1927:
1002:
1758:
1237:
1446:
714:
441:
261:
584:
1292:
1971:
1615:
1092:
637:
885:
849:
813:
777:
185:
145:
96:
1771:
893:
1328:
1641:
652:
if its curvature vanishes: Ω = 0. Equivalently, a connection is flat if the structure group can be reduced to the same underlying group but with the discrete topology.
1181:
1158:
1135:
737:
284:
320:
1646:
668:
is a vector bundle, then one can also think of ω as a matrix of 1-forms and the above formula becomes the structure equation of E. Cartan:
1949:
2033:
2080:
1189:
1347:
2168:
2105:
2001:
2055:
674:
340:
2247:
197:
1976:
1966:
1451:
1247:
1106:
331:
1991:
1463:
479:
2135:
2075:
1259:
2242:
2011:
2198:
2173:
2095:
1986:
1459:
1112:
1028:
598:
48:
1485:
2145:
2026:
1037:
323:
123:
2150:
2140:
2047:
1024:
604:
28:
854:
818:
782:
746:
1922:{\displaystyle \sigma \Omega (X,Y)=\sigma d\omega (X,Y)=X\omega (Y)-Y\omega (X)-\omega ()=-\omega ().}
997:{\displaystyle \Omega _{j}^{i}=d{\omega ^{i}}_{j}+\sum _{k}{\omega ^{i}}_{k}\wedge {\omega ^{k}}_{j}.}
166:
126:
115:
77:
2188:
2160:
2115:
1941:
1012:
640:
287:
52:
1303:
2120:
2070:
2019:
1953:
648:
1620:
1996:
597:, on the right we identified a vertical vector field and a Lie algebra element generating it (
149:
1166:
1143:
1120:
722:
2183:
2085:
1338:
107:
44:
2178:
2090:
1981:
1455:
1334:
1161:
1138:
40:
2211:
2125:
1945:
1008:
740:
269:
293:
2236:
2130:
639:
is the inverse of the normalization factor used by convention in the formula for the
2006:
2216:
72:
17:
1753:{\displaystyle d\omega (X,Y)={\frac {1}{2}}(X\omega (Y)-Y\omega (X)-\omega ())}
1643:
Kobayashi convention for the exterior derivative of a one form which is then
2065:
2043:
68:
36:
2221:
1952:, Vol.I, Chapter 2.5 Curvature form and structure equation, p 75,
1097:
using the standard notation for the
Riemannian curvature tensor.
1232:{\displaystyle \Theta =d\theta +\omega \wedge \theta =D\theta ,}
2015:
1441:{\displaystyle R_{abmn;\ell }+R_{ab\ell m;n}+R_{abn\ell ;m}=0.}
1137:
is the canonical vector-valued 1-form on the frame bundle, the
1027:. In this case the form Ω is an alternative description of the
1183:
is the vector-valued 2-form defined by the structure equation
1344:
The
Bianchi identities can be written in tensor notation as:
1972:
Basic introduction to the mathematics of curved spacetime
1113:
Riemann curvature tensor § Symmetries and identities
709:{\displaystyle \,\Omega =d\omega +\omega \wedge \omega ,}
1019:) and Ω is a 2-form with values in the Lie algebra of O(
436:{\displaystyle \,\Omega (X,Y)=d\omega (X,Y)+{1 \over 2}}
815:
denote components of ω and Ω correspondingly, (so each
256:{\displaystyle \Omega =d\omega +{1 \over 2}=D\omega .}
1774:
1649:
1623:
1488:
1350:
1306:
1262:
1192:
1169:
1146:
1123:
1040:
896:
857:
821:
785:
749:
725:
677:
607:
482:
343:
296:
272:
200:
169:
129:
80:
2197:
2159:
2104:
2054:
266:(In another convention, 1/2 does not appear.) Here
1921:
1752:
1635:
1609:
1440:
1322:
1286:
1231:
1175:
1152:
1129:
1086:
996:
879:
843:
807:
771:
731:
708:
631:
579:{\displaystyle \sigma \Omega (X,Y)=-\omega ()=-+h}
578:
435:
314:
278:
255:
179:
139:
90:
1287:{\displaystyle D\Theta =\Omega \wedge \theta .}
2027:
8:
626:
614:
1297:The second Bianchi identity takes the form
461:There is also another expression for Ω: if
2034:
2020:
2012:
1253:The first Bianchi identity takes the form
1773:
1674:
1648:
1622:
1522:
1487:
1411:
1383:
1355:
1349:
1307:
1305:
1261:
1191:
1168:
1145:
1122:
1041:
1039:
985:
978:
973:
963:
956:
951:
944:
931:
924:
919:
906:
901:
895:
871:
864:
859:
856:
835:
828:
823:
820:
799:
792:
787:
784:
763:
756:
751:
748:
724:
678:
676:
606:
481:
390:
344:
342:
295:
271:
216:
199:
171:
170:
168:
131:
130:
128:
82:
81:
79:
1610:{\displaystyle (X,Y)={\frac {1}{2}}(-)}
1475:
1087:{\displaystyle \,R(X,Y)=\Omega (X,Y),}
55:can be considered as a special case.
7:
1950:Foundations of Differential Geometry
1333:and is valid more generally for any
172:
132:
83:
2081:Radius of curvature (applications)
1778:
1311:
1272:
1266:
1193:
1147:
1063:
898:
861:
789:
679:
632:{\displaystyle \sigma \in \{1,2\}}
593:means the horizontal component of
486:
345:
201:
25:
2169:Curvature of Riemannian manifolds
2002:Curvature of Riemannian manifolds
880:{\displaystyle {\Omega ^{i}}_{j}}
844:{\displaystyle {\omega ^{i}}_{j}}
808:{\displaystyle {\Omega ^{i}}_{j}}
772:{\displaystyle {\omega ^{i}}_{j}}
656:Curvature form in a vector bundle
469:are horizontal vector fields on
180:{\displaystyle {\mathfrak {g}}}
140:{\displaystyle {\mathfrak {g}}}
91:{\displaystyle {\mathfrak {g}}}
1913:
1910:
1898:
1895:
1883:
1880:
1868:
1865:
1856:
1850:
1838:
1832:
1820:
1808:
1793:
1781:
1747:
1744:
1741:
1729:
1726:
1717:
1711:
1699:
1693:
1684:
1668:
1656:
1604:
1601:
1598:
1592:
1583:
1577:
1571:
1565:
1562:
1556:
1547:
1541:
1535:
1532:
1516:
1504:
1501:
1489:
1078:
1066:
1057:
1045:
573:
561:
552:
540:
531:
528:
516:
513:
501:
489:
430:
427:
421:
412:
406:
400:
384:
372:
360:
348:
309:
297:
238:
226:
1:
1977:Contracted Bianchi identities
1967:Connection (principal bundle)
1452:contracted Bianchi identities
1248:exterior covariant derivative
1107:Contracted Bianchi identities
332:exterior covariant derivative
1992:General theory of relativity
1464:general theory of relativity
1323:{\displaystyle \,D\Omega =0}
1015:, the structure group is O(
851:is a usual 1-form and each
646:A connection is said to be
322:is defined in the article "
2264:
1110:
1104:
1636:{\displaystyle \sigma =2}
2199:Curvature of connections
2174:Riemann curvature tensor
2096:Total absolute curvature
1987:Einstein field equations
1460:Einstein field equations
887:is a usual 2-form) then
599:fundamental vector field
49:Riemann curvature tensor
2146:Second fundamental form
2136:Gauss–Codazzi equations
1617:. Here we use also the
1454:are used to derive the
1176:{\displaystyle \omega }
1153:{\displaystyle \Theta }
1130:{\displaystyle \theta }
732:{\displaystyle \wedge }
454:are tangent vectors to
324:Lie algebra-valued form
2151:Third fundamental form
2141:First fundamental form
2106:Differential geometry
2076:Frenet–Serret formulas
2056:Differential geometry
1923:
1754:
1637:
1611:
1442:
1324:
1288:
1233:
1177:
1154:
1131:
1088:
1025:antisymmetric matrices
998:
881:
845:
809:
773:
733:
710:
633:
580:
437:
316:
280:
257:
181:
141:
92:
2248:Differential geometry
2048:differential geometry
1924:
1755:
1638:
1612:
1443:
1325:
1289:
1234:
1178:
1155:
1132:
1089:
1007:For example, for the
999:
882:
846:
810:
774:
743:. More precisely, if
734:
711:
634:
581:
438:
317:
281:
258:
182:
142:
93:
29:differential geometry
2116:Principal curvatures
1772:
1647:
1621:
1486:
1348:
1304:
1260:
1190:
1167:
1144:
1121:
1038:
894:
855:
819:
783:
747:
723:
675:
605:
480:
341:
294:
270:
198:
167:
127:
116:Ehresmann connection
78:
2189:Sectional curvature
2161:Riemannian geometry
2042:Various notions of
1942:Shoshichi Kobayashi
1013:Riemannian manifold
911:
641:exterior derivative
288:exterior derivative
53:Riemannian geometry
2121:Gaussian curvature
2071:Torsion of a curve
1954:Wiley Interscience
1919:
1750:
1633:
1607:
1438:
1320:
1284:
1229:
1173:
1150:
1127:
1101:Bianchi identities
1084:
994:
949:
897:
877:
841:
805:
769:
729:
706:
629:
576:
433:
334:. In other terms,
312:
276:
253:
187:-valued 2-form on
177:
137:
88:
2243:Curvature tensors
2230:
2229:
1997:Chern-Simons form
1682:
1530:
940:
398:
279:{\displaystyle d}
224:
16:(Redirected from
2255:
2184:Scalar curvature
2086:Affine curvature
2036:
2029:
2022:
2013:
1929:
1928:
1926:
1925:
1920:
1766:
1760:
1759:
1757:
1756:
1751:
1683:
1675:
1642:
1640:
1639:
1634:
1616:
1614:
1613:
1608:
1531:
1523:
1480:
1447:
1445:
1444:
1439:
1431:
1430:
1403:
1402:
1375:
1374:
1339:principal bundle
1329:
1327:
1326:
1321:
1293:
1291:
1290:
1285:
1238:
1236:
1235:
1230:
1182:
1180:
1179:
1174:
1159:
1157:
1156:
1151:
1136:
1134:
1133:
1128:
1093:
1091:
1090:
1085:
1029:curvature tensor
1003:
1001:
1000:
995:
990:
989:
984:
983:
982:
968:
967:
962:
961:
960:
948:
936:
935:
930:
929:
928:
910:
905:
886:
884:
883:
878:
876:
875:
870:
869:
868:
850:
848:
847:
842:
840:
839:
834:
833:
832:
814:
812:
811:
806:
804:
803:
798:
797:
796:
778:
776:
775:
770:
768:
767:
762:
761:
760:
738:
736:
735:
730:
715:
713:
712:
707:
638:
636:
635:
630:
585:
583:
582:
577:
442:
440:
439:
434:
399:
391:
321:
319:
318:
315:{\displaystyle }
313:
285:
283:
282:
277:
262:
260:
259:
254:
225:
217:
186:
184:
183:
178:
176:
175:
146:
144:
143:
138:
136:
135:
97:
95:
94:
89:
87:
86:
45:principal bundle
21:
18:Bianchi identity
2263:
2262:
2258:
2257:
2256:
2254:
2253:
2252:
2233:
2232:
2231:
2226:
2193:
2179:Ricci curvature
2155:
2107:
2100:
2091:Total curvature
2057:
2050:
2040:
1982:Einstein tensor
1963:
1938:
1933:
1932:
1770:
1769:
1767:
1763:
1645:
1644:
1619:
1618:
1484:
1483:
1481:
1477:
1472:
1456:Einstein tensor
1407:
1379:
1351:
1346:
1345:
1302:
1301:
1258:
1257:
1242:where as above
1188:
1187:
1165:
1164:
1162:connection form
1142:
1141:
1119:
1118:
1115:
1109:
1103:
1036:
1035:
974:
972:
952:
950:
920:
918:
892:
891:
860:
858:
853:
852:
824:
822:
817:
816:
788:
786:
781:
780:
752:
750:
745:
744:
721:
720:
673:
672:
658:
603:
602:
478:
477:
339:
338:
292:
291:
268:
267:
196:
195:
165:
164:
125:
124:
114:. Let ω be an
76:
75:
61:
23:
22:
15:
12:
11:
5:
2261:
2259:
2251:
2250:
2245:
2235:
2234:
2228:
2227:
2225:
2224:
2219:
2214:
2212:Torsion tensor
2209:
2207:Curvature form
2203:
2201:
2195:
2194:
2192:
2191:
2186:
2181:
2176:
2171:
2165:
2163:
2157:
2156:
2154:
2153:
2148:
2143:
2138:
2133:
2128:
2126:Mean curvature
2123:
2118:
2112:
2110:
2102:
2101:
2099:
2098:
2093:
2088:
2083:
2078:
2073:
2068:
2062:
2060:
2052:
2051:
2041:
2039:
2038:
2031:
2024:
2016:
2010:
2009:
2004:
1999:
1994:
1989:
1984:
1979:
1974:
1969:
1962:
1959:
1958:
1957:
1946:Katsumi Nomizu
1937:
1934:
1931:
1930:
1918:
1915:
1912:
1909:
1906:
1903:
1900:
1897:
1894:
1891:
1888:
1885:
1882:
1879:
1876:
1873:
1870:
1867:
1864:
1861:
1858:
1855:
1852:
1849:
1846:
1843:
1840:
1837:
1834:
1831:
1828:
1825:
1822:
1819:
1816:
1813:
1810:
1807:
1804:
1801:
1798:
1795:
1792:
1789:
1786:
1783:
1780:
1777:
1761:
1749:
1746:
1743:
1740:
1737:
1734:
1731:
1728:
1725:
1722:
1719:
1716:
1713:
1710:
1707:
1704:
1701:
1698:
1695:
1692:
1689:
1686:
1681:
1678:
1673:
1670:
1667:
1664:
1661:
1658:
1655:
1652:
1632:
1629:
1626:
1606:
1603:
1600:
1597:
1594:
1591:
1588:
1585:
1582:
1579:
1576:
1573:
1570:
1567:
1564:
1561:
1558:
1555:
1552:
1549:
1546:
1543:
1540:
1537:
1534:
1529:
1526:
1521:
1518:
1515:
1512:
1509:
1506:
1503:
1500:
1497:
1494:
1491:
1474:
1473:
1471:
1468:
1462:, the bulk of
1437:
1434:
1429:
1426:
1423:
1420:
1417:
1414:
1410:
1406:
1401:
1398:
1395:
1392:
1389:
1386:
1382:
1378:
1373:
1370:
1367:
1364:
1361:
1358:
1354:
1331:
1330:
1319:
1316:
1313:
1310:
1295:
1294:
1283:
1280:
1277:
1274:
1271:
1268:
1265:
1240:
1239:
1228:
1225:
1222:
1219:
1216:
1213:
1210:
1207:
1204:
1201:
1198:
1195:
1172:
1149:
1126:
1102:
1099:
1095:
1094:
1083:
1080:
1077:
1074:
1071:
1068:
1065:
1062:
1059:
1056:
1053:
1050:
1047:
1044:
1009:tangent bundle
1005:
1004:
993:
988:
981:
977:
971:
966:
959:
955:
947:
943:
939:
934:
927:
923:
917:
914:
909:
904:
900:
874:
867:
863:
838:
831:
827:
802:
795:
791:
766:
759:
755:
728:
717:
716:
705:
702:
699:
696:
693:
690:
687:
684:
681:
657:
654:
628:
625:
622:
619:
616:
613:
610:
587:
586:
575:
572:
569:
566:
563:
560:
557:
554:
551:
548:
545:
542:
539:
536:
533:
530:
527:
524:
521:
518:
515:
512:
509:
506:
503:
500:
497:
494:
491:
488:
485:
444:
443:
432:
429:
426:
423:
420:
417:
414:
411:
408:
405:
402:
397:
394:
389:
386:
383:
380:
377:
374:
371:
368:
365:
362:
359:
356:
353:
350:
347:
311:
308:
305:
302:
299:
275:
264:
263:
252:
249:
246:
243:
240:
237:
234:
231:
228:
223:
220:
215:
212:
209:
206:
203:
174:
161:curvature form
134:
85:
60:
57:
33:curvature form
24:
14:
13:
10:
9:
6:
4:
3:
2:
2260:
2249:
2246:
2244:
2241:
2240:
2238:
2223:
2220:
2218:
2215:
2213:
2210:
2208:
2205:
2204:
2202:
2200:
2196:
2190:
2187:
2185:
2182:
2180:
2177:
2175:
2172:
2170:
2167:
2166:
2164:
2162:
2158:
2152:
2149:
2147:
2144:
2142:
2139:
2137:
2134:
2132:
2131:Darboux frame
2129:
2127:
2124:
2122:
2119:
2117:
2114:
2113:
2111:
2109:
2103:
2097:
2094:
2092:
2089:
2087:
2084:
2082:
2079:
2077:
2074:
2072:
2069:
2067:
2064:
2063:
2061:
2059:
2053:
2049:
2045:
2037:
2032:
2030:
2025:
2023:
2018:
2017:
2014:
2008:
2005:
2003:
2000:
1998:
1995:
1993:
1990:
1988:
1985:
1983:
1980:
1978:
1975:
1973:
1970:
1968:
1965:
1964:
1960:
1955:
1951:
1947:
1943:
1940:
1939:
1935:
1916:
1907:
1904:
1901:
1892:
1889:
1886:
1877:
1874:
1871:
1862:
1859:
1853:
1847:
1844:
1841:
1835:
1829:
1826:
1823:
1817:
1814:
1811:
1805:
1802:
1799:
1796:
1790:
1787:
1784:
1775:
1765:
1762:
1738:
1735:
1732:
1723:
1720:
1714:
1708:
1705:
1702:
1696:
1690:
1687:
1679:
1676:
1671:
1665:
1662:
1659:
1653:
1650:
1630:
1627:
1624:
1595:
1589:
1586:
1580:
1574:
1568:
1559:
1553:
1550:
1544:
1538:
1527:
1524:
1519:
1513:
1510:
1507:
1498:
1495:
1492:
1479:
1476:
1469:
1467:
1465:
1461:
1457:
1453:
1448:
1435:
1432:
1427:
1424:
1421:
1418:
1415:
1412:
1408:
1404:
1399:
1396:
1393:
1390:
1387:
1384:
1380:
1376:
1371:
1368:
1365:
1362:
1359:
1356:
1352:
1342:
1340:
1336:
1317:
1314:
1308:
1300:
1299:
1298:
1281:
1278:
1275:
1269:
1263:
1256:
1255:
1254:
1251:
1249:
1245:
1226:
1223:
1220:
1217:
1214:
1211:
1208:
1205:
1202:
1199:
1196:
1186:
1185:
1184:
1170:
1163:
1140:
1124:
1114:
1108:
1100:
1098:
1081:
1075:
1072:
1069:
1060:
1054:
1051:
1048:
1042:
1034:
1033:
1032:
1030:
1026:
1022:
1018:
1014:
1010:
991:
986:
979:
975:
969:
964:
957:
953:
945:
941:
937:
932:
925:
921:
915:
912:
907:
902:
890:
889:
888:
872:
865:
836:
829:
825:
800:
793:
764:
757:
753:
742:
741:wedge product
726:
703:
700:
697:
694:
691:
688:
685:
682:
671:
670:
669:
667:
663:
655:
653:
651:
650:
644:
642:
623:
620:
617:
611:
608:
600:
596:
592:
570:
567:
564:
558:
555:
549:
546:
543:
537:
534:
525:
522:
519:
510:
507:
504:
498:
495:
492:
483:
476:
475:
474:
472:
468:
464:
459:
457:
453:
449:
424:
418:
415:
409:
403:
395:
392:
387:
381:
378:
375:
369:
366:
363:
357:
354:
351:
337:
336:
335:
333:
329:
325:
306:
303:
300:
289:
273:
250:
247:
244:
241:
235:
232:
229:
221:
218:
213:
210:
207:
204:
194:
193:
192:
190:
162:
157:
155:
151:
148:
121:
117:
113:
111:
105:
101:
74:
70:
66:
58:
56:
54:
50:
46:
42:
38:
34:
30:
19:
2206:
2007:Gauge theory
1764:
1478:
1449:
1343:
1332:
1296:
1252:
1246:denotes the
1243:
1241:
1116:
1096:
1023:), i.e. the
1020:
1016:
1006:
718:
665:
661:
659:
647:
645:
594:
590:
588:
470:
466:
462:
460:
455:
451:
447:
445:
330:denotes the
327:
265:
188:
160:
158:
153:
122:(which is a
119:
109:
103:
99:
64:
62:
32:
26:
2217:Cocurvature
2108:of surfaces
2046:defined in
286:stands for
191:defined by
73:Lie algebra
2237:Categories
1936:References
1335:connection
1111:See also:
1105:See also:
108:principal
59:Definition
41:connection
35:describes
2066:Curvature
2058:of curves
2044:curvature
1893:ω
1890:−
1863:ω
1860:−
1848:ω
1842:−
1830:ω
1806:ω
1800:σ
1779:Ω
1776:σ
1724:ω
1721:−
1709:ω
1703:−
1691:ω
1654:ω
1625:σ
1590:ω
1575:ω
1569:−
1554:ω
1539:ω
1499:ω
1496:∧
1493:ω
1422:ℓ
1391:ℓ
1372:ℓ
1312:Ω
1279:θ
1276:∧
1273:Ω
1267:Θ
1224:θ
1215:θ
1212:∧
1209:ω
1203:θ
1194:Θ
1171:ω
1148:Θ
1125:θ
1064:Ω
976:ω
970:∧
954:ω
942:∑
922:ω
899:Ω
862:Ω
826:ω
790:Ω
754:ω
727:∧
701:ω
698:∧
695:ω
689:ω
680:Ω
612:∈
609:σ
538:−
511:ω
508:−
487:Ω
484:σ
419:ω
404:ω
370:ω
346:Ω
307:⋅
304:∧
301:⋅
248:ω
236:ω
233:∧
230:ω
211:ω
202:Ω
159:Then the
69:Lie group
37:curvature
2222:Holonomy
1961:See also
150:one-form
1948:(1963)
1768:Proof:
1458:in the
1160:of the
1139:torsion
1031:, i.e.
739:is the
601:), and
473:, then
163:is the
147:-valued
112:-bundle
1482:since
719:where
589:where
446:where
326:" and
98:, and
47:. The
31:, the
1470:Notes
1337:in a
1011:of a
106:be a
71:with
67:be a
43:on a
39:of a
1944:and
1450:The
779:and
649:flat
156:).
63:Let
1117:If
660:If
152:on
118:on
51:in
27:In
2239::
1466:.
1436:0.
1341:.
1250:.
664:→
643:.
591:hZ
465:,
458:.
450:,
290:,
102:→
2035:e
2028:t
2021:v
1956:.
1917:.
1914:)
1911:]
1908:Y
1905:,
1902:X
1899:[
1896:(
1887:=
1884:)
1881:]
1878:Y
1875:,
1872:X
1869:[
1866:(
1857:)
1854:X
1851:(
1845:Y
1839:)
1836:Y
1833:(
1827:X
1824:=
1821:)
1818:Y
1815:,
1812:X
1809:(
1803:d
1797:=
1794:)
1791:Y
1788:,
1785:X
1782:(
1748:)
1745:)
1742:]
1739:Y
1736:,
1733:X
1730:[
1727:(
1718:)
1715:X
1712:(
1706:Y
1700:)
1697:Y
1694:(
1688:X
1685:(
1680:2
1677:1
1672:=
1669:)
1666:Y
1663:,
1660:X
1657:(
1651:d
1631:2
1628:=
1605:)
1602:]
1599:)
1596:X
1593:(
1587:,
1584:)
1581:Y
1578:(
1572:[
1566:]
1563:)
1560:Y
1557:(
1551:,
1548:)
1545:X
1542:(
1536:[
1533:(
1528:2
1525:1
1520:=
1517:)
1514:Y
1511:,
1508:X
1505:(
1502:]
1490:[
1433:=
1428:m
1425:;
1419:n
1416:b
1413:a
1409:R
1405:+
1400:n
1397:;
1394:m
1388:b
1385:a
1381:R
1377:+
1369:;
1366:n
1363:m
1360:b
1357:a
1353:R
1318:0
1315:=
1309:D
1282:.
1270:=
1264:D
1244:D
1227:,
1221:D
1218:=
1206:+
1200:d
1197:=
1082:,
1079:)
1076:Y
1073:,
1070:X
1067:(
1061:=
1058:)
1055:Y
1052:,
1049:X
1046:(
1043:R
1021:n
1017:n
992:.
987:j
980:k
965:k
958:i
946:k
938:+
933:j
926:i
916:d
913:=
908:i
903:j
873:j
866:i
837:j
830:i
801:j
794:i
765:j
758:i
704:,
692:+
686:d
683:=
666:B
662:E
627:}
624:2
621:,
618:1
615:{
595:Z
574:]
571:Y
568:,
565:X
562:[
559:h
556:+
553:]
550:Y
547:,
544:X
541:[
535:=
532:)
529:]
526:Y
523:,
520:X
517:[
514:(
505:=
502:)
499:Y
496:,
493:X
490:(
471:P
467:Y
463:X
456:P
452:Y
448:X
431:]
428:)
425:Y
422:(
416:,
413:)
410:X
407:(
401:[
396:2
393:1
388:+
385:)
382:Y
379:,
376:X
373:(
367:d
364:=
361:)
358:Y
355:,
352:X
349:(
328:D
310:]
298:[
274:d
251:.
245:D
242:=
239:]
227:[
222:2
219:1
214:+
208:d
205:=
189:P
173:g
154:P
133:g
120:P
110:G
104:B
100:P
84:g
65:G
20:)
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