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