
1290:homotopy equivalent
253:simple closed curve
249:connected component
243:(more generally, a
158:(more generally, a
113:periodicity theorem
86:James Clerk Maxwell
7180:Secondary calculus
7134:Singularity theory
7089:Parallel transport
6857:De Rham cohomology
6496:Generalized Stokes
6064:10.1007/bf02698544
5956:Roe, John (1998).
5875:MorseâSmale system
5788:
5758:
5730:
5703:
5676:
5649:
5537:
5500:
5480:
5454:
5434:
5410:
5391:
5362:
5343:
5314:
5294:
5274:
5264:is oriented, then
5254:
5227:
5146:
5120:
5090:
5062:
5036:
5016:
4992:
4956:
4904:
4843:
4664:Morse inequalities
4652:{\displaystyle M.}
4649:
4626:
4606:
4567:
4550:{\displaystyle M.}
4547:
4524:
4504:
4484:
4464:
4441:
4414:
4349:
4315:{\displaystyle f.}
4312:
4289:
4269:
4249:
4215:
4201:Morse inequalities
4186:{\displaystyle n.}
4183:
4160:
4133:
4113:
4076:
4046:{\displaystyle p.}
4043:
4020:
3963:
3925:
3902:
3882:
3862:
3842:
3815:
3795:
3765:
3732:
3705:
3671:
3647:{\displaystyle b.}
3644:
3621:
3597:
3553:
3528:{\displaystyle M,}
3525:
3502:
3475:
3455:
3387:
3360:
3327:
3307:MorseâPalais lemma
3287:
3267:
3247:
3230:{\displaystyle U.}
3227:
3204:
3189:
3159:
3141:
3117:
3070:
3050:
3008:
2988:
2965:
2893:
2858:
2826:
2802:
2782:
2758:
2738:
2705:
2663:
2643:
2617:
2575:
2555:
2529:
2500:
2481:
2456:
2434:
2362:For the functions
2339:
2309:
2285:
2261:
2241:
2217:
2196:{\displaystyle M,}
2193:
2167:
2140:For a real-valued
2136:Formal development
2122:
2096:
2070:
2019:
1999:
1975:
1938:{\displaystyle 0.}
1935:
1915:
1895:
1871:{\displaystyle f,}
1868:
1845:
1825:
1780:
1744:{\displaystyle f.}
1741:
1718:
1698:
1671:
1651:
1631:
1601:
1578:{\displaystyle s,}
1575:
1552:
1528:
1501:
1451:{\displaystyle r,}
1448:
1425:
1405:
1378:
1328:{\displaystyle q,}
1325:
1302:
1274:
1247:
1206:{\displaystyle p,}
1203:
1180:
1160:
1133:
1095:
1075:
1055:
1020:
1000:
973:
961:
911:
888:
876:
826:
792:
772:
745:
721:
701:{\displaystyle 2,}
698:
675:
646:
622:
602:
574:
554:
534:
514:
491:
448:
428:
400:
380:
360:
284:
272:
229:
203:
172:
148:
136:
41:, specifically in
7215:
7214:
7097:
7096:
6862:Differential form
6516:Whitney embedding
6450:Differential form
6213:978-0-387-98593-0
6175:978-0-486-46244-8
5823:spectral sequence
5586:MorseâBott theory
5568:Riemannian metric
5503:{\displaystyle g}
5457:{\displaystyle N}
5437:{\displaystyle g}
5413:{\displaystyle M}
5365:{\displaystyle M}
5324:handles: thus if
5317:{\displaystyle g}
5297:{\displaystyle g}
5277:{\displaystyle M}
5257:{\displaystyle M}
5093:{\displaystyle M}
5039:{\displaystyle M}
5019:{\displaystyle f}
4527:{\displaystyle n}
4507:{\displaystyle M}
4487:{\displaystyle n}
4325:cellular homology
4292:{\displaystyle M}
4163:{\displaystyle n}
3905:{\displaystyle f}
3885:{\displaystyle p}
3865:{\displaystyle M}
3845:{\displaystyle f}
3818:{\displaystyle a}
3624:{\displaystyle a}
3505:{\displaystyle f}
3478:{\displaystyle a}
3330:{\displaystyle M}
3290:{\displaystyle p}
3270:{\displaystyle f}
3073:{\displaystyle i}
3011:{\displaystyle p}
2991:{\displaystyle U}
2861:{\displaystyle p}
2829:{\displaystyle f}
2814:negative definite
2805:{\displaystyle p}
2785:{\displaystyle M}
2761:{\displaystyle f}
2741:{\displaystyle p}
2666:{\displaystyle f}
2578:{\displaystyle f}
2503:{\displaystyle f}
2342:{\displaystyle p}
2312:{\displaystyle p}
2288:{\displaystyle p}
2264:{\displaystyle f}
2244:{\displaystyle f}
2220:{\displaystyle f}
2022:{\displaystyle f}
2002:{\displaystyle f}
1918:{\displaystyle a}
1848:{\displaystyle 0}
1654:{\displaystyle a}
1555:{\displaystyle a}
1428:{\displaystyle a}
1305:{\displaystyle a}
1183:{\displaystyle a}
1098:{\displaystyle a}
1078:{\displaystyle 2}
1023:{\displaystyle s}
795:{\displaystyle a}
748:{\displaystyle f}
724:{\displaystyle M}
649:{\displaystyle f}
625:{\displaystyle p}
605:{\displaystyle M}
577:{\displaystyle f}
557:{\displaystyle p}
537:{\displaystyle f}
451:{\displaystyle 0}
431:{\displaystyle f}
403:{\displaystyle a}
383:{\displaystyle a}
325:
323:
287:{\displaystyle a}
175:{\displaystyle f}
151:{\displaystyle M}
16:(Redirected from
7245:
7238:Smooth functions
7207:Stratified space
7165:Fréchet manifold
6879:Interior product
6772:
6469:
6365:
6358:
6351:
6342:
6337:
6325:
6314:
6308:
6300:
6291:
6289:
6276:
6254:
6244:
6234:
6225:
6187:
6153:
6144:
6142:
6129:
6117:
6107:
6105:
6075:
6028:
6027:
6007:
6001:
6000:
5988:
5978:
5972:
5971:
5953:
5947:
5946:
5944:
5919:
5898:
5889:
5880:
5871:
5862:
5797:
5795:
5794:
5789:
5784:
5783:
5767:
5765:
5764:
5759:
5757:
5756:
5739:
5737:
5736:
5731:
5729:
5728:
5712:
5710:
5709:
5704:
5702:
5701:
5685:
5683:
5682:
5677:
5675:
5674:
5658:
5656:
5655:
5650:
5645:
5641:
5640:
5639:
5627:
5626:
5596:
5595:
5564:smooth manifolds
5546:
5544:
5543:
5538:
5533:
5532:
5527:
5509:
5507:
5506:
5501:
5489:
5487:
5486:
5481:
5463:
5461:
5460:
5455:
5443:
5441:
5440:
5435:
5419:
5417:
5416:
5411:
5400:
5398:
5397:
5392:
5371:
5369:
5368:
5363:
5352:
5350:
5349:
5344:
5323:
5321:
5320:
5315:
5303:
5301:
5300:
5295:
5283:
5281:
5280:
5275:
5263:
5261:
5260:
5255:
5236:
5234:
5233:
5228:
5223:
5222:
5207:
5206:
5188:
5187:
5155:
5153:
5152:
5147:
5129:
5127:
5126:
5121:
5116:
5115:
5099:
5097:
5096:
5091:
5071:
5069:
5068:
5063:
5045:
5043:
5042:
5037:
5025:
5023:
5022:
5017:
5001:
4999:
4998:
4993:
4991:
4965:
4963:
4962:
4957:
4943:
4942:
4930:
4929:
4913:
4911:
4910:
4905:
4852:
4850:
4849:
4844:
4830:
4829:
4820:
4819:
4783:
4782:
4755:
4754:
4742:
4741:
4732:
4731:
4704:
4703:
4685:
4684:
4666:
4665:
4658:
4656:
4655:
4650:
4635:
4633:
4632:
4627:
4615:
4613:
4612:
4607:
4596:
4595:
4576:
4574:
4573:
4568:
4556:
4554:
4553:
4548:
4533:
4531:
4530:
4525:
4513:
4511:
4510:
4505:
4493:
4491:
4490:
4485:
4473:
4471:
4470:
4465:
4450:
4448:
4447:
4442:
4440:
4439:
4423:
4421:
4420:
4415:
4397:
4396:
4387:
4386:
4358:
4356:
4355:
4350:
4321:
4319:
4318:
4313:
4298:
4296:
4295:
4290:
4278:
4276:
4275:
4270:
4258:
4256:
4255:
4250:
4248:
4224:
4222:
4221:
4216:
4192:
4190:
4189:
4184:
4169:
4167:
4166:
4161:
4142:
4140:
4139:
4134:
4122:
4120:
4119:
4114:
4112:
4111:
4085:
4083:
4082:
4077:
4075:
4074:
4052:
4050:
4049:
4044:
4029:
4027:
4026:
4021:
3992:
3991:
3972:
3970:
3969:
3964:
3934:
3932:
3931:
3926:
3911:
3909:
3908:
3903:
3891:
3889:
3888:
3883:
3871:
3869:
3868:
3863:
3851:
3849:
3848:
3843:
3824:
3822:
3821:
3816:
3804:
3802:
3801:
3796:
3794:
3793:
3774:
3772:
3771:
3766:
3761:
3760:
3741:
3739:
3738:
3733:
3731:
3730:
3714:
3712:
3711:
3706:
3701:
3700:
3680:
3678:
3677:
3672:
3670:
3669:
3653:
3651:
3650:
3645:
3630:
3628:
3627:
3622:
3606:
3604:
3603:
3598:
3581:
3580:
3562:
3560:
3559:
3554:
3534:
3532:
3531:
3526:
3511:
3509:
3508:
3503:
3484:
3482:
3481:
3476:
3464:
3462:
3461:
3456:
3436:
3435:
3420:
3419:
3396:
3394:
3393:
3388:
3386:
3385:
3369:
3367:
3366:
3361:
3359:
3336:
3334:
3333:
3328:
3296:
3294:
3293:
3288:
3276:
3274:
3273:
3268:
3256:
3254:
3253:
3248:
3236:
3234:
3233:
3228:
3213:
3211:
3210:
3205:
3202:
3197:
3178:
3173:
3154:
3149:
3130:
3125:
3079:
3077:
3076:
3071:
3059:
3057:
3056:
3051:
3034:
3033:
3017:
3015:
3014:
3009:
2997:
2995:
2994:
2989:
2974:
2972:
2971:
2966:
2964:
2960:
2959:
2958:
2940:
2939:
2927:
2926:
2902:
2900:
2899:
2894:
2867:
2865:
2864:
2859:
2835:
2833:
2832:
2827:
2811:
2809:
2808:
2803:
2791:
2789:
2788:
2783:
2767:
2765:
2764:
2759:
2747:
2745:
2744:
2739:
2714:
2712:
2711:
2706:
2698:
2697:
2672:
2670:
2669:
2664:
2652:
2650:
2649:
2644:
2626:
2624:
2623:
2618:
2610:
2609:
2584:
2582:
2581:
2576:
2564:
2562:
2561:
2556:
2538:
2536:
2535:
2530:
2509:
2507:
2506:
2501:
2490:
2488:
2487:
2482:
2477:
2465:
2463:
2462:
2457:
2455:
2443:
2441:
2440:
2435:
2427:
2426:
2411:
2410:
2356:
2355:
2348:
2346:
2345:
2340:
2326:
2325:
2318:
2316:
2315:
2310:
2294:
2292:
2291:
2286:
2270:
2268:
2267:
2262:
2250:
2248:
2247:
2242:
2226:
2224:
2223:
2218:
2202:
2200:
2199:
2194:
2176:
2174:
2173:
2168:
2166:
2131:
2129:
2128:
2123:
2105:
2103:
2102:
2097:
2079:
2077:
2076:
2071:
2060:
2059:
2028:
2026:
2025:
2020:
2008:
2006:
2005:
2000:
1984:
1982:
1981:
1976:
1959:
1944:
1942:
1941:
1936:
1924:
1922:
1921:
1916:
1904:
1902:
1901:
1896:
1894:
1893:
1877:
1875:
1874:
1869:
1854:
1852:
1851:
1846:
1834:
1832:
1831:
1826:
1821:
1820:
1789:
1787:
1786:
1781:
1779:
1750:
1748:
1747:
1742:
1727:
1725:
1724:
1719:
1707:
1705:
1704:
1699:
1697:
1696:
1680:
1678:
1677:
1672:
1660:
1658:
1657:
1652:
1640:
1638:
1637:
1632:
1630:
1629:
1610:
1608:
1607:
1602:
1600:
1599:
1584:
1582:
1581:
1576:
1561:
1559:
1558:
1553:
1537:
1535:
1534:
1529:
1527:
1526:
1510:
1508:
1507:
1502:
1457:
1455:
1454:
1449:
1434:
1432:
1431:
1426:
1414:
1412:
1411:
1406:
1404:
1403:
1387:
1385:
1384:
1379:
1334:
1332:
1331:
1326:
1311:
1309:
1308:
1303:
1283:
1281:
1280:
1275:
1273:
1272:
1256:
1254:
1253:
1248:
1212:
1210:
1209:
1204:
1189:
1187:
1186:
1181:
1169:
1167:
1166:
1161:
1159:
1158:
1142:
1140:
1139:
1134:
1104:
1102:
1101:
1096:
1084:
1082:
1081:
1076:
1064:
1062:
1061:
1056:
1029:
1027:
1026:
1021:
1009:
1007:
1006:
1001:
970:
968:
967:
962:
920:
918:
917:
912:
910:
909:
885:
883:
882:
877:
835:
833:
832:
827:
825:
824:
801:
799:
798:
793:
781:
779:
778:
773:
771:
770:
754:
752:
751:
746:
730:
728:
727:
722:
707:
705:
704:
699:
684:
682:
681:
676:
655:
653:
652:
647:
631:
629:
628:
623:
611:
609:
608:
603:
583:
581:
580:
575:
563:
561:
560:
555:
543:
541:
540:
535:
500:
498:
497:
492:
490:
489:
457:
455:
454:
449:
437:
435:
434:
429:
409:
407:
406:
401:
389:
387:
386:
381:
369:
367:
366:
361:
341:
340:
327:
326:
324:
321:
319:
314:
310:
309:
293:
291:
290:
285:
238:
236:
235:
230:
228:
212:
210:
209:
204:
202:
181:
179:
178:
173:
157:
155:
154:
149:
111:'s proof of his
21:
18:Morse inequality
7253:
7252:
7248:
7247:
7246:
7244:
7243:
7242:
7218:
7217:
7216:
7211:
7150:Banach manifold
7143:Generalizations
7138:
7093:
7030:
6927:
6889:Ricci curvature
6845:Cotangent space
6823:
6761:
6603:
6597:
6556:Exponential map
6520:
6465:
6459:
6379:
6369:
6334:
6317:
6301:
6294:
6287:
6280:
6273:
6257:
6253:(269): 421â427.
6242:
6237:
6228:
6214:
6204:Springer-Verlag
6190:
6176:
6156:
6147:
6132:
6128:(120): 264â268.
6115:
6110:
6078:
6040:
6037:
6035:Further reading
6032:
6031:
6024:
6009:
6008:
6004:
5997:
5980:
5979:
5975:
5968:
5955:
5954:
5950:
5921:
5920:
5916:
5911:
5906:
5896:
5887:
5878:
5869:
5860:
5831:
5813:Round functions
5775:
5770:
5769:
5748:
5743:
5742:
5720:
5715:
5714:
5693:
5688:
5687:
5666:
5661:
5660:
5631:
5618:
5617:
5613:
5608:
5607:
5593:
5592:
5588:
5553:
5519:
5514:
5513:
5492:
5491:
5466:
5465:
5446:
5445:
5426:
5425:
5402:
5401:
5374:
5373:
5354:
5353:
5326:
5325:
5306:
5305:
5286:
5285:
5266:
5265:
5246:
5245:
5242:
5211:
5192:
5179:
5174:
5173:
5170:de Rham complex
5132:
5131:
5107:
5102:
5101:
5082:
5081:
5072:was studied by
5048:
5047:
5028:
5027:
5008:
5007:
5002:with precisely
4970:
4969:
4934:
4921:
4916:
4915:
4857:
4856:
4821:
4811:
4768:
4746:
4733:
4723:
4689:
4676:
4671:
4670:
4663:
4662:
4638:
4637:
4618:
4617:
4587:
4582:
4581:
4559:
4558:
4536:
4535:
4516:
4515:
4496:
4495:
4476:
4475:
4453:
4452:
4431:
4426:
4425:
4388:
4378:
4361:
4360:
4332:
4331:
4301:
4300:
4281:
4280:
4261:
4260:
4227:
4226:
4207:
4206:
4203:
4172:
4171:
4152:
4151:
4143:-cell attached.
4125:
4124:
4097:
4092:
4091:
4060:
4055:
4054:
4032:
4031:
3980:
3975:
3974:
3937:
3936:
3914:
3913:
3894:
3893:
3874:
3873:
3854:
3853:
3834:
3833:
3807:
3806:
3785:
3780:
3779:
3752:
3747:
3746:
3722:
3717:
3716:
3692:
3687:
3686:
3661:
3656:
3655:
3633:
3632:
3613:
3612:
3569:
3564:
3563:
3536:
3535:
3514:
3513:
3494:
3493:
3467:
3466:
3424:
3411:
3406:
3405:
3377:
3372:
3371:
3344:
3343:
3319:
3318:
3315:
3279:
3278:
3259:
3258:
3239:
3238:
3216:
3215:
3082:
3081:
3062:
3061:
3025:
3020:
3019:
3000:
2999:
2980:
2979:
2950:
2931:
2918:
2917:
2913:
2908:
2907:
2870:
2869:
2850:
2849:
2846:
2838:Sylvester's Law
2818:
2817:
2794:
2793:
2774:
2773:
2750:
2749:
2730:
2729:
2689:
2675:
2674:
2673:is of the form
2655:
2654:
2629:
2628:
2601:
2587:
2586:
2585:is of the form
2567:
2566:
2541:
2540:
2512:
2511:
2492:
2491:
2468:
2467:
2446:
2445:
2418:
2402:
2364:
2363:
2353:
2352:
2331:
2330:
2323:
2322:
2301:
2300:
2277:
2276:
2273:critical values
2253:
2252:
2233:
2232:
2229:critical points
2209:
2208:
2182:
2181:
2145:
2144:
2142:smooth function
2138:
2108:
2107:
2082:
2081:
2051:
2031:
2030:
2011:
2010:
1991:
1990:
1952:
1947:
1946:
1927:
1926:
1907:
1906:
1885:
1880:
1879:
1857:
1856:
1837:
1836:
1812:
1792:
1791:
1764:
1763:
1757:Euclidean space
1730:
1729:
1710:
1709:
1688:
1683:
1682:
1663:
1662:
1643:
1642:
1621:
1616:
1615:
1591:
1586:
1585:
1564:
1563:
1544:
1543:
1518:
1513:
1512:
1460:
1459:
1437:
1436:
1417:
1416:
1395:
1390:
1389:
1337:
1336:
1314:
1313:
1294:
1293:
1264:
1259:
1258:
1215:
1214:
1192:
1191:
1172:
1171:
1150:
1145:
1144:
1107:
1106:
1087:
1086:
1067:
1066:
1032:
1031:
1012:
1011:
977:
976:
923:
922:
901:
896:
895:
838:
837:
816:
811:
810:
784:
783:
762:
757:
756:
737:
736:
713:
712:
687:
686:
658:
657:
638:
637:
614:
613:
594:
593:
566:
565:
546:
545:
526:
525:
518:critical points
481:
476:
475:
460:Jacobian matrix
440:
439:
420:
419:
392:
391:
372:
371:
329:
301:
296:
295:
276:
275:
219:
218:
187:
186:
164:
163:
140:
139:
128:
98:critical points
35:
32:Morse potential
28:
23:
22:
15:
12:
11:
5:
7251:
7249:
7241:
7240:
7235:
7230:
7220:
7219:
7213:
7212:
7210:
7209:
7204:
7199:
7194:
7189:
7188:
7187:
7177:
7172:
7167:
7162:
7157:
7152:
7146:
7144:
7140:
7139:
7137:
7136:
7131:
7126:
7121:
7116:
7111:
7105:
7103:
7099:
7098:
7095:
7094:
7092:
7091:
7086:
7081:
7076:
7071:
7066:
7061:
7056:
7051:
7046:
7040:
7038:
7032:
7031:
7029:
7028:
7023:
7018:
7013:
7008:
7003:
6998:
6988:
6983:
6978:
6968:
6963:
6958:
6953:
6948:
6943:
6937:
6935:
6929:
6928:
6926:
6925:
6920:
6915:
6914:
6913:
6903:
6898:
6897:
6896:
6886:
6881:
6876:
6871:
6870:
6869:
6859:
6854:
6853:
6852:
6842:
6837:
6831:
6829:
6825:
6824:
6822:
6821:
6816:
6811:
6806:
6805:
6804:
6794:
6789:
6784:
6778:
6776:
6769:
6763:
6762:
6760:
6759:
6754:
6744:
6739:
6725:
6720:
6715:
6710:
6705:
6703:Parallelizable
6700:
6695:
6690:
6689:
6688:
6678:
6673:
6668:
6663:
6658:
6653:
6648:
6643:
6638:
6633:
6623:
6613:
6607:
6605:
6599:
6598:
6596:
6595:
6590:
6585:
6583:Lie derivative
6580:
6578:Integral curve
6575:
6570:
6565:
6564:
6563:
6553:
6548:
6547:
6546:
6539:Diffeomorphism
6536:
6530:
6528:
6522:
6521:
6519:
6518:
6513:
6508:
6503:
6498:
6493:
6488:
6483:
6478:
6472:
6470:
6461:
6460:
6458:
6457:
6452:
6447:
6442:
6437:
6432:
6427:
6422:
6417:
6416:
6415:
6410:
6400:
6399:
6398:
6387:
6385:
6384:Basic concepts
6381:
6380:
6370:
6368:
6367:
6360:
6353:
6345:
6339:
6338:
6332:
6326:. BirkhÀuser.
6322:Morse Homology
6315:
6292:
6278:
6271:
6255:
6235:
6226:
6212:
6188:
6174:
6154:
6145:
6130:
6108:
6096:(2): 331â358.
6076:
6036:
6033:
6030:
6029:
6022:
6002:
5995:
5973:
5966:
5948:
5935:(4): 661â692.
5913:
5912:
5910:
5907:
5905:
5904:
5899:
5890:
5881:
5872:
5863:
5854:
5849:
5844:
5838:
5832:
5830:
5827:
5819:Morse homology
5787:
5782:
5778:
5755:
5751:
5727:
5723:
5700:
5696:
5673:
5669:
5648:
5644:
5638:
5634:
5630:
5625:
5621:
5616:
5587:
5584:
5580:Floer homology
5556:Morse homology
5552:
5551:Morse homology
5549:
5536:
5531:
5526:
5523:
5499:
5479:
5476:
5473:
5453:
5433:
5409:
5390:
5387:
5384:
5381:
5361:
5342:
5339:
5336:
5333:
5313:
5293:
5273:
5253:
5241:
5238:
5226:
5221:
5218:
5214:
5210:
5205:
5202:
5199:
5195:
5191:
5186:
5182:
5145:
5142:
5139:
5119:
5114:
5110:
5089:
5061:
5058:
5055:
5035:
5015:
4990:
4986:
4983:
4980:
4977:
4955:
4952:
4949:
4946:
4941:
4937:
4933:
4928:
4924:
4903:
4900:
4897:
4894:
4891:
4888:
4885:
4882:
4879:
4876:
4873:
4870:
4867:
4864:
4842:
4839:
4836:
4833:
4828:
4824:
4818:
4814:
4810:
4807:
4804:
4801:
4798:
4795:
4792:
4789:
4786:
4781:
4778:
4775:
4771:
4767:
4764:
4761:
4758:
4753:
4749:
4745:
4740:
4736:
4730:
4726:
4722:
4719:
4716:
4713:
4710:
4707:
4702:
4699:
4696:
4692:
4688:
4683:
4679:
4667:
4648:
4645:
4625:
4605:
4602:
4599:
4594:
4590:
4566:
4546:
4543:
4523:
4503:
4483:
4463:
4460:
4438:
4434:
4413:
4410:
4407:
4404:
4401:
4395:
4391:
4385:
4381:
4377:
4374:
4371:
4368:
4348:
4345:
4342:
4339:
4311:
4308:
4288:
4268:
4247:
4243:
4240:
4237:
4234:
4214:
4202:
4199:
4182:
4179:
4159:
4145:
4144:
4132:
4110:
4107:
4104:
4100:
4073:
4070:
4067:
4063:
4042:
4039:
4019:
4016:
4013:
4010:
4007:
4004:
4001:
3998:
3995:
3990:
3987:
3983:
3962:
3959:
3956:
3953:
3950:
3947:
3944:
3924:
3921:
3901:
3881:
3861:
3841:
3814:
3792:
3788:
3776:
3775:
3764:
3759:
3755:
3729:
3725:
3704:
3699:
3695:
3668:
3664:
3643:
3640:
3620:
3596:
3593:
3590:
3587:
3584:
3579:
3576:
3572:
3552:
3549:
3546:
3543:
3524:
3521:
3501:
3474:
3454:
3451:
3448:
3445:
3442:
3439:
3434:
3431:
3427:
3423:
3418:
3414:
3384:
3380:
3358:
3354:
3351:
3339:Morse function
3326:
3314:
3311:
3286:
3266:
3246:
3226:
3223:
3201:
3196:
3192:
3188:
3185:
3182:
3177:
3172:
3169:
3166:
3162:
3158:
3153:
3148:
3144:
3140:
3137:
3134:
3129:
3124:
3120:
3116:
3113:
3110:
3107:
3104:
3101:
3098:
3095:
3092:
3089:
3069:
3049:
3046:
3043:
3040:
3037:
3032:
3028:
3007:
2987:
2963:
2957:
2953:
2949:
2946:
2943:
2938:
2934:
2930:
2925:
2921:
2916:
2892:
2889:
2886:
2883:
2880:
2877:
2857:
2845:
2842:
2825:
2801:
2781:
2757:
2737:
2704:
2701:
2696:
2692:
2688:
2685:
2682:
2662:
2642:
2639:
2636:
2616:
2613:
2608:
2604:
2600:
2597:
2594:
2574:
2554:
2551:
2548:
2528:
2525:
2522:
2519:
2499:
2480:
2476:
2454:
2433:
2430:
2425:
2421:
2417:
2414:
2409:
2405:
2401:
2398:
2395:
2392:
2389:
2386:
2383:
2380:
2377:
2374:
2371:
2357:
2338:
2327:
2308:
2297:Hessian matrix
2284:
2260:
2240:
2216:
2192:
2189:
2165:
2161:
2158:
2155:
2152:
2137:
2134:
2121:
2118:
2115:
2095:
2092:
2089:
2069:
2066:
2063:
2058:
2054:
2050:
2047:
2044:
2041:
2038:
2018:
1998:
1985:âthat is, the
1974:
1971:
1968:
1965:
1962:
1958:
1955:
1934:
1914:
1892:
1888:
1867:
1864:
1844:
1824:
1819:
1815:
1811:
1808:
1805:
1802:
1799:
1778:
1774:
1771:
1740:
1737:
1717:
1695:
1691:
1670:
1650:
1628:
1624:
1598:
1594:
1574:
1571:
1551:
1525:
1521:
1500:
1497:
1494:
1491:
1488:
1485:
1482:
1479:
1476:
1473:
1470:
1467:
1447:
1444:
1424:
1402:
1398:
1377:
1374:
1371:
1368:
1365:
1362:
1359:
1356:
1353:
1350:
1347:
1344:
1324:
1321:
1301:
1271:
1267:
1246:
1243:
1240:
1237:
1234:
1231:
1228:
1225:
1222:
1202:
1199:
1179:
1157:
1153:
1132:
1129:
1126:
1123:
1120:
1117:
1114:
1094:
1074:
1054:
1051:
1048:
1045:
1042:
1039:
1019:
999:
996:
993:
990:
987:
984:
960:
957:
954:
951:
948:
945:
942:
939:
936:
933:
930:
908:
904:
875:
872:
869:
866:
863:
860:
857:
854:
851:
848:
845:
823:
819:
791:
769:
765:
744:
720:
708:respectively.
697:
694:
674:
671:
668:
665:
645:
621:
601:
573:
553:
533:
488:
484:
468:tangent spaces
447:
427:
412:critical point
399:
379:
359:
356:
353:
350:
347:
344:
339:
336:
332:
318:
308:
304:
283:
227:
217:of a point in
201:
197:
194:
171:
147:
134:A saddle point
127:
126:Basic concepts
124:
80:Before Morse,
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
7250:
7239:
7236:
7234:
7231:
7229:
7226:
7225:
7223:
7208:
7205:
7203:
7202:Supermanifold
7200:
7198:
7195:
7193:
7190:
7186:
7183:
7182:
7181:
7178:
7176:
7173:
7171:
7168:
7166:
7163:
7161:
7158:
7156:
7153:
7151:
7148:
7147:
7145:
7141:
7135:
7132:
7130:
7127:
7125:
7122:
7120:
7117:
7115:
7112:
7110:
7107:
7106:
7104:
7100:
7090:
7087:
7085:
7082:
7080:
7077:
7075:
7072:
7070:
7067:
7065:
7062:
7060:
7057:
7055:
7052:
7050:
7047:
7045:
7042:
7041:
7039:
7037:
7033:
7027:
7024:
7022:
7019:
7017:
7014:
7012:
7009:
7007:
7004:
7002:
6999:
6997:
6993:
6989:
6987:
6984:
6982:
6979:
6977:
6973:
6969:
6967:
6964:
6962:
6959:
6957:
6954:
6952:
6949:
6947:
6944:
6942:
6939:
6938:
6936:
6934:
6930:
6924:
6923:Wedge product
6921:
6919:
6916:
6912:
6909:
6908:
6907:
6904:
6902:
6899:
6895:
6892:
6891:
6890:
6887:
6885:
6882:
6880:
6877:
6875:
6872:
6868:
6867:Vector-valued
6865:
6864:
6863:
6860:
6858:
6855:
6851:
6848:
6847:
6846:
6843:
6841:
6838:
6836:
6833:
6832:
6830:
6826:
6820:
6817:
6815:
6812:
6810:
6807:
6803:
6800:
6799:
6798:
6797:Tangent space
6795:
6793:
6790:
6788:
6785:
6783:
6780:
6779:
6777:
6773:
6770:
6768:
6764:
6758:
6755:
6753:
6749:
6745:
6743:
6740:
6738:
6734:
6730:
6726:
6724:
6721:
6719:
6716:
6714:
6711:
6709:
6706:
6704:
6701:
6699:
6696:
6694:
6691:
6687:
6684:
6683:
6682:
6679:
6677:
6674:
6672:
6669:
6667:
6664:
6662:
6659:
6657:
6654:
6652:
6649:
6647:
6644:
6642:
6639:
6637:
6634:
6632:
6628:
6624:
6622:
6618:
6614:
6612:
6609:
6608:
6606:
6600:
6594:
6591:
6589:
6586:
6584:
6581:
6579:
6576:
6574:
6571:
6569:
6566:
6562:
6561:in Lie theory
6559:
6558:
6557:
6554:
6552:
6549:
6545:
6542:
6541:
6540:
6537:
6535:
6532:
6531:
6529:
6527:
6523:
6517:
6514:
6512:
6509:
6507:
6504:
6502:
6499:
6497:
6494:
6492:
6489:
6487:
6484:
6482:
6479:
6477:
6474:
6473:
6471:
6468:
6464:Main results
6462:
6456:
6453:
6451:
6448:
6446:
6445:Tangent space
6443:
6441:
6438:
6436:
6433:
6431:
6428:
6426:
6423:
6421:
6418:
6414:
6411:
6409:
6406:
6405:
6404:
6401:
6397:
6394:
6393:
6392:
6389:
6388:
6386:
6382:
6377:
6373:
6366:
6361:
6359:
6354:
6352:
6347:
6346:
6343:
6335:
6333:9780817629045
6329:
6324:
6323:
6316:
6312:
6306:
6298:
6293:
6286:
6285:
6279:
6274:
6272:0-691-08008-9
6268:
6264:
6260:
6256:
6252:
6248:
6241:
6236:
6232:
6227:
6223:
6219:
6215:
6209:
6205:
6201:
6197:
6193:
6189:
6185:
6181:
6177:
6171:
6167:
6163:
6159:
6155:
6151:
6146:
6141:
6136:
6131:
6127:
6123:
6122:
6114:
6109:
6104:
6099:
6095:
6091:
6090:
6085:
6081:
6077:
6073:
6069:
6065:
6061:
6057:
6053:
6052:
6047:
6043:
6039:
6038:
6034:
6025:
6023:9781439831601
6019:
6016:. CRC Press.
6015:
6014:
6006:
6003:
5998:
5992:
5987:
5986:
5977:
5974:
5969:
5963:
5959:
5952:
5949:
5943:
5938:
5934:
5931:
5930:
5925:
5918:
5915:
5908:
5903:
5900:
5894:
5891:
5885:
5882:
5876:
5873:
5867:
5864:
5858:
5855:
5853:
5850:
5848:
5845:
5842:
5839:
5837:
5834:
5833:
5828:
5826:
5824:
5820:
5816:
5814:
5810:
5808:
5804:
5798:
5785:
5780:
5776:
5753:
5749:
5725:
5721:
5698:
5694:
5671:
5667:
5646:
5642:
5636:
5632:
5628:
5623:
5619:
5614:
5604:
5601:
5597:
5585:
5583:
5581:
5577:
5573:
5572:Betti numbers
5569:
5565:
5561:
5557:
5550:
5548:
5534:
5529:
5512:
5497:
5477:
5474:
5471:
5451:
5431:
5423:
5422:connected sum
5407:
5388:
5385:
5382:
5379:
5359:
5340:
5337:
5334:
5331:
5311:
5291:
5271:
5251:
5239:
5237:
5224:
5219:
5216:
5212:
5208:
5203:
5200:
5197:
5193:
5189:
5184:
5180:
5171:
5167:
5166:Edward Witten
5163:
5159:
5143:
5140:
5137:
5117:
5112:
5108:
5087:
5079:
5076:in 1952; the
5075:
5059:
5056:
5053:
5033:
5013:
5005:
4981:
4978:
4975:
4966:
4953:
4947:
4939:
4935:
4931:
4926:
4922:
4901:
4895:
4892:
4889:
4886:
4883:
4880:
4877:
4874:
4871:
4865:
4862:
4853:
4840:
4834:
4826:
4822:
4816:
4808:
4805:
4799:
4796:
4793:
4787:
4779:
4776:
4773:
4769:
4765:
4759:
4751:
4747:
4743:
4738:
4734:
4728:
4720:
4717:
4711:
4708:
4705:
4700:
4697:
4694:
4690:
4686:
4681:
4677:
4668:
4661:
4646:
4643:
4623:
4600:
4592:
4588:
4580:
4564:
4544:
4541:
4521:
4501:
4481:
4461:
4458:
4436:
4432:
4408:
4402:
4399:
4393:
4389:
4383:
4375:
4372:
4366:
4343:
4337:
4330:
4326:
4309:
4306:
4286:
4266:
4238:
4235:
4232:
4212:
4200:
4198:
4196:
4180:
4177:
4157:
4148:
4130:
4108:
4105:
4102:
4098:
4089:
4071:
4068:
4065:
4061:
4040:
4037:
4014:
4011:
4008:
4005:
4002:
3999:
3996:
3988:
3985:
3981:
3960:
3957:
3954:
3948:
3942:
3922:
3919:
3899:
3879:
3859:
3839:
3831:
3828:
3827:
3826:
3812:
3805:changes when
3790:
3786:
3762:
3757:
3753:
3744:
3727:
3723:
3702:
3697:
3693:
3684:
3683:diffeomorphic
3666:
3662:
3641:
3638:
3618:
3610:
3591:
3588:
3585:
3577:
3574:
3570:
3550:
3547:
3544:
3541:
3522:
3519:
3499:
3491:
3488:
3487:
3486:
3472:
3449:
3446:
3440:
3432:
3429:
3425:
3421:
3416:
3412:
3402:
3400:
3382:
3378:
3349:
3340:
3324:
3312:
3310:
3308:
3304:
3300:
3284:
3264:
3244:
3224:
3221:
3199:
3194:
3190:
3186:
3183:
3180:
3175:
3170:
3167:
3164:
3160:
3156:
3151:
3146:
3142:
3138:
3135:
3132:
3127:
3122:
3118:
3114:
3108:
3102:
3099:
3093:
3087:
3067:
3047:
3044:
3038:
3030:
3026:
3005:
2985:
2978:
2961:
2955:
2951:
2947:
2944:
2941:
2936:
2932:
2928:
2923:
2919:
2914:
2906:
2890:
2887:
2881:
2878:
2875:
2855:
2843:
2841:
2839:
2823:
2815:
2799:
2779:
2771:
2770:tangent space
2755:
2735:
2727:
2726:
2720:
2718:
2717:monkey saddle
2702:
2699:
2694:
2690:
2686:
2683:
2680:
2660:
2640:
2637:
2634:
2614:
2611:
2606:
2602:
2598:
2595:
2592:
2572:
2552:
2549:
2546:
2526:
2523:
2520:
2517:
2497:
2478:
2431:
2428:
2423:
2419:
2415:
2412:
2407:
2403:
2399:
2396:
2393:
2390:
2387:
2384:
2381:
2375:
2369:
2360:
2358:
2351:
2336:
2328:
2321:
2306:
2298:
2282:
2274:
2258:
2238:
2230:
2214:
2206:
2190:
2187:
2180:
2156:
2153:
2150:
2143:
2135:
2133:
2119:
2116:
2113:
2093:
2090:
2087:
2067:
2064:
2061:
2056:
2052:
2048:
2042:
2036:
2016:
1996:
1988:
1972:
1969:
1963:
1956:
1953:
1932:
1912:
1890:
1886:
1865:
1862:
1842:
1822:
1817:
1813:
1809:
1803:
1797:
1772:
1769:
1760:
1758:
1754:
1738:
1735:
1715:
1693:
1689:
1668:
1648:
1626:
1622:
1612:
1596:
1592:
1572:
1569:
1549:
1541:
1523:
1519:
1498:
1492:
1486:
1483:
1480:
1477:
1471:
1465:
1445:
1442:
1422:
1400:
1396:
1375:
1369:
1363:
1360:
1357:
1354:
1348:
1342:
1322:
1319:
1299:
1291:
1287:
1269:
1265:
1244:
1238:
1232:
1229:
1226:
1223:
1220:
1200:
1197:
1177:
1155:
1151:
1130:
1127:
1124:
1118:
1112:
1105:is less than
1092:
1072:
1052:
1049:
1046:
1043:
1040:
1037:
1017:
997:
994:
991:
988:
985:
982:
955:
949:
946:
943:
940:
934:
928:
906:
902:
892:
870:
864:
861:
858:
855:
849:
843:
821:
817:
807:
803:
789:
767:
763:
742:
734:
718:
709:
695:
692:
672:
669:
666:
663:
643:
635:
632:on which the
619:
599:
591:
590:tangent space
587:
571:
551:
531:
523:
519:
510:
506:
504:
503:mountain pass
486:
482:
473:
469:
465:
461:
445:
425:
417:
413:
397:
377:
354:
351:
345:
337:
334:
330:
316:
306:
302:
281:
268:
264:
262:
261:saddle points
258:
254:
250:
246:
242:
216:
215:inverse image
192:
185:
169:
161:
145:
132:
125:
123:
121:
116:
114:
110:
106:
103:
99:
95:
91:
87:
83:
82:Arthur Cayley
78:
76:
72:
68:
67:CW structures
64:
63:Marston Morse
60:
56:
52:
48:
44:
40:
33:
19:
7228:Morse theory
7129:Moving frame
7124:Morse theory
7123:
7114:Gauge theory
6906:Tensor field
6835:Closed/Exact
6814:Vector field
6782:Distribution
6723:Hypercomplex
6718:Quaternionic
6455:Vector field
6413:Smooth atlas
6321:
6296:
6283:
6263:Morse Theory
6262:
6259:Milnor, John
6250:
6246:
6230:
6195:
6161:
6149:
6140:math/0104155
6125:
6119:
6093:
6087:
6055:
6049:
6012:
6005:
5984:
5976:
5957:
5951:
5932:
5927:
5917:
5893:Sard's lemma
5817:
5811:
5799:
5713:is equal to
5605:
5600:critical set
5591:
5589:
5578:is known as
5554:
5243:
5157:
5080:states that
5074:Georges Reeb
5003:
4967:
4854:
4660:
4579:Betti number
4204:
4149:
4146:
3829:
3777:
3489:
3403:
3338:
3316:
2977:neighborhood
2847:
2723:
2721:
2361:
2350:
2320:
2319:is called a
2205:differential
2139:
1761:
1613:
974:
710:
515:
462:acting as a
414:, where the
273:
257:double point
241:contour line
137:
117:
79:
57:by studying
47:Morse theory
46:
36:
7074:Levi-Civita
7064:Generalized
7036:Connections
6986:Lie algebra
6918:Volume form
6819:Vector flow
6792:Pushforward
6787:Lie bracket
6686:Lie algebra
6651:G-structure
6440:Pushforward
6420:Submanifold
6192:Lang, Serge
6080:Bott, Raoul
6042:Bott, Raoul
5444:2-tori. If
5046:? The case
3465:changes as
3214:throughout
2844:Morse lemma
2271:are called
1288:, which is
522:independent
39:mathematics
7222:Categories
7197:Stratifold
7155:Diffeology
6951:Associated
6752:Symplectic
6737:Riemannian
6666:Hyperbolic
6593:Submersion
6501:HopfâRinow
6435:Submersion
6430:Smooth map
6092:. (N.S.).
6058:: 99â114.
5996:0824717090
5967:0582325021
5909:References
5852:Jacobi set
5803:Raoul Bott
5164:. In 1982
4534:-cells in
3018:such that
2653:(that is,
2565:(that is,
464:linear map
109:Raoul Bott
105:functional
90:topography
7079:Principal
7054:Ehresmann
7011:Subbundle
7001:Principal
6976:Fibration
6956:Cotangent
6828:Covectors
6681:Lie group
6661:Hermitian
6604:manifolds
6573:Immersion
6568:Foliation
6506:Noether's
6491:Frobenius
6486:De Rham's
6481:Darboux's
6372:Manifolds
6305:cite book
6184:853621933
5754:−
5726:−
5672:−
5624:−
5198:−
5130:The case
5026:restrict
4985:→
4940:γ
4932:≥
4927:γ
4893:
4878:…
4866:∈
4863:γ
4817:γ
4806:−
4797:⋯
4794:±
4777:−
4774:γ
4766:−
4752:γ
4744:≥
4729:γ
4718:−
4709:⋯
4706:±
4698:−
4695:γ
4687:−
4682:γ
4624:γ
4593:γ
4565:γ
4459:γ
4437:γ
4403:χ
4394:γ
4384:γ
4373:−
4367:∑
4338:χ
4267:γ
4242:→
4213:γ
4131:γ
4109:ε
4106:−
4072:ε
4015:ε
4003:ε
4000:−
3986:−
3935:and that
3920:γ
3912:of index
3575:−
3444:∞
3441:−
3430:−
3353:→
3245:γ
3184:⋯
3165:γ
3147:γ
3139:−
3136:⋯
3133:−
3115:−
2945:…
2885:→
2703:⋯
2615:⋯
2550:≠
2432:⋯
2160:→
2114:ϵ
2088:ϵ
2065:ϵ
1716:γ
1669:γ
586:dimension
512:The torus
349:∞
346:−
335:−
245:level set
196:→
94:geodesics
7175:Orbifold
7170:K-theory
7160:Diffiety
6884:Pullback
6698:Oriented
6676:Kenmotsu
6656:Hadamard
6602:Types of
6551:Geodesic
6376:Glossary
6261:(1963).
6222:39379395
6194:(1999).
6082:(1982).
6072:54005577
6044:(1988).
5829:See also
5560:homology
4914:one has
3973:Suppose
3832:Suppose
3830:Theorem.
3492:Suppose
3490:Theorem.
3299:isolated
3060:for all
1957:″
1790:and let
1753:embedded
1708:, where
1540:cylinder
466:between
416:gradient
247:). Each
184:function
160:manifold
75:homology
55:manifold
51:topology
7119:History
7102:Related
7016:Tangent
6994:)
6974:)
6941:Adjoint
6933:Bundles
6911:density
6809:Torsion
6775:Vectors
6767:Tensors
6750:)
6735:)
6731:,
6729:Pseudoâ
6708:Poisson
6641:Finsler
6636:Fibered
6631:Contact
6629:)
6621:Complex
6619:)
6588:Section
4123:with a
3609:compact
3399:generic
3370:in the
1987:Hessian
1925:passes
802:rises.
634:Hessian
584:is the
182:is the
100:of the
7233:Lemmas
7084:Vector
7069:Koszul
7049:Cartan
7044:Affine
7026:Vector
7021:Tensor
7006:Spinor
6996:Normal
6992:Stable
6946:Affine
6850:bundle
6802:bundle
6748:Almost
6671:KĂ€hler
6627:Almost
6617:Almost
6611:Closed
6511:Sard's
6467:(list)
6330:
6269:
6220:
6210:
6182:
6172:
6070:
6020:
5993:
5964:
5659:where
4424:where
162:). If
102:energy
7192:Sheaf
6966:Fiber
6742:Rizza
6713:Prime
6544:Local
6534:Curve
6396:Atlas
6288:(PDF)
6243:(PDF)
6135:arXiv
6116:(PDF)
6068:S2CID
4053:Then
3745:onto
3654:Then
3337:is a
3237:Here
2975:in a
2905:chart
2725:index
2444:from
2349:is a
2177:on a
1835:Then
1511:then
1388:then
1284:is a
1257:then
1213:when
1143:then
921:when
836:when
733:torus
731:be a
239:is a
53:of a
7059:Form
6961:Dual
6894:flow
6757:Tame
6733:Subâ
6646:Flat
6526:Maps
6328:ISBN
6311:link
6267:ISBN
6218:OCLC
6208:ISBN
6180:OCLC
6170:ISBN
6018:ISBN
5991:ISBN
5962:ISBN
5768:and
5475:>
5383:>
3872:and
3715:and
3631:and
3545:<
3080:and
2848:Let
2722:The
2117:<
2091:>
1484:<
1478:<
1458:and
1361:<
1355:<
1335:and
1286:disk
1230:<
1224:<
1065:and
1010:and
947:<
941:<
862:<
856:<
685:and
472:rank
84:and
69:and
6981:Jet
6098:doi
6060:doi
5937:doi
5562:of
5424:of
4890:dim
4225:of
4090:to
4086:is
3685:to
3681:is
3607:is
3309:).
3277:at
2998:of
2792:at
2772:to
2748:of
2466:to
2231:of
2207:of
2132:).
2029:to
1989:of
1755:in
636:of
612:at
592:to
564:of
438:is
418:of
322:def
37:In
7224::
6972:Co
6307:}}
6303:{{
6251:40
6249:.
6245:.
6216:.
6206:.
6198:.
6178:.
6164:.
6126:18
6124:.
6118:.
6086:.
6066:.
6056:68
6054:.
6048:.
5933:17
5926:.
5809:.
5582:.
4669::
2840:.
2719:.
2359:.
1933:0.
122:.
115:.
77:.
45:,
6990:(
6970:(
6746:(
6727:(
6625:(
6615:(
6378:)
6374:(
6364:e
6357:t
6350:v
6336:.
6313:)
6290:.
6275:.
6233:.
6224:.
6186:.
6143:.
6137::
6106:.
6100::
6094:7
6074:.
6062::
6026:.
5999:.
5970:.
5945:.
5939::
5786:.
5781:+
5777:i
5750:i
5722:i
5699:+
5695:i
5668:i
5647:,
5643:)
5637:+
5633:i
5629:,
5620:i
5615:(
5535:.
5530:2
5525:P
5522:R
5498:g
5478:0
5472:g
5452:N
5432:g
5408:M
5389:,
5386:0
5380:g
5360:M
5341:,
5338:0
5335:=
5332:g
5312:g
5292:g
5272:M
5252:M
5225:.
5220:f
5217:t
5213:e
5209:d
5204:f
5201:t
5194:e
5190:=
5185:t
5181:d
5158:M
5144:3
5141:=
5138:k
5118:.
5113:n
5109:S
5088:M
5060:2
5057:=
5054:k
5034:M
5014:f
5004:k
4989:R
4982:M
4979::
4976:f
4954:.
4951:)
4948:M
4945:(
4936:b
4923:C
4902:,
4899:}
4896:M
4887:=
4884:n
4881:,
4875:,
4872:0
4869:{
4841:.
4838:)
4835:M
4832:(
4827:0
4823:b
4813:)
4809:1
4803:(
4800:+
4791:)
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4785:(
4780:1
4770:b
4763:)
4760:M
4757:(
4748:b
4739:0
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4725:)
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4712:+
4701:1
4691:C
4678:C
4647:.
4644:M
4604:)
4601:M
4598:(
4589:b
4545:.
4542:M
4522:n
4502:M
4482:n
4462:.
4433:C
4412:)
4409:M
4406:(
4400:=
4390:C
4380:)
4376:1
4370:(
4347:)
4344:M
4341:(
4310:.
4307:f
4287:M
4246:R
4239:M
4236::
4233:f
4181:.
4178:n
4158:n
4103:q
4099:M
4069:+
4066:q
4062:M
4041:.
4038:p
4018:]
4012:+
4009:q
4006:,
3997:q
3994:[
3989:1
3982:f
3961:.
3958:q
3955:=
3952:)
3949:p
3946:(
3943:f
3923:,
3900:f
3880:p
3860:M
3840:f
3813:a
3791:a
3787:M
3763:.
3758:a
3754:M
3728:b
3724:M
3703:,
3698:b
3694:M
3667:a
3663:M
3642:.
3639:b
3619:a
3595:]
3592:b
3589:,
3586:a
3583:[
3578:1
3571:f
3551:,
3548:b
3542:a
3523:,
3520:M
3500:f
3473:a
3453:]
3450:a
3447:,
3438:(
3433:1
3426:f
3422:=
3417:a
3413:M
3383:2
3379:C
3357:R
3350:M
3325:M
3285:p
3265:f
3225:.
3222:U
3200:2
3195:n
3191:x
3187:+
3181:+
3176:2
3171:1
3168:+
3161:x
3157:+
3152:2
3143:x
3128:2
3123:1
3119:x
3112:)
3109:p
3106:(
3103:f
3100:=
3097:)
3094:x
3091:(
3088:f
3068:i
3048:0
3045:=
3042:)
3039:p
3036:(
3031:i
3027:x
3006:p
2986:U
2962:)
2956:n
2952:x
2948:,
2942:,
2937:2
2933:x
2929:,
2924:1
2920:x
2915:(
2891:.
2888:R
2882:M
2879::
2876:f
2856:p
2824:f
2800:p
2780:M
2756:f
2736:p
2700:+
2695:3
2691:x
2687:d
2684:+
2681:a
2661:f
2641:0
2638:=
2635:c
2612:+
2607:2
2603:x
2599:c
2596:+
2593:a
2573:f
2553:0
2547:c
2527:,
2524:0
2521:=
2518:b
2498:f
2479:,
2475:R
2453:R
2429:+
2424:3
2420:x
2416:d
2413:+
2408:2
2404:x
2400:c
2397:+
2394:x
2391:b
2388:+
2385:a
2382:=
2379:)
2376:x
2373:(
2370:f
2337:p
2307:p
2283:p
2259:f
2239:f
2215:f
2191:,
2188:M
2164:R
2157:M
2154::
2151:f
2120:0
2094:0
2068:x
2062:+
2057:3
2053:x
2049:=
2046:)
2043:x
2040:(
2037:f
2017:f
1997:f
1973:0
1970:=
1967:)
1964:0
1961:(
1954:f
1913:a
1891:a
1887:M
1866:,
1863:f
1843:0
1823:.
1818:3
1814:x
1810:=
1807:)
1804:x
1801:(
1798:f
1777:R
1773:=
1770:M
1739:.
1736:f
1694:a
1690:M
1649:a
1627:a
1623:M
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1593:M
1573:,
1570:s
1550:a
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1520:M
1499:,
1496:)
1493:s
1490:(
1487:f
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1475:)
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1469:(
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1376:,
1373:)
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1131:,
1128:0
1125:=
1122:)
1119:p
1116:(
1113:f
1093:a
1073:2
1053:,
1050:1
1047:,
1044:1
1041:,
1038:0
1018:s
998:,
995:r
992:,
989:q
986:,
983:p
959:)
956:s
953:(
950:f
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938:)
935:r
932:(
929:f
907:a
903:M
874:)
871:r
868:(
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859:a
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850:q
847:(
844:f
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818:M
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764:M
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693:2
673:,
670:1
667:,
664:0
644:f
620:p
600:M
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552:p
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487:a
483:M
446:0
426:f
398:a
378:a
358:]
355:a
352:,
343:(
338:1
331:f
317:=
307:a
303:M
282:a
226:R
200:R
193:M
170:f
146:M
96:(
34:.
20:)
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