594:
775:
843:
406:
362:
129:
994:
959:
221:
680:
498:
472:
654:
634:
446:
924:
714:
894:
541:
291:
248:
179:
867:
518:
426:
311:
268:
156:
93:
1891:
1082:
1886:
1173:
1197:
1392:
72:. In this section, we discuss how to describe an Ehresmann connection in sheaf-theoretic terms as a Grothendieck connection.
1262:
1488:
1541:
1069:
1825:
1934:
546:
1590:
1573:
1182:
21:
722:
780:
1785:
1192:
370:
49:
1770:
1493:
1267:
1939:
1815:
777:
given by projection the respective factors of the
Cartesian product, which restrict to give projections
1820:
1790:
1498:
1454:
1435:
1202:
1146:
364:
The problem is thus to obtain an intrinsic description of the sheaf of sections of this vector bundle.
41:
316:
1357:
1222:
1020: â Function which tells how a certain variable changes as it moves along certain points in space
1742:
1607:
1299:
1141:
1017:
135:
29:
1439:
1409:
1333:
1323:
1279:
1109:
1062:
102:
964:
929:
184:
1780:
1399:
1294:
1207:
1114:
659:
477:
451:
96:
1429:
1424:
639:
606:
431:
69:
899:
689:
1760:
1698:
1546:
1250:
1240:
1212:
1187:
1097:
872:
683:
61:
45:
8:
1898:
1580:
1458:
1443:
1372:
1131:
293:
With the latter interpretation, an
Ehresmann connection is a section of the bundle (over
132:
65:
1871:
523:
273:
230:
161:
1840:
1795:
1692:
1563:
1367:
1055:
852:
503:
411:
296:
253:
141:
78:
17:
1377:
64:
of algebraic geometry. Thus the connection in a certain sense must live in a natural
1775:
1755:
1750:
1657:
1568:
1382:
1362:
1217:
1156:
1001:
1913:
1707:
1662:
1585:
1556:
1414:
1347:
1342:
1337:
1327:
1119:
1102:
597:
1856:
1765:
1595:
1551:
1317:
846:
1722:
1647:
1617:
1515:
1508:
1448:
1419:
1289:
1284:
1245:
1000:
is a specified isomorphism between these two spaces. One may proceed to define
1928:
1908:
1732:
1727:
1712:
1702:
1652:
1629:
1503:
1463:
1404:
1352:
1151:
32:
in terms of descent data from infinitesimal neighbourhoods of the diagonal.
1835:
1830:
1672:
1639:
1612:
1520:
1161:
1678:
1667:
1624:
1525:
1126:
1005:
500:
which vanish along the diagonal. Much of the infinitesimal geometry of
1903:
1861:
1687:
1600:
1232:
1136:
1047:
224:
57:
1717:
1682:
1387:
1274:
1881:
1876:
1866:
1257:
1078:
1473:
367:
Grothendieck's solution is to consider the diagonal embedding
1037:
Katz, N., "Nilpotent connections and the monodromy theorem",
1030:
Osserman, B., "Connections, curvature, and p-curvature",
52:. The construction itself must satisfy a requirement of
44:
constructed in a manner analogous to that in which the
40:
967:
932:
902:
875:
855:
783:
725:
692:
662:
642:
609:
549:
526:
506:
480:
454:
434:
414:
373:
319:
299:
276:
256:
233:
187:
164:
144:
105:
81:
988:
953:
926:In general, there is no canonical way to identify
918:
888:
861:
837:
769:
708:
674:
648:
628:
588:
535:
512:
492:
466:
440:
420:
400:
356:
305:
285:
262:
242:
215:
173:
150:
123:
87:
1926:
589:{\displaystyle \Delta ^{*}\left(I,I^{2}\right)}
60:for a wider class of structures including the
35:
1063:
716:(See below for a coordinate description.)
1070:
1056:
770:{\displaystyle p_{1},p_{2}:M\times M\to M}
838:{\displaystyle p_{1},p_{2}:M^{(2)}\to M.}
56:, which may be regarded as the analog of
1077:
401:{\displaystyle \Delta :M\to M\times M.}
1927:
1008:of a connection in the same language.
602:first-order infinitesimal neighborhood
250:This may be regarded as a bundle over
1051:
686:corresponding to the sheaf of ideals
270:or a bundle over the total space of
13:
643:
551:
435:
374:
14:
1951:
719:There are a pair of projections
596:is the sheaf of sections of the
357:{\displaystyle J^{1}(M,E)\to E.}
22:synthetic differential geometry
1110:Differentiable/Smooth manifold
826:
821:
815:
761:
621:
615:
383:
345:
342:
330:
210:
198:
115:
1:
1024:
520:can be realized in terms of
7:
1816:Classification of manifolds
1011:
124:{\displaystyle \pi :E\to M}
36:Introduction and motivation
10:
1956:
989:{\displaystyle p_{2}^{*}E}
954:{\displaystyle p_{1}^{*}E}
869:along one or the other of
216:{\displaystyle J^{1}(M,E)}
158:is a manifold fibred over
1892:over commutative algebras
1849:
1808:
1741:
1638:
1534:
1481:
1472:
1308:
1231:
1170:
1090:
675:{\displaystyle M\times M}
493:{\displaystyle M\times M}
474:consists of functions on
467:{\displaystyle M\times M}
1935:Connection (mathematics)
1608:Riemann curvature tensor
1018:Connection (mathematics)
998:Grothendieck connection
649:{\displaystyle \Delta }
629:{\displaystyle M^{(2)}}
441:{\displaystyle \Delta }
26:Grothendieck connection
1400:Manifold with boundary
1115:Differential structure
990:
955:
920:
919:{\displaystyle p_{2}.}
890:
863:
839:
771:
710:
709:{\displaystyle I^{2}.}
676:
650:
630:
590:
537:
514:
494:
468:
442:
422:
402:
358:
307:
287:
264:
244:
217:
175:
152:
125:
89:
42:GaussâManin connection
991:
956:
921:
891:
889:{\displaystyle p_{1}}
864:
845:One may now form the
840:
772:
711:
677:
651:
631:
591:
538:
515:
495:
469:
443:
423:
403:
359:
308:
288:
265:
245:
218:
176:
153:
126:
90:
70:Grothendieck topology
1547:Covariant derivative
1098:Topological manifold
996:with each other. A
965:
930:
900:
873:
853:
781:
723:
690:
660:
640:
607:
600:. One may define a
547:
524:
504:
478:
452:
432:
412:
371:
317:
297:
274:
254:
231:
185:
162:
142:
103:
79:
54:geometric invariance
46:Ehresmann connection
28:is a way of viewing
1581:Exterior derivative
1183:AtiyahâSinger index
1132:Riemannian manifold
982:
947:
849:of the fibre space
223:be the first-order
1940:Algebraic geometry
1887:Secondary calculus
1841:Singularity theory
1796:Parallel transport
1564:De Rham cohomology
1203:Generalized Stokes
986:
968:
951:
933:
916:
886:
859:
835:
767:
706:
672:
646:
626:
586:
536:{\displaystyle I.}
533:
510:
490:
464:
438:
418:
398:
354:
303:
286:{\displaystyle E.}
283:
260:
243:{\displaystyle E.}
240:
213:
174:{\displaystyle M.}
171:
148:
121:
85:
18:algebraic geometry
1922:
1921:
1804:
1803:
1569:Differential form
1223:Whitney embedding
1157:Differential form
862:{\displaystyle E}
513:{\displaystyle M}
421:{\displaystyle I}
306:{\displaystyle E}
263:{\displaystyle M}
151:{\displaystyle E}
88:{\displaystyle M}
50:Koszul connection
1947:
1914:Stratified space
1872:Fréchet manifold
1586:Interior product
1479:
1478:
1176:
1072:
1065:
1058:
1049:
1048:
1039:IHES Publ. Math.
995:
993:
992:
987:
981:
976:
960:
958:
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952:
946:
941:
925:
923:
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911:
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844:
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841:
836:
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793:
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748:
747:
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734:
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707:
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701:
681:
679:
678:
673:
655:
653:
652:
647:
635:
633:
632:
627:
625:
624:
598:cotangent bundle
595:
593:
592:
587:
585:
581:
580:
579:
559:
558:
542:
540:
539:
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519:
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473:
471:
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363:
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329:
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269:
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197:
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180:
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172:
157:
155:
154:
149:
130:
128:
127:
122:
94:
92:
91:
86:
48:generalizes the
1955:
1954:
1950:
1949:
1948:
1946:
1945:
1944:
1925:
1924:
1923:
1918:
1857:Banach manifold
1850:Generalizations
1845:
1800:
1737:
1634:
1596:Ricci curvature
1552:Cotangent space
1530:
1468:
1310:
1304:
1263:Exponential map
1227:
1172:
1166:
1086:
1076:
1044:(1970) 175â232.
1027:
1014:
977:
972:
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963:
962:
942:
937:
931:
928:
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814:
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521:
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453:
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433:
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429:
413:
410:
409:
372:
369:
368:
324:
320:
318:
315:
314:
298:
295:
294:
275:
272:
271:
255:
252:
251:
232:
229:
228:
227:of sections of
192:
188:
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182:
163:
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143:
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139:
104:
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100:
80:
77:
76:
38:
12:
11:
5:
1953:
1943:
1942:
1937:
1920:
1919:
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1911:
1906:
1901:
1896:
1895:
1894:
1884:
1879:
1874:
1869:
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1859:
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1801:
1799:
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1715:
1710:
1705:
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1660:
1655:
1650:
1644:
1642:
1636:
1635:
1633:
1632:
1627:
1622:
1621:
1620:
1610:
1605:
1604:
1603:
1593:
1588:
1583:
1578:
1577:
1576:
1566:
1561:
1560:
1559:
1549:
1544:
1538:
1536:
1532:
1531:
1529:
1528:
1523:
1518:
1513:
1512:
1511:
1501:
1496:
1491:
1485:
1483:
1476:
1470:
1469:
1467:
1466:
1461:
1451:
1446:
1432:
1427:
1422:
1417:
1412:
1410:Parallelizable
1407:
1402:
1397:
1396:
1395:
1385:
1380:
1375:
1370:
1365:
1360:
1355:
1350:
1345:
1340:
1330:
1320:
1314:
1312:
1306:
1305:
1303:
1302:
1297:
1292:
1290:Lie derivative
1287:
1285:Integral curve
1282:
1277:
1272:
1271:
1270:
1260:
1255:
1254:
1253:
1246:Diffeomorphism
1243:
1237:
1235:
1229:
1228:
1226:
1225:
1220:
1215:
1210:
1205:
1200:
1195:
1190:
1185:
1179:
1177:
1168:
1167:
1165:
1164:
1159:
1154:
1149:
1144:
1139:
1134:
1129:
1124:
1123:
1122:
1117:
1107:
1106:
1105:
1094:
1092:
1091:Basic concepts
1088:
1087:
1075:
1074:
1067:
1060:
1052:
1046:
1045:
1035:
1026:
1023:
1022:
1021:
1013:
1010:
985:
980:
975:
971:
950:
945:
940:
936:
915:
910:
906:
883:
879:
858:
834:
831:
828:
823:
820:
817:
813:
809:
804:
800:
796:
791:
787:
766:
763:
760:
757:
754:
751:
746:
742:
738:
733:
729:
705:
700:
696:
671:
668:
665:
645:
623:
620:
617:
613:
584:
578:
574:
570:
567:
563:
557:
553:
543:For instance,
532:
529:
509:
489:
486:
483:
463:
460:
457:
437:
417:
397:
394:
391:
388:
385:
382:
379:
376:
353:
350:
347:
344:
341:
338:
335:
332:
327:
323:
302:
282:
279:
259:
239:
236:
212:
209:
206:
203:
200:
195:
191:
170:
167:
147:
120:
117:
114:
111:
108:
84:
37:
34:
9:
6:
4:
3:
2:
1952:
1941:
1938:
1936:
1933:
1932:
1930:
1915:
1912:
1910:
1909:Supermanifold
1907:
1905:
1902:
1900:
1897:
1893:
1890:
1889:
1888:
1885:
1883:
1880:
1878:
1875:
1873:
1870:
1868:
1865:
1863:
1860:
1858:
1855:
1854:
1852:
1848:
1842:
1839:
1837:
1834:
1832:
1829:
1827:
1824:
1822:
1819:
1817:
1814:
1813:
1811:
1807:
1797:
1794:
1792:
1789:
1787:
1784:
1782:
1779:
1777:
1774:
1772:
1769:
1767:
1764:
1762:
1759:
1757:
1754:
1752:
1749:
1748:
1746:
1744:
1740:
1734:
1731:
1729:
1726:
1724:
1721:
1719:
1716:
1714:
1711:
1709:
1706:
1704:
1700:
1696:
1694:
1691:
1689:
1686:
1684:
1680:
1676:
1674:
1671:
1669:
1666:
1664:
1661:
1659:
1656:
1654:
1651:
1649:
1646:
1645:
1643:
1641:
1637:
1631:
1630:Wedge product
1628:
1626:
1623:
1619:
1616:
1615:
1614:
1611:
1609:
1606:
1602:
1599:
1598:
1597:
1594:
1592:
1589:
1587:
1584:
1582:
1579:
1575:
1574:Vector-valued
1572:
1571:
1570:
1567:
1565:
1562:
1558:
1555:
1554:
1553:
1550:
1548:
1545:
1543:
1540:
1539:
1537:
1533:
1527:
1524:
1522:
1519:
1517:
1514:
1510:
1507:
1506:
1505:
1504:Tangent space
1502:
1500:
1497:
1495:
1492:
1490:
1487:
1486:
1484:
1480:
1477:
1475:
1471:
1465:
1462:
1460:
1456:
1452:
1450:
1447:
1445:
1441:
1437:
1433:
1431:
1428:
1426:
1423:
1421:
1418:
1416:
1413:
1411:
1408:
1406:
1403:
1401:
1398:
1394:
1391:
1390:
1389:
1386:
1384:
1381:
1379:
1376:
1374:
1371:
1369:
1366:
1364:
1361:
1359:
1356:
1354:
1351:
1349:
1346:
1344:
1341:
1339:
1335:
1331:
1329:
1325:
1321:
1319:
1316:
1315:
1313:
1307:
1301:
1298:
1296:
1293:
1291:
1288:
1286:
1283:
1281:
1278:
1276:
1273:
1269:
1268:in Lie theory
1266:
1265:
1264:
1261:
1259:
1256:
1252:
1249:
1248:
1247:
1244:
1242:
1239:
1238:
1236:
1234:
1230:
1224:
1221:
1219:
1216:
1214:
1211:
1209:
1206:
1204:
1201:
1199:
1196:
1194:
1191:
1189:
1186:
1184:
1181:
1180:
1178:
1175:
1171:Main results
1169:
1163:
1160:
1158:
1155:
1153:
1152:Tangent space
1150:
1148:
1145:
1143:
1140:
1138:
1135:
1133:
1130:
1128:
1125:
1121:
1118:
1116:
1113:
1112:
1111:
1108:
1104:
1101:
1100:
1099:
1096:
1095:
1093:
1089:
1084:
1080:
1073:
1068:
1066:
1061:
1059:
1054:
1053:
1050:
1043:
1040:
1036:
1033:
1029:
1028:
1019:
1016:
1015:
1009:
1007:
1003:
999:
983:
978:
973:
969:
948:
943:
938:
934:
913:
908:
904:
881:
877:
856:
848:
832:
829:
818:
811:
807:
802:
798:
794:
789:
785:
764:
758:
755:
752:
749:
744:
740:
736:
731:
727:
717:
703:
698:
694:
685:
669:
666:
663:
618:
611:
603:
599:
582:
576:
572:
568:
565:
561:
555:
530:
527:
507:
487:
484:
481:
461:
458:
455:
428:of ideals of
415:
395:
392:
389:
386:
380:
377:
365:
351:
348:
339:
336:
333:
325:
321:
300:
280:
277:
257:
237:
234:
226:
207:
204:
201:
193:
189:
168:
165:
145:
137:
134:
118:
112:
109:
106:
98:
82:
73:
71:
67:
63:
59:
55:
51:
47:
43:
33:
31:
27:
23:
19:
1836:Moving frame
1831:Morse theory
1821:Gauge theory
1613:Tensor field
1542:Closed/Exact
1521:Vector field
1489:Distribution
1430:Hypercomplex
1425:Quaternionic
1162:Vector field
1120:Smooth atlas
1041:
1038:
1031:
997:
718:
601:
366:
74:
53:
39:
25:
15:
1781:Levi-Civita
1771:Generalized
1743:Connections
1693:Lie algebra
1625:Volume form
1526:Vector flow
1499:Pushforward
1494:Lie bracket
1393:Lie algebra
1358:G-structure
1147:Pushforward
1127:Submanifold
1006:p-curvature
30:connections
1929:Categories
1904:Stratifold
1862:Diffeology
1658:Associated
1459:Symplectic
1444:Riemannian
1373:Hyperbolic
1300:Submersion
1208:HopfâRinow
1142:Submersion
1137:Smooth map
1025:References
682:to be the
408:The sheaf
225:jet bundle
138:, so that
136:submersion
133:surjective
58:covariance
1786:Principal
1761:Ehresmann
1718:Subbundle
1708:Principal
1683:Fibration
1663:Cotangent
1535:Covectors
1388:Lie group
1368:Hermitian
1311:manifolds
1280:Immersion
1275:Foliation
1213:Noether's
1198:Frobenius
1193:De Rham's
1188:Darboux's
1079:Manifolds
1002:curvature
979:∗
944:∗
827:→
762:→
756:×
684:subscheme
667:×
644:Δ
556:∗
552:Δ
485:×
459:×
436:Δ
390:×
384:→
375:Δ
346:→
116:→
107:π
1882:Orbifold
1877:K-theory
1867:Diffiety
1591:Pullback
1405:Oriented
1383:Kenmotsu
1363:Hadamard
1309:Types of
1258:Geodesic
1083:Glossary
1032:preprint
1012:See also
847:pullback
97:manifold
1826:History
1809:Related
1723:Tangent
1701:)
1681:)
1648:Adjoint
1640:Bundles
1618:density
1516:Torsion
1482:Vectors
1474:Tensors
1457:)
1442:)
1438:,
1436:Pseudoâ
1415:Poisson
1348:Finsler
1343:Fibered
1338:Contact
1336:)
1328:Complex
1326:)
1295:Section
62:schemes
1791:Vector
1776:Koszul
1756:Cartan
1751:Affine
1733:Vector
1728:Tensor
1713:Spinor
1703:Normal
1699:Stable
1653:Affine
1557:bundle
1509:bundle
1455:Almost
1378:KĂ€hler
1334:Almost
1324:Almost
1318:Closed
1218:Sard's
1174:(list)
1899:Sheaf
1673:Fiber
1449:Rizza
1420:Prime
1251:Local
1241:Curve
1103:Atlas
95:be a
68:on a
66:sheaf
1766:Form
1668:Dual
1601:flow
1464:Tame
1440:Subâ
1353:Flat
1233:Maps
1004:and
961:and
181:Let
99:and
75:Let
24:, a
20:and
1688:Jet
896:or
656:in
636:of
448:in
16:In
1931::
1679:Co
1042:39
313:)
131:a
1697:(
1677:(
1453:(
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1332:(
1322:(
1085:)
1081:(
1071:e
1064:t
1057:v
1034:.
984:E
974:2
970:p
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939:1
935:p
914:.
909:2
905:p
882:1
878:p
857:E
833:.
830:M
822:)
819:2
816:(
812:M
808::
803:2
799:p
795:,
790:1
786:p
765:M
759:M
753:M
750::
745:2
741:p
737:,
732:1
728:p
704:.
699:2
695:I
670:M
664:M
622:)
619:2
616:(
612:M
583:)
577:2
573:I
569:,
566:I
562:(
531:.
528:I
508:M
488:M
482:M
462:M
456:M
416:I
396:.
393:M
387:M
381:M
378::
352:.
349:E
343:)
340:E
337:,
334:M
331:(
326:1
322:J
301:E
281:.
278:E
258:M
238:.
235:E
211:)
208:E
205:,
202:M
199:(
194:1
190:J
169:.
166:M
146:E
119:M
113:E
110::
83:M
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