74:
1041:
It is still possible to define the index for a vector field with nonisolated zeroes. A construction of this index and the extension of
Poincaré–Hopf theorem for vector fields with nonisolated zeroes is outlined in Section 1.1.2 of
788:
972:. Extend the vector field to this neighborhood so that it still has the same zeroes and the zeroes have the same indices. In addition, make sure that the extended vector field at the boundary of
913:
without boundary, this amounts to saying that the integral is 0. But by examining vector fields in a sufficiently small neighborhood of a source or sink, we see that sources and sinks contribute
469:
1055:
415:
893:. Thus, this theorem establishes a deep link between two seemingly unrelated areas of mathematics. It is perhaps as interesting that the proof of this theorem relies heavily on
592:
1005:. Technique: cut away all zeroes of the vector field with small neighborhoods. Then use the fact that the degree of a map from the boundary of an n-dimensional manifold to an
840:
77:
According to the
Poincare-Hopf theorem, closed trajectories can encircle two centres and one saddle or one centre, but never just the saddle. (Here for in case of a
525:
864:
811:
713:
689:
669:
645:
618:
493:
376:
356:
332:
312:
292:
272:
252:
232:
208:
188:
168:
148:
128:
108:
917:
amounts (known as the index) to the total, and they must all sum to 0. This result may be considered one of the earliest of a whole series of theorems (e.g.
1147:
926:
1128:
721:
918:
1152:
418:
1111:
424:
1106:
381:
1001:
sphere. Thus, the sum of the indices is independent of the actual vector field, and depends only on the manifold
955:
866:. A particularly useful corollary is when there is a non-vanishing vector field implying Euler characteristic 0.
1025:
624:
1101:
934:
910:
890:
692:
530:
40:
922:
843:
902:
982:
The sum of indices of the zeroes of the old (and new) vector field is equal to the degree of the
78:
59:
898:
1124:
906:
870:
816:
211:
44:
498:
715:
be pointing in the outward normal direction along the boundary. Then we have the formula
849:
796:
698:
674:
654:
630:
603:
478:
361:
341:
317:
297:
277:
257:
237:
217:
193:
173:
153:
133:
113:
93:
16:
Counts 0s of a vector field on a differentiable manifold using its Euler characteristic
1141:
621:
1060:
1032:
for which it is clear that the sum of indices is equal to the Euler characteristic.
648:
335:
63:
909:
is equal to the integral of that form over the boundary. In the special case of a
20:
874:
48:
1079:
Henri
Poincaré, On curves defined by differential equations (1881–1882)
983:
73:
1013:
sphere, that can be extended to the whole n-dimensional manifold, is zero.
941:
concepts. They play an important role in the modern study of both fields.
930:
894:
886:
472:
67:
1088:
938:
914:
1016:
Finally, identify this sum of indices as the Euler characteristic of
783:{\displaystyle \sum _{i}\operatorname {index} _{x_{i}}(v)=\chi (M)\,}
72:
793:
where the sum of the indices is over all the isolated zeroes of
885:
The Euler characteristic of a closed surface is a purely
58:
theorem is often illustrated by the special case of the
1119:
Brasselet, Jean-Paul; Seade, José; Suwa, Tatsuo (2009).
1020:. To do that, construct a very specific vector field on
889:
concept, whereas the index of a vector field is purely
852:
819:
799:
724:
701:
677:
657:
633:
606:
533:
501:
481:
427:
384:
364:
344:
320:
300:
280:
260:
240:
220:
196:
176:
156:
136:
116:
96:
954:in some high-dimensional Euclidean space. (Use the
1043:
858:
834:
805:
782:
707:
683:
663:
639:
612:
586:
519:
487:
464:{\displaystyle u:\partial D\to \mathbb {S} ^{n-1}}
463:
409:
370:
350:
326:
306:
286:
266:
246:
226:
202:
182:
162:
142:
122:
102:
1056:Eisenbud–Levine–Khimshiashvili signature formula
873:and later generalized to higher dimensions by
62:, which simply states that there is no smooth
869:The theorem was proven for two dimensions by
410:{\displaystyle \operatorname {index} _{x}(v)}
8:
581:
566:
110:be a differentiable manifold, of dimension
929:) establishing deep relationships between
39:) is an important theorem that is used in
851:
818:
798:
779:
744:
739:
729:
723:
700:
676:
656:
632:
605:
561:
532:
500:
480:
449:
445:
444:
426:
389:
383:
363:
343:
319:
299:
279:
259:
239:
219:
195:
175:
155:
135:
115:
95:
901:, which states that the integral of the
1072:
7:
1121:Vector fields on singular varieties
587:{\displaystyle u(z)=v(z)/\|v(z)\|}
434:
14:
1148:Theorems in differential topology
927:Grothendieck–Riemann–Roch theorem
1044:Brasselet, Seade & Suwa 2009
829:
823:
776:
770:
761:
755:
578:
572:
558:
552:
543:
537:
514:
502:
440:
404:
398:
1:
961:Take a small neighborhood of
70:having no sources or sinks.
1107:Encyclopedia of Mathematics
919:Atiyah–Singer index theorem
33:Poincaré–Hopf index theorem
29:Poincaré–Hopf index formula
1169:
1092:(1926), pp. 209–221.
965:in that Euclidean space,
956:Whitney embedding theorem
671:with isolated zeroes. If
1123:. Heidelberg: Springer.
835:{\displaystyle \chi (M)}
417:, can be defined as the
1102:"Poincaré–Hopf theorem"
625:differentiable manifold
190:is an isolated zero of
66:on an even-dimensional
897:, and, in particular,
860:
836:
807:
784:
709:
695:, then we insist that
685:
665:
641:
614:
588:
521:
489:
465:
411:
372:
352:
328:
308:
288:
268:
248:
228:
204:
184:
164:
144:
124:
104:
82:
1153:Differential topology
986:from the boundary of
979:is directed outwards.
861:
837:
808:
785:
710:
686:
666:
642:
615:
589:
522:
520:{\displaystyle (n-1)}
490:
466:
412:
373:
353:
329:
309:
289:
269:
249:
234:. Pick a closed ball
229:
205:
185:
165:
145:
125:
105:
76:
41:differential topology
25:Poincaré–Hopf theorem
850:
844:Euler characteristic
817:
797:
722:
699:
675:
655:
631:
604:
531:
499:
479:
425:
382:
362:
342:
318:
298:
294:is the only zero of
278:
258:
238:
218:
194:
174:
154:
134:
114:
94:
43:. It is named after
903:exterior derivative
27:(also known as the
856:
832:
803:
780:
734:
705:
681:
661:
637:
610:
584:
527:-sphere given by
517:
485:
461:
407:
368:
348:
324:
304:
284:
264:
244:
224:
200:
180:
160:
150:a vector field on
140:
120:
100:
83:
79:Hamiltonian system
60:hairy ball theorem
37:Hopf index theorem
1130:978-3-642-05205-7
923:De Rham's theorem
907:differential form
859:{\displaystyle M}
806:{\displaystyle v}
725:
708:{\displaystyle v}
684:{\displaystyle M}
664:{\displaystyle M}
640:{\displaystyle v}
613:{\displaystyle M}
488:{\displaystyle D}
371:{\displaystyle x}
351:{\displaystyle v}
327:{\displaystyle D}
307:{\displaystyle v}
287:{\displaystyle x}
267:{\displaystyle x}
247:{\displaystyle D}
227:{\displaystyle x}
212:local coordinates
203:{\displaystyle v}
183:{\displaystyle x}
163:{\displaystyle M}
143:{\displaystyle v}
123:{\displaystyle n}
103:{\displaystyle M}
1160:
1134:
1115:
1093:
1086:
1080:
1077:
1012:
1000:
865:
863:
862:
857:
841:
839:
838:
833:
812:
810:
809:
804:
789:
787:
786:
781:
751:
750:
749:
748:
733:
714:
712:
711:
706:
690:
688:
687:
682:
670:
668:
667:
662:
646:
644:
643:
638:
619:
617:
616:
611:
593:
591:
590:
585:
565:
526:
524:
523:
518:
494:
492:
491:
486:
470:
468:
467:
462:
460:
459:
448:
416:
414:
413:
408:
394:
393:
377:
375:
374:
369:
357:
355:
354:
349:
333:
331:
330:
325:
313:
311:
310:
305:
293:
291:
290:
285:
273:
271:
270:
265:
253:
251:
250:
245:
233:
231:
230:
225:
209:
207:
206:
201:
189:
187:
186:
181:
169:
167:
166:
161:
149:
147:
146:
141:
129:
127:
126:
121:
109:
107:
106:
101:
86:Formal statement
1168:
1167:
1163:
1162:
1161:
1159:
1158:
1157:
1138:
1137:
1131:
1118:
1100:
1097:
1096:
1087:
1083:
1078:
1074:
1069:
1052:
1039:
1011:–1)-dimensional
1006:
999:–1)-dimensional
994:
992:
978:
971:
947:
945:Sketch of proof
899:Stokes' theorem
883:
848:
847:
815:
814:
795:
794:
740:
735:
720:
719:
697:
696:
673:
672:
653:
652:
629:
628:
602:
601:
529:
528:
497:
496:
477:
476:
443:
423:
422:
385:
380:
379:
360:
359:
340:
339:
316:
315:
296:
295:
276:
275:
256:
255:
236:
235:
216:
215:
210:, and fix some
192:
191:
172:
171:
170:. Suppose that
152:
151:
132:
131:
112:
111:
92:
91:
88:
17:
12:
11:
5:
1166:
1164:
1156:
1155:
1150:
1140:
1139:
1136:
1135:
1129:
1116:
1095:
1094:
1081:
1071:
1070:
1068:
1065:
1064:
1063:
1058:
1051:
1048:
1038:
1037:Generalization
1035:
1034:
1033:
1014:
990:
980:
976:
969:
959:
946:
943:
882:
879:
871:Henri Poincaré
855:
831:
828:
825:
822:
802:
791:
790:
778:
775:
772:
769:
766:
763:
760:
757:
754:
747:
743:
738:
732:
728:
704:
680:
660:
636:
609:
583:
580:
577:
574:
571:
568:
564:
560:
557:
554:
551:
548:
545:
542:
539:
536:
516:
513:
510:
507:
504:
484:
458:
455:
452:
447:
442:
439:
436:
433:
430:
406:
403:
400:
397:
392:
388:
367:
347:
323:
303:
283:
263:
243:
223:
199:
179:
159:
139:
119:
99:
87:
84:
45:Henri Poincaré
15:
13:
10:
9:
6:
4:
3:
2:
1165:
1154:
1151:
1149:
1146:
1145:
1143:
1132:
1126:
1122:
1117:
1113:
1109:
1108:
1103:
1099:
1098:
1091:
1085:
1082:
1076:
1073:
1066:
1062:
1059:
1057:
1054:
1053:
1049:
1047:
1045:
1036:
1031:
1027:
1026:triangulation
1023:
1019:
1015:
1010:
1004:
998:
989:
985:
981:
975:
968:
964:
960:
957:
953:
949:
948:
944:
942:
940:
936:
932:
928:
924:
920:
916:
912:
908:
904:
900:
896:
892:
888:
880:
878:
876:
872:
867:
853:
845:
826:
820:
800:
773:
767:
764:
758:
752:
745:
741:
736:
730:
726:
718:
717:
716:
702:
694:
678:
658:
650:
634:
626:
623:
607:
599:
595:
575:
569:
562:
555:
549:
546:
540:
534:
511:
508:
505:
482:
474:
456:
453:
450:
437:
431:
428:
420:
401:
395:
390:
386:
365:
345:
337:
321:
301:
281:
261:
241:
221:
213:
197:
177:
157:
137:
117:
97:
85:
80:
75:
71:
69:
65:
61:
57:
56:Poincaré–Hopf
52:
50:
46:
42:
38:
34:
30:
26:
22:
1120:
1105:
1089:
1084:
1075:
1061:Hopf theorem
1040:
1029:
1021:
1017:
1008:
1002:
996:
987:
973:
966:
962:
951:
884:
881:Significance
868:
792:
649:vector field
597:
596:
254:centered at
89:
64:vector field
55:
53:
36:
32:
28:
24:
18:
895:integration
887:topological
421:of the map
334:. Then the
21:mathematics
1142:Categories
1067:References
935:analytical
875:Heinz Hopf
274:, so that
49:Heinz Hopf
1112:EMS Press
984:Gauss map
931:geometric
821:χ
768:χ
753:
727:∑
582:‖
567:‖
509:−
471:from the
454:−
441:→
435:∂
396:
1050:See also
1024:using a
939:physical
911:manifold
891:analytic
693:boundary
598:Theorem.
473:boundary
68:n-sphere
1114:, 2001
993:to the
915:integer
842:is the
627:. Let
622:compact
495:to the
1127:
950:Embed
419:degree
130:, and
23:, the
905:of a
737:index
647:be a
620:be a
387:index
336:index
214:near
35:, or
1125:ISBN
933:and
813:and
691:has
600:Let
90:Let
54:The
47:and
1046:).
1028:of
937:or
846:of
651:on
475:of
358:at
338:of
314:in
19:In
1144::
1110:,
1104:,
1090:96
958:.)
925:,
921:,
877:.
594:.
378:,
51:.
31:,
1133:.
1042:(
1030:M
1022:M
1018:M
1009:n
1007:(
1003:M
997:n
995:(
991:ε
988:N
977:ε
974:N
970:ε
967:N
963:M
952:M
854:M
830:)
827:M
824:(
801:v
777:)
774:M
771:(
765:=
762:)
759:v
756:(
746:i
742:x
731:i
703:v
679:M
659:M
635:v
608:M
579:)
576:z
573:(
570:v
563:/
559:)
556:z
553:(
550:v
547:=
544:)
541:z
538:(
535:u
515:)
512:1
506:n
503:(
483:D
457:1
451:n
446:S
438:D
432::
429:u
405:)
402:v
399:(
391:x
366:x
346:v
322:D
302:v
282:x
262:x
242:D
222:x
198:v
178:x
158:M
138:v
118:n
98:M
81:)
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