84:
74:
53:
22:
521:
represent vector fields (sections of the tangent bundle) and so since dz-ydx is a one-form, it can be evaulated on these vectors fields. In fact, dz-ydx is zero when evaluated on these vector fields, and so at a point of the manifold, these vectors span a plane in the "kernel" of the one-form. When
166:
hoping to get a basic understanding of the topic. However, the first sentence was completely impenetrable to a mere mortal such as myself. I'm not a math genius by any means, but I did get through
Calculus 2 without too much trouble, so I'd expect to be able to glean something about the topic. The
945:
253:
I'm not so sure I would classify the result that all three-manifolds possess a contact structure as an application of contact geometry to low-dimensional topology. It's really an application of contact geometry to contact geometry. A true "application" should be used to prove something interesting
210:
There certainly are differences between contact and symplectic structures. The dimensions they exist for example. But this statement is about contact structures in dimension three and symplectic structures in any even dimension. If you restrict your attention to the two dimensional world, then there
177:
Thank you for your comment! Indeed, this article is far from accessible, and can be improved. It's a long-standing tradition here to begin an article with a one-sentence definition of the subject of the title. You should skip that first sentence, which is necessarily technical, and read on. Be aware
450:
There certainly are a few things that could be done to make some of the points more clear. It's good to have the perspective of someone new to the subject. I would recommend to you that you get a good book on differential geometry and study up if you are serious about leaning more about contact
290:
I'm trying to learn contact geometry, but I am having trouble with the section "Contact forms and structures". I understand few of the terms in the first part of the section, and there are no links to help with understanding. Terms like "kernel of a contact form", "hyperplane field", "symplectic
959:
need not exist globally. That is, there does not necessarily exist a globally defined 1-form whose kernel is the contact structure. However, if your contact structure is co-orientable then there is such a global 1-form. By co-orientable I mean that there is a nonzero transverse vector field or,
552:
You should be careful about trusting experts to recommend good books. :) In all seriousness, many of the "standard" texts of differential geometry tend to be very heavy and difficult to use for learning the subject for the first time. I would put my money on John Lee's
167:
only thing I understood was parallel parking, but I still have no idea what it's got to do with contact geometry. Can at least the first paragraph be dumbed down to avoid terms like "manifold", "hyperplane", "tangent bundle", "non-degeneracy", "non-integrability", etc.?
235:
Not to be too pedantic about it, but there is sort of a cohomological restriction to the existence of symplectic structures, even for two-dimensional manifolds. They do need to be orientable. But as I said above, I certainly agree that the sentence was misleading.
946:
http://books.google.com/books?id=RERR4zMDYRgC&pg=PA57&lpg=PA57&dq=Global+contact+structure&source=bl&ots=N9eS_srfup&sig=SJwHvyhXYArR8JWm_14FOfaiZtU&hl=en&ei=-Ua6SdfjDcPQkAX3g_ikCA&sa=X&oi=book_result&resnum=1&ct=result
1003:
Needs section on tight vs. overtwisted, among other things. The history could be expanded. Needs illustrations, better history, concise description of applications, wikilinks to the concepts used in definitions... I've downrated it to 'Start' class.
254:
about low-dimensional toplogy outside of the subset of facts already related to contact geometry. For example, Cerf's
Theorem (that any diffeomorphism of the 3-sphere extends to the 4-ball) was reproven by Eliahsberg using contact techniques.
561:. Everything in this book is done in two and three dimensions so you can visualize the results. That would prepare you to read Lee's book from the "geometry" point of view. You might need a bit of basic topology as well.
214:
I would suggest to delete this sentence or write a complete section about what the cohomological restrictions are . Either way, I would not declare this statement to be a difference between symplectic and contact geometry.
685:
522:
I get some time, I'll put some of these things on my to-do list. As you correctly point out, at the very least, some of these items should be linked to places where more information is available.
294:
I was hoping the part beginning with "As a prime example" would orient me, but I am still having trouble. I understand the 1-form dz-ydx, but it then says the contact plane is spanned by vectors
207:"One difference between contact and symplectic geometry is that every 3-manifold admits a contact structure while there are cohomological obstructions to the existence of symplectic structures."
388:
182:, probably, 4 or 5 advanced mathematics courses beyond basic calculus, so quite a bit more mathematical knowledge is needed to understand this and similar articles. You may want to try learning
211:
is no restriction on the existence of symplectic structures as well. On the other hand, there are cohomological restrictions to the existence of contact structures in dimensions above three !!
140:
519:
273:
Nice job! These are even better than my suggestion since they were not already theorems before contact geometry came along. Of course, it doesn't hurt that Gompf is your advisor. :)
332:
878:
814:
557:. I don't know your current level so I don't know how basic you need things. If you need a more undergraduate text as a prerequisite for Lee's book, you can try Do Carmo's
476:
777:
748:
408:
846:
435:
708:
1033:
130:
535:
Thanks - I'm big on having the right book. Are there one or two books on differential geometry that people agree are head and shoulders above the rest?
410:
symbols come from and what do they mean? Are they related to dx and dy somehow? Notice that the x and z variables are interchanged in the definition of
586:
by Roger Balian. It involves contact geometry and symplectic geometry, so if that changes any recommendation, let me know. Thanks again for your help.
1028:
106:
967:
932:
690:
Despite reasonable exposure to forms, I'm uncertain what the exponent is supposed to indicate here. Clearly it doesn't mean "apply
97:
58:
997:, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.
640:
33:
337:
907:
715:
481:
971:
21:
936:
163:
903:
711:
297:
291:
bundle". What is the difference between a "contact structure ε on a manifold" and a "contact manifold"?
179:
39:
960:
equivalently, that the quotient of the tangent bundle by the contact hyperplane field is trivial. ---
83:
963:
913:
183:
168:
851:
790:
454:
105:
on
Knowledge. If you would like to participate, please visit the project page, where you can join
89:
73:
52:
931:
I saw some books defined contact structure \alpha locally, i.e. \alpha need not exist globally.
757:
728:
393:
634:
I have seen here on
Knowledge and elsewhere the definition for the contact 1-form written as
889:
831:
612:
562:
523:
274:
265:
255:
237:
223:
413:
693:
1022:
994:
1016:
Last edited at 06:02, 30 May 2007 (UTC). Substituted at 01:55, 5 May 2016 (UTC)
1005:
885:
187:
102:
186:
first, and come back to contact geometry later (not necessarily on wikipedia).
437:
as compared to the definition of the 1 form. Is that correct, and if so, why?
79:
1008:
975:
940:
917:
893:
719:
615:
590:
565:
539:
526:
444:
277:
268:
258:
240:
226:
190:
171:
587:
536:
441:
583:
195:
P.S. The link at
Coefficient of friction was wrong, and I have fixed it.
787:
on even-degree forms, so the result is not necessarily 0 (unlike
440:
Can anyone write this with a few more clues to follow? Thanks -
902:
Thanks. Looks like it's been clarified in the article as well.
710:
n times," else the expression would vanish. What does it mean?
15:
611:
I'm going to carry this conversation into your talk page.
204:
I find the following statement in the article misleading.
582:
Ok, Thanks. The bottom line is I am trying to understand
989:
854:
834:
793:
760:
731:
696:
680:{\displaystyle \alpha \wedge (d\alpha )^{n}\neq 0.\ }
643:
484:
457:
416:
396:
340:
300:
101:, a collaborative effort to improve the coverage of
872:
840:
808:
771:
742:
702:
679:
513:
470:
429:
402:
383:{\displaystyle x_{2}=\partial _{x}+y\partial _{z}}
382:
326:
783:-form. Note that the exterior multiplication is
993:, and are posted here for posterity. Following
987:The comment(s) below were originally left at
8:
559:Differential Geometry of Curves and Surfaces
514:{\displaystyle \partial _{x}+y\partial _{z}}
19:
47:
859:
853:
833:
798:
792:
759:
730:
695:
663:
642:
505:
489:
483:
462:
456:
421:
415:
395:
374:
358:
345:
339:
318:
305:
299:
630:Notation in Definition of Contact Form?
49:
955:is a global structure, but a contact
222:I agree 100%. Consider it removed.
7:
95:This article is within the scope of
327:{\displaystyle x_{1}=\partial _{y}}
178:that contact geometry is a part of
38:It is of interest to the following
502:
486:
459:
397:
371:
355:
315:
14:
1034:Mid-priority mathematics articles
995:several discussions in past years
115:Knowledge:WikiProject Mathematics
1029:Start-Class mathematics articles
555:Introduction to Smooth Manifolds
118:Template:WikiProject Mathematics
82:
72:
51:
20:
135:This article has been rated as
990:Talk:Contact geometry/Comments
873:{\displaystyle d^{n}\alpha \ }
779:is a 2-form, the result is a 2
660:
650:
1:
809:{\displaystyle \alpha ^{n}\ }
616:08:58, 15 December 2006 (UTC)
591:05:32, 15 December 2006 (UTC)
566:04:12, 15 December 2006 (UTC)
540:23:23, 14 December 2006 (UTC)
527:20:34, 14 December 2006 (UTC)
471:{\displaystyle \partial _{y}}
445:02:09, 14 December 2006 (UTC)
109:and see a list of open tasks.
976:22:54, 4 December 2009 (UTC)
880:and indeed would vanish for
725:It means "exterior multiply
162:I reached this article from
1050:
941:11:46, 13 March 2009 (UTC)
848:" would have been denoted
772:{\displaystyle d\alpha \ }
743:{\displaystyle d\alpha \ }
241:05:11, 23 April 2007 (UTC)
227:03:10, 23 April 2007 (UTC)
1002:
918:19:29, 28 June 2008 (UTC)
894:00:52, 27 June 2008 (UTC)
720:05:25, 26 June 2008 (UTC)
403:{\displaystyle \partial }
278:00:24, 14 July 2006 (UTC)
269:20:58, 13 July 2006 (UTC)
259:09:45, 13 July 2006 (UTC)
191:01:40, 18 July 2007 (UTC)
172:20:42, 17 July 2007 (UTC)
134:
67:
46:
1009:06:02, 30 May 2007 (UTC)
451:geometry. For example,
141:project's priority scale
841:{\displaystyle \alpha }
164:Coefficient of friction
98:WikiProject Mathematics
874:
842:
810:
773:
744:
704:
681:
515:
472:
431:
404:
384:
328:
28:This article is rated
875:
843:
816:, which vanishes for
811:
774:
745:
705:
682:
516:
473:
432:
430:{\displaystyle x_{2}}
405:
385:
329:
180:differential geometry
852:
832:
791:
758:
729:
694:
641:
482:
455:
414:
394:
338:
298:
184:analytical mechanics
121:mathematics articles
264:Here's my attempt.
983:Assessment comment
870:
838:
806:
769:
740:
700:
677:
511:
468:
427:
400:
380:
324:
158:Hard to understand
90:Mathematics portal
34:content assessment
1014:
1013:
966:comment added by
920:
869:
805:
768:
739:
703:{\displaystyle d}
676:
155:
154:
151:
150:
147:
146:
1041:
1000:
999:
992:
978:
911:
879:
877:
876:
871:
868:
864:
863:
847:
845:
844:
839:
815:
813:
812:
807:
804:
803:
802:
778:
776:
775:
770:
767:
749:
747:
746:
741:
738:
709:
707:
706:
701:
686:
684:
683:
678:
675:
668:
667:
520:
518:
517:
512:
510:
509:
494:
493:
477:
475:
474:
469:
467:
466:
436:
434:
433:
428:
426:
425:
409:
407:
406:
401:
390:. Where did the
389:
387:
386:
381:
379:
378:
363:
362:
350:
349:
333:
331:
330:
325:
323:
322:
310:
309:
218:best wishes. S.
123:
122:
119:
116:
113:
92:
87:
86:
76:
69:
68:
63:
55:
48:
31:
25:
24:
16:
1049:
1048:
1044:
1043:
1042:
1040:
1039:
1038:
1019:
1018:
988:
985:
968:131.215.108.142
961:
929:
927:Local vs Global
904:Trevorgoodchild
855:
850:
849:
830:
829:
794:
789:
788:
756:
755:
727:
726:
712:Trevorgoodchild
692:
691:
659:
639:
638:
632:
501:
485:
480:
479:
458:
453:
452:
417:
412:
411:
392:
391:
370:
354:
341:
336:
335:
314:
301:
296:
295:
288:
251:
202:
160:
120:
117:
114:
111:
110:
88:
81:
61:
32:on Knowledge's
29:
12:
11:
5:
1047:
1045:
1037:
1036:
1031:
1021:
1020:
1012:
1011:
984:
981:
980:
979:
928:
925:
924:
923:
922:
921:
897:
896:
867:
862:
858:
837:
801:
797:
766:
763:
754:times". Since
737:
734:
699:
688:
687:
674:
671:
666:
662:
658:
655:
652:
649:
646:
631:
628:
627:
626:
625:
624:
623:
622:
621:
620:
619:
618:
600:
599:
598:
597:
596:
595:
594:
593:
573:
572:
571:
570:
569:
568:
545:
544:
543:
542:
530:
529:
508:
504:
500:
497:
492:
488:
465:
461:
424:
420:
399:
377:
373:
369:
366:
361:
357:
353:
348:
344:
321:
317:
313:
308:
304:
287:
284:
283:
282:
281:
280:
250:
247:
246:
245:
244:
243:
230:
229:
201:
198:
197:
196:
193:
159:
156:
153:
152:
149:
148:
145:
144:
133:
127:
126:
124:
107:the discussion
94:
93:
77:
65:
64:
56:
44:
43:
37:
26:
13:
10:
9:
6:
4:
3:
2:
1046:
1035:
1032:
1030:
1027:
1026:
1024:
1017:
1010:
1007:
1001:
998:
996:
991:
982:
977:
973:
969:
965:
958:
954:
950:
949:
948:
947:
943:
942:
938:
934:
933:211.99.194.53
926:
919:
916:was added at
915:
909:
905:
901:
900:
899:
898:
895:
891:
887:
883:
865:
860:
856:
835:
827:
823:
820:≥ 2). "Apply
819:
799:
795:
786:
782:
764:
761:
753:
735:
732:
724:
723:
722:
721:
717:
713:
697:
672:
669:
664:
656:
653:
647:
644:
637:
636:
635:
629:
617:
614:
610:
609:
608:
607:
606:
605:
604:
603:
602:
601:
592:
589:
585:
581:
580:
579:
578:
577:
576:
575:
574:
567:
564:
560:
556:
551:
550:
549:
548:
547:
546:
541:
538:
534:
533:
532:
531:
528:
525:
506:
498:
495:
490:
463:
449:
448:
447:
446:
443:
438:
422:
418:
375:
367:
364:
359:
351:
346:
342:
319:
311:
306:
302:
292:
285:
279:
276:
272:
271:
270:
267:
263:
262:
261:
260:
257:
248:
242:
239:
234:
233:
232:
231:
228:
225:
221:
220:
219:
216:
212:
208:
205:
199:
194:
192:
189:
185:
181:
176:
175:
174:
173:
170:
165:
157:
142:
138:
132:
129:
128:
125:
108:
104:
100:
99:
91:
85:
80:
78:
75:
71:
70:
66:
60:
57:
54:
50:
45:
41:
35:
27:
23:
18:
17:
1015:
986:
956:
952:
944:
930:
881:
825:
821:
817:
784:
780:
751:
689:
633:
558:
554:
439:
293:
289:
252:
249:Applications
217:
213:
209:
206:
203:
161:
137:Mid-priority
136:
96:
62:Mid‑priority
40:WikiProjects
962:—Preceding
912:—Preceding
785:commutative
613:VectorPosse
563:VectorPosse
524:VectorPosse
275:VectorPosse
266:Orthografer
256:VectorPosse
238:VectorPosse
224:VectorPosse
112:Mathematics
103:mathematics
59:Mathematics
30:Start-class
1023:Categories
951:A contact
750:by itself
584:this paper
200:Misleading
953:structure
169:JesterXXV
964:unsigned
914:comment
139:on the
1006:Arcfrk
886:Arcfrk
884:≥ 2.
824:times
188:Arcfrk
36:scale.
972:talk
957:form
937:talk
908:talk
890:talk
716:talk
478:and
334:and
286:Help
910:)
828:to
588:PAR
537:PAR
442:PAR
131:Mid
1025::
974:)
939:)
892:)
866:α
836:α
796:α
765:α
736:α
718:)
673:0.
670:≠
657:α
648:∧
645:α
503:∂
487:∂
460:∂
398:∂
372:∂
356:∂
316:∂
970:(
935:(
906:(
888:(
882:n
861:n
857:d
826:d
822:n
818:n
800:n
781:n
762:d
752:n
733:d
714:(
698:d
665:n
661:)
654:d
651:(
507:z
499:y
496:+
491:x
464:y
423:2
419:x
376:z
368:y
365:+
360:x
352:=
347:2
343:x
320:y
312:=
307:1
303:x
143:.
42::
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
