1298:
31:
2041:). Note that this defines the Levi-Civita connection without making any explicit reference to any metric tensor (although the metric tensor can be understood to be a special case of a solder form, as it establishes a mapping between the tangent and cotangent bundles of the base space, i.e. between the horizontal and vertical subspaces of the frame bundle).
1076:. Excluding trivial cases, there are an infinite number of horizontal subspaces at each point. Also note that arbitrary choices of horizontal space at each point will not, in general, form a smooth vector bundle; they must also vary in an appropriately smooth way.
1929:, that is vanishing in the vertical bundle, and is such that ω+σ is another connection 1-form that is torsion-free. The resulting one-form ω+σ is nothing other than the Levi-Civita connection. One can take this as a definition: since the torsion is given by
1886:
vanishes on the horizontal bundle, and is non-zero only on the vertical bundle. In this way, the connection form can be used to define the horizontal bundle: The horizontal bundle is the kernel of the connection
1977:
2027:
1720:
806:. The name is motivated by low-dimensional examples like the trivial line bundle over a circle, which is sometimes depicted as a vertical cylinder projecting to a horizontal circle. A subspace
624:
1825:
1140:
335:
194:
1877:
1636:
1074:
562:
724:
138:
659:
2071:
1750:
1361:
1028:
962:
932:
898:
864:
834:
784:
754:
520:
494:
458:
421:
391:
361:
78:
105:
679:
1898:
vanishes on the vertical bundle and is non-zero only on the horizontal bundle. By definition, the solder form takes its values entirely in the horizontal bundle.
270:
240:
217:
1331:
804:
290:
52:
1034:, their disjoint union is a horizontal bundle. The use of the words "the" and "a" here is intentional: each vertical subspace is unique, defined explicitly by
140:, the vertical fiber is unique. It is the tangent space to the fiber. The horizontal fiber is non-unique. It merely has to be transverse to the vertical fiber.
1333:
on the strip, the projection map projects it towards the middle ring, and the fiber is perpendicular to the middle ring. The vertical bundle at this point
460:. The vertical bundle consists of all vectors that are tangent to the fibers, while the horizontal bundle requires some choice of complementary subbundle.
1913:, i.e. to make a connection be torsionless. Indeed, if one writes θ for the solder form, then the torsion tensor Θ is given by Θ = D θ (with D the
1909:
vanishes on the vertical bundle, and can be used to define exactly that part that needs to be added to an arbitrary connection to turn it into a
1541:
In both cases, the product structure gives a natural choice of horizontal bundle, and hence an
Ehresmann connection: the horizontal bundle of
2139:
2150:
1932:
2246:
2218:
2196:
2163:
2080:
1297:
1575:
take on specific properties on the vertical and horizontal bundles, or even can be defined in terms of them. Some of these are:
1982:
2236:
1641:
1914:
1104:
901:
567:
1421:. Applying the definition in the paragraph above to find the vertical bundle, we consider first a point (m,n) in
1763:
2049:
1118:
1170:
2241:
2034:
1910:
424:
299:
2123:
1895:
1572:
1209:
167:
1830:
1598:
1231:
1080:
1037:
525:
684:
2155:
110:
1313:
over the circle, and the circle can be pictured as the middle ring of the strip. At each point
629:
2214:
2192:
2159:
2135:
2076:
1926:
1730:
1568:
1367:
1735:
1336:
1092:
1003:
937:
907:
873:
839:
809:
759:
729:
499:
469:
433:
396:
366:
340:
57:
2029:, and it is not hard to show that σ must vanish on the vertical bundle, and that σ must be
83:
1883:
1163:
664:
1306:
252:
222:
199:
2207:
2128:
1316:
968:
789:
275:
247:
37:
2230:
428:
157:
2174:
30:
1906:
1902:
1584:
1112:
1084:
161:
2052:
must necessarily live in the vertical bundle, and vanish in any horizontal bundle.
2099:
1891:
1310:
1182:
293:
145:
17:
1259:
1224:
661:) is a linear surjection whose kernel has the same dimension as the fibers of
243:
1371:
1564:
2033:-invariant on each fibre (more precisely, that σ transforms in the
1296:
29:
1250:, called the horizontal bundle of the connection. At each point
1972:{\displaystyle \Theta =D\theta =d\theta +\omega \wedge \theta }
1079:
The horizontal bundle is one way to formulate the notion of an
2022:{\displaystyle d\theta =-(\omega +\sigma )\wedge \theta }
1587:
that is in the vertical bundle. That is, for each point
1979:, the vanishing of the torsion is equivalent to having
1715:{\displaystyle V_{e}E\subset T_{e}E=T_{e}(E_{\pi (e)})}
1301:
Vertical and horizontal subspaces for the Möbius strip.
1115:
associated to some vector bundle, which is a principal
1099:, then the horizontal bundle is usually required to be
1985:
1935:
1833:
1766:
1738:
1644:
1601:
1339:
1319:
1121:
1040:
1006:
940:
910:
876:
842:
812:
792:
762:
732:
687:
667:
632:
570:
528:
502:
472:
436:
399:
369:
343:
302:
278:
255:
225:
202:
170:
113:
86:
60:
40:
2206:
2127:
2021:
1971:
1871:
1819:
1744:
1714:
1630:
1355:
1325:
1134:
1068:
1022:
956:
926:
892:
858:
828:
798:
778:
748:
718:
673:
653:
618:
556:
514:
488:
452:
415:
385:
355:
329:
284:
264:
234:
211:
188:
132:
99:
72:
46:
2173:Kolář, Ivan; Michor, Peter; Slovák, Jan (1993),
2098:Kolář, Ivan; Michor, Peter; Slovák, Jan (1993),
619:{\displaystyle d\pi _{e}\colon T_{e}E\to T_{b}B}
1366:A simple example of a smooth fiber bundle is a
34:Here, we have a fiber bundle over a base space
2148:Kobayashi, Shoshichi; Nomizu, Katsumi (1996).
164:. More precisely, given a smooth fiber bundle
1103:-invariant: such a choice is equivalent to a
8:
2151:Foundations of Differential Geometry, Vol. 1
1000:. Likewise, provided the horizontal spaces
2176:Natural Operations in Differential Geometry
2101:Natural Operations in Differential Geometry
1917:). For any given connection ω, there is a
1820:{\displaystyle \alpha (v_{1},...,v_{r})=0}
1433:is m. The preimage of m under this same pr
1238:is a choice of a complementary subbundle H
2075:(1981) Addison-Wesely Publishing Company
1984:
1934:
1863:
1838:
1832:
1802:
1777:
1765:
1737:
1694:
1681:
1665:
1649:
1643:
1619:
1606:
1600:
1344:
1338:
1318:
1126:
1120:
1057:
1039:
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1005:
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731:
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374:
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342:
301:
277:
254:
224:
201:
169:
124:
112:
91:
85:
59:
39:
2187:Krupka, Demeter; Janyška, Josef (1990),
2093:
2091:
2072:Gauge Theory and Variational Principles
2062:
1429:. Then the image of this point under pr
1135:{\displaystyle \operatorname {GL} _{n}}
107:of points. At each point in the fiber
1827:whenever at least one of the vectors
1476:). If we take the other projection pr
7:
1523:) then the vertical bundle will be V
1219:. Furthermore, the vertical bundle V
2191:, Univerzita J. E. Purkyně V Brně,
2189:Lectures on differential invariants
1363:is the tangent space to the fiber.
756:consists of exactly the vectors in
337:. This means that, over each point
330:{\displaystyle VE\oplus HE\cong TE}
1936:
1105:connection on the principal bundle
25:
2126:; DeWitt-Morette, Cécile (1977),
463:To make this precise, define the
189:{\displaystyle \pi \colon E\to B}
2048:is a principal bundle, then the
1722:is the vertical vector space at
1162:be a smooth fiber bundle over a
2130:Analysis, Manifolds and Physics
1872:{\displaystyle v_{1},...,v_{r}}
1631:{\displaystyle v_{e}\in V_{e}E}
1453:. The vertical bundle is then V
1069:{\displaystyle \ker(d\pi _{e})}
996:this is the vertical bundle of
557:{\displaystyle \ker(d\pi _{e})}
2213:, Cambridge University Press,
2010:
1998:
1808:
1770:
1709:
1704:
1698:
1687:
1063:
1047:
719:{\displaystyle F=\pi ^{-1}(b)}
713:
707:
648:
642:
600:
551:
535:
180:
1:
1915:exterior covariant derivative
1169:. The vertical bundle is the
1468:, which is a subbundle of T(
1204:is surjective at each point
564:. That is, the differential
2209:The geometry of jet bundles
1504:to define the fiber bundle
1393:) with bundle projection pr
1258:, the two subspaces form a
1107:. This notably occurs when
2263:
1548:is the vertical bundle of
786:which are also tangent to
133:{\displaystyle p\in p_{x}}
1087:. Thus, for example, if
654:{\displaystyle b=\pi (e)}
2247:Connection (mathematics)
2050:fundamental vector field
971:of the vertical spaces V
2205:Saunders, D.J. (1989),
2134:, Amsterdam: Elsevier,
1745:{\displaystyle \alpha }
1595:, one chooses a vector
425:complementary subspaces
80:corresponds to a fiber
2124:Choquet-Bruhat, Yvonne
2035:adjoint representation
2023:
1973:
1911:Levi-Civita connection
1873:
1821:
1746:
1716:
1632:
1374:. Consider the bundle
1357:
1356:{\displaystyle V_{e}E}
1327:
1302:
1136:
1070:
1024:
1023:{\displaystyle H_{e}E}
958:
957:{\displaystyle H_{e}E}
928:
927:{\displaystyle V_{e}E}
894:
893:{\displaystyle T_{e}E}
860:
859:{\displaystyle T_{e}E}
830:
829:{\displaystyle H_{e}E}
800:
780:
779:{\displaystyle T_{e}E}
750:
749:{\displaystyle V_{e}E}
720:
675:
655:
620:
558:
516:
515:{\displaystyle e\in E}
490:
489:{\displaystyle V_{e}E}
454:
453:{\displaystyle T_{e}E}
417:
416:{\displaystyle H_{e}E}
387:
386:{\displaystyle V_{e}E}
357:
356:{\displaystyle e\in E}
331:
286:
266:
236:
219:and horizontal bundle
213:
196:, the vertical bundle
190:
141:
134:
101:
74:
73:{\displaystyle x\in X}
48:
2237:Differential topology
2024:
1974:
1896:tautological one-form
1874:
1822:
1747:
1717:
1633:
1581:vertical vector field
1573:differential geometry
1358:
1328:
1300:
1137:
1071:
1025:
959:
929:
895:
861:
831:
801:
781:
751:
721:
676:
656:
621:
559:
517:
491:
455:
418:
388:
358:
332:
287:
267:
237:
214:
191:
135:
102:
100:{\displaystyle p_{x}}
75:
49:
33:
1983:
1933:
1831:
1764:
1736:
1642:
1599:
1337:
1317:
1232:Ehresmann connection
1119:
1081:Ehresmann connection
1038:
1004:
938:
908:
874:
840:
810:
790:
760:
730:
685:
674:{\displaystyle \pi }
665:
630:
568:
526:
500:
470:
434:
397:
367:
341:
300:
276:
253:
223:
200:
168:
111:
84:
58:
38:
2084:(See theorem 1.2.4)
1177: := ker(d
1030:vary smoothly with
162:smooth fiber bundle
27:Mathematics concept
2156:Wiley Interscience
2044:In the case where
2019:
1969:
1901:For the case of a
1869:
1817:
1742:
1712:
1628:
1569:differential forms
1563:Various important
1534: ×
1484: ×
1385: ×
1353:
1323:
1303:
1132:
1066:
1020:
988:is the subbundle V
954:
924:
890:
856:
826:
796:
776:
746:
716:
671:
651:
616:
554:
512:
486:
450:
413:
383:
353:
327:
282:
265:{\displaystyle TE}
262:
235:{\displaystyle HE}
232:
212:{\displaystyle VE}
209:
186:
142:
130:
97:
70:
44:
2182:, Springer-Verlag
2141:978-0-7204-0494-4
2107:, Springer-Verlag
1927:contorsion tensor
1729:A differentiable
1368:Cartesian product
1326:{\displaystyle e}
1146:Formal definition
799:{\displaystyle F}
285:{\displaystyle E}
154:horizontal bundle
54:. Each bas point
47:{\displaystyle X}
16:(Redirected from
2254:
2223:
2212:
2201:
2183:
2181:
2169:
2154:(New ed.).
2144:
2133:
2110:
2108:
2106:
2095:
2086:
2069:David Bleecker,
2067:
2028:
2026:
2025:
2020:
1978:
1976:
1975:
1970:
1878:
1876:
1875:
1870:
1868:
1867:
1843:
1842:
1826:
1824:
1823:
1818:
1807:
1806:
1782:
1781:
1756:is said to be a
1751:
1749:
1748:
1743:
1721:
1719:
1718:
1713:
1708:
1707:
1686:
1685:
1670:
1669:
1654:
1653:
1637:
1635:
1634:
1629:
1624:
1623:
1611:
1610:
1555:and vice versa.
1449:) = {m} × T
1362:
1360:
1359:
1354:
1349:
1348:
1332:
1330:
1329:
1324:
1141:
1139:
1138:
1133:
1131:
1130:
1075:
1073:
1072:
1067:
1062:
1061:
1029:
1027:
1026:
1021:
1016:
1015:
963:
961:
960:
955:
950:
949:
933:
931:
930:
925:
920:
919:
899:
897:
896:
891:
886:
885:
868:horizontal space
865:
863:
862:
857:
852:
851:
835:
833:
832:
827:
822:
821:
805:
803:
802:
797:
785:
783:
782:
777:
772:
771:
755:
753:
752:
747:
742:
741:
725:
723:
722:
717:
706:
705:
680:
678:
677:
672:
660:
658:
657:
652:
625:
623:
622:
617:
612:
611:
596:
595:
583:
582:
563:
561:
560:
555:
550:
549:
521:
519:
518:
513:
495:
493:
492:
487:
482:
481:
459:
457:
456:
451:
446:
445:
422:
420:
419:
414:
409:
408:
392:
390:
389:
384:
379:
378:
362:
360:
359:
354:
336:
334:
333:
328:
291:
289:
288:
283:
271:
269:
268:
263:
241:
239:
238:
233:
218:
216:
215:
210:
195:
193:
192:
187:
160:associated to a
139:
137:
136:
131:
129:
128:
106:
104:
103:
98:
96:
95:
79:
77:
76:
71:
53:
51:
50:
45:
21:
2262:
2261:
2257:
2256:
2255:
2253:
2252:
2251:
2227:
2226:
2221:
2204:
2199:
2186:
2179:
2172:
2166:
2147:
2142:
2122:
2119:
2114:
2113:
2104:
2097:
2096:
2089:
2068:
2064:
2059:
1981:
1980:
1931:
1930:
1921:one-form σ on T
1884:connection form
1859:
1834:
1829:
1828:
1798:
1773:
1762:
1761:
1758:horizontal form
1734:
1733:
1690:
1677:
1661:
1645:
1640:
1639:
1615:
1602:
1597:
1596:
1561:
1554:
1547:
1529:
1522:
1510:
1479:
1459:
1444:
1437:is {m} ×
1436:
1432:
1396:
1392:
1381: := (
1380:
1340:
1335:
1334:
1315:
1314:
1295:
1285:
1276:
1267:
1262:, such that T
1203:
1164:smooth manifold
1148:
1122:
1117:
1116:
1053:
1036:
1035:
1007:
1002:
1001:
976:
941:
936:
935:
911:
906:
905:
877:
872:
871:
843:
838:
837:
813:
808:
807:
788:
787:
763:
758:
757:
733:
728:
727:
694:
683:
682:
663:
662:
628:
627:
603:
587:
574:
566:
565:
541:
524:
523:
498:
497:
473:
468:
467:
437:
432:
431:
400:
395:
394:
370:
365:
364:
339:
338:
298:
297:
274:
273:
251:
250:
221:
220:
198:
197:
166:
165:
150:vertical bundle
120:
109:
108:
87:
82:
81:
56:
55:
36:
35:
28:
23:
22:
18:Vertical bundle
15:
12:
11:
5:
2260:
2258:
2250:
2249:
2244:
2239:
2229:
2228:
2225:
2224:
2219:
2202:
2197:
2184:
2170:
2164:
2145:
2140:
2118:
2115:
2112:
2111:
2087:
2061:
2060:
2058:
2055:
2054:
2053:
2042:
2018:
2015:
2012:
2009:
2006:
2003:
2000:
1997:
1994:
1991:
1988:
1968:
1965:
1962:
1959:
1956:
1953:
1950:
1947:
1944:
1941:
1938:
1899:
1888:
1880:
1866:
1862:
1858:
1855:
1852:
1849:
1846:
1841:
1837:
1816:
1813:
1810:
1805:
1801:
1797:
1794:
1791:
1788:
1785:
1780:
1776:
1772:
1769:
1741:
1727:
1711:
1706:
1703:
1700:
1697:
1693:
1689:
1684:
1680:
1676:
1673:
1668:
1664:
1660:
1657:
1652:
1648:
1627:
1622:
1618:
1614:
1609:
1605:
1560:
1557:
1552:
1545:
1527:
1520:
1508:
1500:) →
1492: : (
1477:
1457:
1442:
1434:
1430:
1417:) →
1409: : (
1394:
1390:
1378:
1352:
1347:
1343:
1322:
1294:
1291:
1281:
1272:
1263:
1208:, it yields a
1201:
1193: → T
1189: : T
1147:
1144:
1129:
1125:
1065:
1060:
1056:
1052:
1049:
1046:
1043:
1019:
1014:
1010:
972:
969:disjoint union
953:
948:
944:
923:
918:
914:
889:
884:
880:
855:
850:
846:
825:
820:
816:
795:
775:
770:
766:
745:
740:
736:
715:
712:
709:
704:
701:
697:
693:
690:
681:. If we write
670:
650:
647:
644:
641:
638:
635:
615:
610:
606:
602:
599:
594:
590:
586:
581:
577:
573:
553:
548:
544:
540:
537:
534:
531:
511:
508:
505:
485:
480:
476:
465:vertical space
449:
444:
440:
412:
407:
403:
382:
377:
373:
352:
349:
346:
326:
323:
320:
317:
314:
311:
308:
305:
281:
261:
258:
248:tangent bundle
231:
228:
208:
205:
185:
182:
179:
176:
173:
158:vector bundles
127:
123:
119:
116:
94:
90:
69:
66:
63:
43:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2259:
2248:
2245:
2243:
2242:Fiber bundles
2240:
2238:
2235:
2234:
2232:
2222:
2220:0-521-36948-7
2216:
2211:
2210:
2203:
2200:
2198:80-210-0165-8
2194:
2190:
2185:
2178:
2177:
2171:
2167:
2165:0-471-15733-3
2161:
2157:
2153:
2152:
2146:
2143:
2137:
2132:
2131:
2125:
2121:
2120:
2116:
2103:
2102:
2094:
2092:
2088:
2085:
2082:
2081:0-201-10096-7
2078:
2074:
2073:
2066:
2063:
2056:
2051:
2047:
2043:
2040:
2036:
2032:
2016:
2013:
2007:
2004:
2001:
1995:
1992:
1989:
1986:
1966:
1963:
1960:
1957:
1954:
1951:
1948:
1945:
1942:
1939:
1928:
1925:, called the
1924:
1920:
1916:
1912:
1908:
1904:
1900:
1897:
1893:
1889:
1885:
1881:
1864:
1860:
1856:
1853:
1850:
1847:
1844:
1839:
1835:
1814:
1811:
1803:
1799:
1795:
1792:
1789:
1786:
1783:
1778:
1774:
1767:
1759:
1755:
1739:
1732:
1728:
1725:
1701:
1695:
1691:
1682:
1678:
1674:
1671:
1666:
1662:
1658:
1655:
1650:
1646:
1625:
1620:
1616:
1612:
1607:
1603:
1594:
1590:
1586:
1582:
1578:
1577:
1576:
1574:
1570:
1566:
1558:
1556:
1551:
1544:
1539:
1537:
1533:
1526:
1518:
1514:
1507:
1503:
1499:
1495:
1491:
1488: →
1487:
1483:
1475:
1471:
1467:
1463:
1456:
1452:
1448:
1445:({m} ×
1440:
1428:
1424:
1420:
1416:
1412:
1408:
1404:
1400:
1388:
1384:
1377:
1373:
1369:
1364:
1350:
1345:
1341:
1320:
1312:
1308:
1299:
1292:
1290:
1288:
1284:
1279:
1275:
1270:
1266:
1261:
1257:
1253:
1249:
1245:
1241:
1237:
1233:
1228:
1226:
1222:
1218:
1214:
1212:
1207:
1198:
1196:
1192:
1188:
1184:
1180:
1176:
1172:
1168:
1165:
1161:
1157:
1153:
1145:
1143:
1127:
1123:
1114:
1110:
1106:
1102:
1098:
1096:
1090:
1086:
1082:
1077:
1058:
1054:
1050:
1044:
1041:
1033:
1017:
1012:
1008:
999:
995:
991:
987:
983:
979:
975:
970:
965:
951:
946:
942:
921:
916:
912:
903:
887:
882:
878:
869:
853:
848:
844:
823:
818:
814:
793:
773:
768:
764:
743:
738:
734:
710:
702:
699:
695:
691:
688:
668:
645:
639:
636:
633:
613:
608:
604:
597:
592:
588:
584:
579:
575:
571:
546:
542:
538:
532:
529:
509:
506:
503:
483:
478:
474:
466:
461:
447:
442:
438:
430:
429:tangent space
426:
410:
405:
401:
380:
375:
371:
363:, the fibers
350:
347:
344:
324:
321:
318:
315:
312:
309:
306:
303:
295:
279:
259:
256:
249:
245:
229:
226:
206:
203:
183:
177:
174:
171:
163:
159:
155:
151:
147:
125:
121:
117:
114:
92:
88:
67:
64:
61:
41:
32:
19:
2208:
2188:
2175:
2149:
2129:
2100:
2083:
2070:
2065:
2045:
2038:
2030:
1922:
1918:
1907:torsion form
1903:frame bundle
1879:is vertical.
1757:
1753:
1723:
1592:
1588:
1585:vector field
1580:
1562:
1549:
1542:
1540:
1535:
1531:
1524:
1516:
1512:
1505:
1501:
1497:
1493:
1489:
1485:
1481:
1473:
1469:
1465:
1461:
1454:
1450:
1446:
1438:
1426:
1422:
1418:
1414:
1410:
1406:
1402:
1398:
1386:
1382:
1375:
1365:
1307:Möbius strip
1304:
1286:
1282:
1277:
1273:
1268:
1264:
1255:
1251:
1247:
1243:
1239:
1235:
1229:
1220:
1216:
1210:
1205:
1199:
1194:
1190:
1186:
1178:
1174:
1166:
1159:
1155:
1151:
1149:
1113:frame bundle
1108:
1100:
1094:
1088:
1085:fiber bundle
1078:
1031:
997:
993:
989:
985:
981:
977:
973:
966:
867:
866:is called a
464:
462:
153:
149:
143:
1892:solder form
1441:, so that T
1311:line bundle
1183:tangent map
294:Whitney sum
146:mathematics
2231:Categories
2117:References
1559:Properties
1511: := (
1260:direct sum
1225:integrable
1093:principal
902:direct sum
296:satisfies
244:subbundles
2109:(page 77)
2017:θ
2014:∧
2008:σ
2002:ω
1996:−
1990:θ
1967:θ
1964:∧
1961:ω
1955:θ
1946:θ
1937:Θ
1768:α
1740:α
1696:π
1659:⊂
1613:∈
1372:manifolds
1213:subbundle
1181:) of the
1055:π
1045:
980:for each
700:−
696:π
669:π
640:π
601:→
585::
576:π
543:π
533:
507:∈
348:∈
319:≅
310:⊕
181:→
175::
172:π
118:∈
65:∈
1530:= T
1480: :
1464:× T
1397: :
1223:is also
1200:Since dπ
1142:bundle.
152:and the
1565:tensors
1515:×
1496:,
1425:×
1413:,
1401:×
1370:of two
1293:Example
1211:regular
1111:is the
1097:-bundle
900:is the
726:, then
626:(where
427:of the
246:of the
2217:
2195:
2162:
2138:
2079:
1919:unique
1905:, the
1731:r-form
1638:where
1472:×
1179:π
1171:kernel
522:to be
292:whose
148:, the
2180:(PDF)
2105:(PDF)
2057:Notes
1887:form.
1583:is a
1571:from
1519:, pr
1443:(m,n)
1389:, pr
1309:is a
1091:is a
1083:on a
423:form
2215:ISBN
2193:ISBN
2160:ISBN
2136:ISBN
2077:ISBN
1890:The
1882:The
1567:and
1305:The
1271:= V
1246:in T
1242:to V
1215:of T
1150:Let
992:of T
967:The
934:and
393:and
242:are
156:are
2037:of
1894:or
1760:if
1752:on
1591:of
1280:⊕ H
1254:in
1234:on
1230:An
1042:ker
984:in
904:of
870:if
836:of
530:ker
496:at
272:of
144:In
2233::
2158:.
2090:^
1579:A
1538:.
1460:=
1405:→
1289:.
1227:.
1197:.
1124:GL
994:E;
964:.
2168:.
2046:E
2039:G
2031:G
2011:)
2005:+
1999:(
1993:=
1987:d
1958:+
1952:d
1949:=
1943:D
1940:=
1923:E
1865:r
1861:v
1857:,
1854:.
1851:.
1848:.
1845:,
1840:1
1836:v
1815:0
1812:=
1809:)
1804:r
1800:v
1796:,
1793:.
1790:.
1787:.
1784:,
1779:1
1775:v
1771:(
1754:E
1726:.
1724:e
1710:)
1705:)
1702:e
1699:(
1692:E
1688:(
1683:e
1679:T
1675:=
1672:E
1667:e
1663:T
1656:E
1651:e
1647:V
1626:E
1621:e
1617:V
1608:e
1604:v
1593:E
1589:e
1553:2
1550:B
1546:1
1543:B
1536:N
1532:M
1528:2
1525:B
1521:2
1517:N
1513:M
1509:2
1506:B
1502:y
1498:y
1494:x
1490:N
1486:N
1482:M
1478:2
1474:N
1470:M
1466:N
1462:M
1458:1
1455:B
1451:N
1447:N
1439:N
1435:1
1431:1
1427:N
1423:M
1419:x
1415:y
1411:x
1407:M
1403:N
1399:M
1395:1
1391:1
1387:N
1383:M
1379:1
1376:B
1351:E
1346:e
1342:V
1321:e
1287:E
1283:e
1278:E
1274:e
1269:E
1265:e
1256:E
1252:e
1248:E
1244:E
1240:E
1236:E
1221:E
1217:E
1206:e
1202:e
1195:B
1191:E
1187:π
1185:d
1175:E
1173:V
1167:B
1160:B
1158:→
1156:E
1154::
1152:π
1128:n
1109:E
1101:G
1095:G
1089:E
1064:)
1059:e
1051:d
1048:(
1032:e
1018:E
1013:e
1009:H
998:E
990:E
986:E
982:e
978:E
974:e
952:E
947:e
943:H
922:E
917:e
913:V
888:E
883:e
879:T
854:E
849:e
845:T
824:E
819:e
815:H
794:F
774:E
769:e
765:T
744:E
739:e
735:V
714:)
711:b
708:(
703:1
692:=
689:F
649:)
646:e
643:(
637:=
634:b
614:B
609:b
605:T
598:E
593:e
589:T
580:e
572:d
552:)
547:e
539:d
536:(
510:E
504:e
484:E
479:e
475:V
448:E
443:e
439:T
411:E
406:e
402:H
381:E
376:e
372:V
351:E
345:e
325:E
322:T
316:E
313:H
307:E
304:V
280:E
260:E
257:T
230:E
227:H
207:E
204:V
184:B
178:E
126:x
122:p
115:p
93:x
89:p
68:X
62:x
42:X
20:)
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