85:
1052:
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
799:
983:. 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
924:
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
480:
426:
904:. 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
603:
1016:. 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
851:
88:
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
536:
1066:
875:
822:
724:
700:
680:
656:
629:
504:
387:
367:
343:
323:
303:
283:
263:
243:
219:
199:
179:
159:
139:
119:
928:
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.
17:
732:
1158:
937:
429:
1139:
435:
392:
1012:
sphere. Thus, the sum of the indices is independent of the actual vector field, and depends only on the manifold
929:
877:. A particularly useful corollary is when there is a non-vanishing vector field implying Euler characteristic 0.
1163:
1122:
1112:
1117:
541:
966:
1036:
346:
993:
The sum of indices of the zeroes of the old (and new) vector field is equal to the degree of the
635:
945:
921:
901:
703:
827:
51:
933:
854:
509:
726:
be pointing in the outward normal direction along the boundary. Then we have the formula
8:
913:
483:
909:
860:
807:
709:
685:
665:
641:
614:
489:
372:
352:
328:
308:
288:
268:
248:
228:
204:
184:
164:
144:
124:
104:
89:
70:
27:
Counts 0s of a vector field on a differentiable manifold using its Euler characteristic
1135:
917:
222:
881:
55:
1043:
for which it is clear that the sum of indices is equal to the Euler characteristic.
920:
is equal to the integral of that form over the boundary. In the special case of a
1152:
1090:
Henri
Poincaré, On curves defined by differential equations (1881–1882)
632:
1071:
659:
74:
1024:
sphere, that can be extended to the whole n-dimensional manifold, is zero.
952:
concepts. They play an important role in the modern study of both fields.
31:
1099:
885:
59:
1027:
Finally, identify this sum of indices as the Euler characteristic of
994:
794:{\displaystyle \sum _{i}\operatorname {index} _{x_{i}}(v)=\chi (M)\,}
941:
905:
897:
78:
949:
925:
804:
where the sum of the indices is over all the isolated zeroes of
84:
896:
The Euler characteristic of a closed surface is a purely
69:
theorem is often illustrated by the special case of the
1130:
Brasselet, Jean-Paul; Seade, José; Suwa, Tatsuo (2009).
1031:. To do that, construct a very specific vector field on
900:
concept, whereas the index of a vector field is purely
863:
830:
810:
735:
712:
688:
668:
644:
617:
544:
512:
492:
438:
395:
375:
355:
331:
311:
291:
271:
251:
231:
207:
187:
167:
147:
127:
107:
965:in some high-dimensional Euclidean space. (Use the
1129:
1054:
869:
845:
816:
793:
718:
694:
674:
650:
623:
597:
530:
498:
475:{\displaystyle u:\partial D\to \mathbb {S} ^{n-1}}
474:
420:
381:
361:
337:
317:
297:
277:
257:
237:
213:
193:
173:
153:
133:
113:
1150:
1067:Eisenbud–Levine–Khimshiashvili signature formula
884:and later generalized to higher dimensions by
73:, which simply states that there is no smooth
880:The theorem was proven for two dimensions by
421:{\displaystyle \operatorname {index} _{x}(v)}
592:
577:
121:be a differentiable manifold, of dimension
940:) establishing deep relationships between
50:) is an important theorem that is used in
790:
456:
912:, which states that the integral of the
83:
14:
1151:
1132:Vector fields on singular varieties
96:
24:
955:
598:{\displaystyle u(z)=v(z)/\|v(z)\|}
445:
25:
1175:
1159:Theorems in differential topology
1047:
938:Grothendieck–Riemann–Roch theorem
1055:Brasselet, Seade & Suwa 2009
891:
1093:
1084:
840:
834:
787:
781:
772:
766:
589:
583:
569:
563:
554:
548:
525:
513:
451:
415:
409:
13:
1:
1077:
972:Take a small neighborhood of
81:having no sources or sinks.
7:
1118:Encyclopedia of Mathematics
1060:
930:Atiyah–Singer index theorem
44:Poincaré–Hopf index theorem
40:Poincaré–Hopf index formula
18:Poincaré–Hopf index theorem
10:
1180:
1103:(1926), pp. 209–221.
976:in that Euclidean space,
967:Whitney embedding theorem
682:with isolated zeroes. If
1134:. Heidelberg: Springer.
846:{\displaystyle \chi (M)}
428:, can be defined as the
1113:"Poincaré–Hopf theorem"
636:differentiable manifold
201:is an isolated zero of
77:on an even-dimensional
908:, and, in particular,
871:
847:
818:
795:
720:
706:, then we insist that
696:
676:
652:
625:
599:
532:
500:
476:
422:
383:
363:
339:
319:
299:
279:
259:
239:
215:
195:
175:
155:
135:
115:
93:
1164:Differential topology
997:from the boundary of
990:is directed outwards.
872:
848:
819:
796:
721:
697:
677:
653:
626:
600:
533:
531:{\displaystyle (n-1)}
501:
477:
423:
384:
364:
340:
320:
300:
280:
260:
245:. Pick a closed ball
240:
216:
196:
176:
156:
136:
116:
87:
52:differential topology
36:Poincaré–Hopf theorem
861:
855:Euler characteristic
828:
808:
733:
710:
686:
666:
642:
615:
542:
510:
490:
436:
393:
373:
353:
329:
309:
305:is the only zero of
289:
269:
249:
229:
205:
185:
165:
145:
125:
105:
54:. It is named after
914:exterior derivative
38:(also known as the
867:
843:
814:
791:
745:
716:
692:
672:
648:
621:
595:
538:-sphere given by
528:
496:
472:
418:
379:
359:
335:
315:
295:
275:
255:
235:
211:
191:
171:
161:a vector field on
151:
131:
111:
94:
90:Hamiltonian system
71:hairy ball theorem
48:Hopf index theorem
1141:978-3-642-05205-7
934:De Rham's theorem
918:differential form
870:{\displaystyle M}
817:{\displaystyle v}
736:
719:{\displaystyle v}
695:{\displaystyle M}
675:{\displaystyle M}
651:{\displaystyle v}
624:{\displaystyle M}
499:{\displaystyle D}
382:{\displaystyle x}
362:{\displaystyle v}
338:{\displaystyle D}
318:{\displaystyle v}
298:{\displaystyle x}
278:{\displaystyle x}
258:{\displaystyle D}
238:{\displaystyle x}
223:local coordinates
214:{\displaystyle v}
194:{\displaystyle x}
174:{\displaystyle M}
154:{\displaystyle v}
134:{\displaystyle n}
114:{\displaystyle M}
16:(Redirected from
1171:
1145:
1126:
1104:
1097:
1091:
1088:
1023:
1011:
876:
874:
873:
868:
852:
850:
849:
844:
823:
821:
820:
815:
800:
798:
797:
792:
762:
761:
760:
759:
744:
725:
723:
722:
717:
701:
699:
698:
693:
681:
679:
678:
673:
657:
655:
654:
649:
630:
628:
627:
622:
604:
602:
601:
596:
576:
537:
535:
534:
529:
505:
503:
502:
497:
481:
479:
478:
473:
471:
470:
459:
427:
425:
424:
419:
405:
404:
388:
386:
385:
380:
368:
366:
365:
360:
344:
342:
341:
336:
324:
322:
321:
316:
304:
302:
301:
296:
284:
282:
281:
276:
264:
262:
261:
256:
244:
242:
241:
236:
220:
218:
217:
212:
200:
198:
197:
192:
180:
178:
177:
172:
160:
158:
157:
152:
140:
138:
137:
132:
120:
118:
117:
112:
97:Formal statement
21:
1179:
1178:
1174:
1173:
1172:
1170:
1169:
1168:
1149:
1148:
1142:
1111:
1108:
1107:
1098:
1094:
1089:
1085:
1080:
1063:
1050:
1022:–1)-dimensional
1017:
1010:–1)-dimensional
1005:
1003:
989:
982:
958:
956:Sketch of proof
910:Stokes' theorem
894:
862:
859:
858:
829:
826:
825:
809:
806:
805:
755:
751:
750:
746:
740:
734:
731:
730:
711:
708:
707:
687:
684:
683:
667:
664:
663:
643:
640:
639:
616:
613:
612:
572:
543:
540:
539:
511:
508:
507:
491:
488:
487:
460:
455:
454:
437:
434:
433:
400:
396:
394:
391:
390:
374:
371:
370:
354:
351:
350:
330:
327:
326:
310:
307:
306:
290:
287:
286:
270:
267:
266:
250:
247:
246:
230:
227:
226:
221:, and fix some
206:
203:
202:
186:
183:
182:
181:. Suppose that
166:
163:
162:
146:
143:
142:
126:
123:
122:
106:
103:
102:
99:
28:
23:
22:
15:
12:
11:
5:
1177:
1167:
1166:
1161:
1147:
1146:
1140:
1127:
1106:
1105:
1092:
1082:
1081:
1079:
1076:
1075:
1074:
1069:
1062:
1059:
1049:
1048:Generalization
1046:
1045:
1044:
1025:
1001:
991:
987:
980:
970:
957:
954:
893:
890:
882:Henri Poincaré
866:
842:
839:
836:
833:
813:
802:
801:
789:
786:
783:
780:
777:
774:
771:
768:
765:
758:
754:
749:
743:
739:
715:
691:
671:
647:
620:
594:
591:
588:
585:
582:
579:
575:
571:
568:
565:
562:
559:
556:
553:
550:
547:
527:
524:
521:
518:
515:
495:
469:
466:
463:
458:
453:
450:
447:
444:
441:
417:
414:
411:
408:
403:
399:
378:
358:
334:
314:
294:
274:
254:
234:
210:
190:
170:
150:
130:
110:
98:
95:
56:Henri Poincaré
26:
9:
6:
4:
3:
2:
1176:
1165:
1162:
1160:
1157:
1156:
1154:
1143:
1137:
1133:
1128:
1124:
1120:
1119:
1114:
1110:
1109:
1102:
1096:
1087:
1083:
1073:
1070:
1068:
1065:
1064:
1058:
1056:
1042:
1038:
1037:triangulation
1034:
1030:
1026:
1021:
1015:
1009:
1000:
996:
992:
986:
979:
975:
971:
968:
964:
960:
959:
953:
951:
947:
943:
939:
935:
931:
927:
923:
919:
915:
911:
907:
903:
899:
889:
887:
883:
878:
864:
856:
837:
831:
811:
784:
778:
775:
769:
763:
756:
752:
747:
741:
737:
729:
728:
727:
713:
705:
689:
669:
661:
645:
637:
634:
618:
610:
606:
586:
580:
573:
566:
560:
557:
551:
545:
522:
519:
516:
493:
485:
467:
464:
461:
448:
442:
439:
431:
412:
406:
401:
397:
376:
356:
348:
332:
312:
292:
272:
252:
232:
224:
208:
188:
168:
148:
128:
108:
91:
86:
82:
80:
76:
72:
68:
67:Poincaré–Hopf
63:
61:
57:
53:
49:
45:
41:
37:
33:
19:
1131:
1116:
1100:
1095:
1086:
1072:Hopf theorem
1051:
1040:
1032:
1028:
1019:
1013:
1007:
998:
984:
977:
973:
962:
895:
892:Significance
879:
803:
660:vector field
608:
607:
265:centered at
100:
75:vector field
66:
64:
47:
43:
39:
35:
29:
906:integration
898:topological
432:of the map
345:. Then the
32:mathematics
1153:Categories
1078:References
946:analytical
886:Heinz Hopf
285:, so that
60:Heinz Hopf
1123:EMS Press
995:Gauss map
942:geometric
832:χ
779:χ
764:
738:∑
593:‖
578:‖
520:−
482:from the
465:−
452:→
446:∂
407:
1061:See also
1035:using a
950:physical
922:manifold
902:analytic
704:boundary
609:Theorem.
484:boundary
79:n-sphere
1125:, 2001
1004:to the
926:integer
853:is the
638:. Let
633:compact
506:to the
1138:
961:Embed
430:degree
141:, and
34:, the
916:of a
748:index
658:be a
631:be a
398:index
347:index
225:near
46:, or
1136:ISBN
944:and
824:and
702:has
611:Let
101:Let
65:The
58:and
1057:).
1039:of
948:or
857:of
662:on
486:of
369:at
349:of
325:in
30:In
1155::
1121:,
1115:,
1101:96
969:.)
936:,
932:,
888:.
605:.
389:,
62:.
42:,
1144:.
1053:(
1041:M
1033:M
1029:M
1020:n
1018:(
1014:M
1008:n
1006:(
1002:ε
999:N
988:ε
985:N
981:ε
978:N
974:M
963:M
865:M
841:)
838:M
835:(
812:v
788:)
785:M
782:(
776:=
773:)
770:v
767:(
757:i
753:x
742:i
714:v
690:M
670:M
646:v
619:M
590:)
587:z
584:(
581:v
574:/
570:)
567:z
564:(
561:v
558:=
555:)
552:z
549:(
546:u
526:)
523:1
517:n
514:(
494:D
468:1
462:n
457:S
449:D
443::
440:u
416:)
413:v
410:(
402:x
377:x
357:v
333:D
313:v
293:x
273:x
253:D
233:x
209:v
189:x
169:M
149:v
129:n
109:M
92:)
20:)
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