2231:
1005:
in 1902. These planes are characterized by the fact that they have a 4-dimensional automorphism group. They are not isomorphic to a smooth plane. More generally, all non-classical compact 2-dimensional planes
1309:
1141:
1070:
1390:
1258:
2148:
483:
453:
371:
316:
153:
1892:
832:
2073:
1764:
1655:
1478:
1446:
1924:
1796:
527:
197:
2255:
2172:
2041:
2017:
1968:
1732:
1708:
1623:
1599:
1570:
1546:
1502:
1414:
1346:
1222:
1189:
1165:
1099:
1028:
995:
623:
423:
395:
286:
53:
2265:
The condition, that the geometric operations of a projective plane are complex analytic, is very restrictive. In fact, it is satisfied only in the classical complex plane.
240:
87:
896:
1988:
1944:
1816:
1684:
1522:
739:
701:
1862:
2112:
1836:
971:
951:
927:
852:
663:
594:
567:
503:
173:
1572:
the connected component of its full automorphism group. The Hughes planes are not smooth. This yields a result similar to the case of 4-dimensional planes:
630:
This shows that there are many compact connected topological projective planes that are not smooth. On the other hand, the following construction yields
637:
of dimension 2, 4, and 8, with a compact group of automorphisms of dimension 1, 4, and 13, respectively: represent points and lines in the usual way by
1260:
have been classified. Up to duality, they are either translation planes or they are isomorphic to a unique so-called shift plane. According to
55:. Its geometric operations of joining two distinct points by a line and of intersecting two lines in a point are not only continuous but even
2181:
2270:
Every complex analytic projective plane is isomorphic as an analytic plane to the complex plane with its standard analytic structure
1272:
1104:
1033:
1359:
242:, because this is true for compact connected projective topological planes. These four cases will be treated separately below.
1227:
2117:
247:
The point manifold of a smooth projective plane is homeomorphic to its classical counterpart, and so is the line manifold
2893:
458:
428:
330:
291:
112:
573:
1867:
1504:
is a translation plane, or a dual translation plane, or a Hughes plane. The latter can be characterized as follows:
570:
260:
play a crucial role in the study of smooth planes. A bijection of the point set of a projective plane is called a
2838:
744:
953:(a closed subset homeomorphic to the circle), and any two distinct points are joined by a unique line. Then
906:
Compact 2-dimensional projective planes can be described in the following way: the point space is a compact
638:
634:
537:
of dimension 8, 16, 35, or 78, respectively. All other smooth planes have much smaller groups. See below.
2046:
1737:
1628:
1451:
1419:
997:, any two distinct lines intersect in a unique point, and the geometric operations are continuous (apply
29:
2873:
1897:
1769:
930:
508:
178:
2236:
2153:
2022:
1998:
1949:
1713:
1689:
1604:
1580:
1551:
1527:
1483:
1395:
1327:
1203:
1170:
1146:
1080:
1009:
976:
604:
404:
376:
267:
34:
319:
213:
62:
2863:
2805:
2759:
2675:
2530:
2430:
2396:
2348:
1002:
907:
2300:
Löwen, R. (1983), "Topology and dimension of stable planes: On a conjecture of H. Freudenthal",
857:
2898:
1264:, Chap. 10), this shift plane is not smooth. Hence, the result on translation planes implies:
631:
546:
1973:
1929:
1801:
1669:
1507:
706:
668:
2797:
2749:
2741:
2665:
2657:
2628:
2522:
2482:
2456:
2422:
2386:
2378:
2340:
1841:
534:
25:
2090:
264:, if it maps lines onto lines. The continuous collineations of a compact projective plane
2877:
2788:
Breitsprecher, S. (1967), "Einzigkeit der reellen und der komplexen projektiven Ebene",
2447:
Immervoll, S. (2003), "Real analytic projective planes with large automorphism groups",
206:
The geometric operations of smooth planes are continuous; hence, each smooth plane is a
1821:
1686:
denote the automorphism group of a compact 16-dimensional topological projective plane
956:
936:
912:
837:
648:
579:
552:
488:
158:
90:
2729:
2487:
2366:
2887:
2809:
2679:
2534:
2434:
2849:
Salzmann, H.; Betten, D.; Grundhöfer, T.; Hähl, H.; Löwen, R.; Stroppel, M. (1995),
2763:
2400:
2352:
261:
257:
210:
topological plane. Smooth planes exist only with point spaces of dimension 2 where
1001:, §31 to the complement of a line). A familiar family of examples was given by
2745:
1269:
A smooth 4-dimensional plane is isomorphic to the classical complex plane, or
642:
94:
2633:
549:
if its automorphism group has a subgroup that fixes each point on some line
203:
and where both geometric operations of joining and intersecting are smooth.
2661:
2460:
2382:
200:
98:
17:
2730:"16-dimensional smooth projective planes with large collineation groups"
2513:
Betten, D. (1971), "2-dimensionale differenzierbare projektive Ebenen",
1990:
are known explicitly. Nevertheless, none of these planes can be smooth:
2801:
2526:
2426:
2344:
2754:
2670:
2391:
2473:
Moulton, F. R. (1902), "A simple non-desarguesian plane geometry",
2226:{\displaystyle \dim \operatorname {Aut} {\mathcal {P}}\leq c_{m}-2}
2868:
28:. The most prominent example of a smooth projective plane is the
101:
are smooth planes. However, these are not the only such planes.
2331:
Kramer, L. (1994), "The topology of smooth projective planes",
2084:
The last four results combine to give the following theorem:
1304:{\displaystyle \dim \operatorname {Aut} {\mathcal {P}}\leq 6}
1136:{\displaystyle \dim \operatorname {Aut} {\mathcal {P}}\geq 3}
1065:{\displaystyle \dim \operatorname {Aut} {\mathcal {P}}\geq 3}
401:
in other words, the group of automorphisms of a smooth plane
2821:
2703:
2606:
2570:
2558:
2500:
2318:
2287:
2242:
2199:
2159:
2135:
2028:
2004:
1955:
1873:
1719:
1695:
1610:
1586:
1557:
1533:
1489:
1401:
1377:
1349:
1333:
1290:
1239:
1209:
1176:
1152:
1122:
1086:
1051:
1015:
998:
982:
610:
470:
440:
410:
382:
336:
303:
273:
118:
40:
1385:{\displaystyle \Sigma =\operatorname {Aut} {\mathcal {P}}}
2413:
Otte, J. (1995), "Smooth
Projective Translation Planes",
533:
The automorphism groups of the four classical planes are
1253:{\displaystyle \operatorname {Aut} {\mathcal {P}}\geq 7}
373:
is a smooth plane, then each continuous collineation of
2648:
Bödi, R. (1999), "Smooth Hughes planes are classical",
2143:{\displaystyle \dim \operatorname {Aut} {\mathcal {P}}}
2239:
2184:
2156:
2120:
2093:
2049:
2025:
2001:
1976:
1952:
1932:
1900:
1870:
1844:
1824:
1804:
1772:
1740:
1716:
1692:
1672:
1631:
1607:
1583:
1554:
1530:
1510:
1486:
1454:
1422:
1398:
1362:
1330:
1275:
1230:
1206:
1173:
1149:
1107:
1083:
1036:
1012:
979:
973:
is homeomorphic to the point space of the real plane
959:
939:
915:
860:
840:
747:
709:
671:
651:
607:
582:
555:
511:
491:
461:
431:
407:
379:
333:
294:
270:
216:
181:
161:
115:
65:
37:
1946:
also fixes an incident point-line pair, but neither
2860:Compact planes, mostly 8-dimensional. A retrospect
2249:
2225:
2166:
2142:
2106:
2067:
2035:
2019:is a 16-dimensional smooth projective plane, then
2011:
1982:
1962:
1938:
1918:
1886:
1856:
1830:
1810:
1790:
1758:
1726:
1702:
1678:
1649:
1617:
1593:
1564:
1540:
1516:
1496:
1472:
1440:
1408:
1384:
1340:
1303:
1252:
1216:
1183:
1159:
1135:
1093:
1064:
1022:
989:
965:
945:
921:
890:
846:
826:
733:
695:
657:
617:
588:
561:
521:
497:
478:{\displaystyle \operatorname {Aut} {\mathcal {P}}}
477:
448:{\displaystyle \operatorname {Aut} {\mathcal {P}}}
447:
417:
389:
366:{\displaystyle {\mathcal {P}}=(P,{\mathfrak {L}})}
365:
311:{\displaystyle \operatorname {Aut} {\mathcal {P}}}
310:
280:
234:
191:
167:
148:{\displaystyle {\mathcal {P}}=(P,{\mathfrak {L}})}
147:
81:
47:
625:is isomorphic to one of the four classical planes
1072:are known explicitly; none of these is smooth:
1887:{\displaystyle {\mathcal {P}}\smallsetminus W}
8:
89:). Similarly, the classical planes over the
318:. This group is taken with the topology of
1734:is the smooth classical octonion plane or
827:{\displaystyle ax+by+cz=t|c|^{2}|z|^{2}cz}
601:Every smooth projective translation plane
2867:
2753:
2669:
2632:
2486:
2390:
2241:
2240:
2238:
2211:
2198:
2197:
2183:
2174:is a non-classical compact 2-dimensional
2158:
2157:
2155:
2134:
2133:
2119:
2098:
2092:
2048:
2027:
2026:
2024:
2003:
2002:
2000:
1975:
1954:
1953:
1951:
1931:
1899:
1872:
1871:
1869:
1843:
1823:
1803:
1771:
1739:
1718:
1717:
1715:
1694:
1693:
1691:
1671:
1630:
1609:
1608:
1606:
1585:
1584:
1582:
1556:
1555:
1553:
1532:
1531:
1529:
1509:
1488:
1487:
1485:
1453:
1421:
1400:
1399:
1397:
1376:
1375:
1361:
1332:
1331:
1329:
1289:
1288:
1274:
1238:
1237:
1229:
1208:
1207:
1205:
1175:
1174:
1172:
1151:
1150:
1148:
1121:
1120:
1106:
1085:
1084:
1082:
1050:
1049:
1035:
1014:
1013:
1011:
981:
980:
978:
958:
938:
914:
880:
869:
861:
859:
839:
812:
807:
798:
792:
787:
778:
746:
708:
670:
650:
609:
608:
606:
581:
554:
513:
512:
510:
490:
469:
468:
460:
439:
438:
430:
409:
408:
406:
381:
380:
378:
354:
353:
335:
334:
332:
302:
301:
293:
272:
271:
269:
215:
183:
182:
180:
160:
136:
135:
117:
116:
114:
73:
64:
39:
38:
36:
2775:
2691:
2594:
1894:and its dual are translation planes. If
1353:
641:over the real or complex numbers or the
485:is a smooth Lie transformation group of
2619:Salzmann, H. (2003), "Baer subplanes",
2367:"Collineations of smooth stable planes"
2280:
1524:leaves some classical complex subplane
1101:is a smooth 2-dimensional plane and if
1601:is a smooth 8-dimensional plane, then
2839:"Smooth stable and projective planes"
1625:is the classical quaternion plane or
1416:is the classical quaternion plane or
1224:with a 4-dimensional point space and
7:
2715:
2582:
2546:
1352:, Chapter 8) and, more recently, in
1261:
854:is a fixed real parameter such that
2068:{\displaystyle \dim \Sigma \leq 38}
2043:is the classical octonion plane or
1759:{\displaystyle \dim \Sigma \leq 40}
1650:{\displaystyle \dim \Sigma \leq 16}
1473:{\displaystyle \dim \Sigma \geq 17}
1441:{\displaystyle \dim \Sigma \leq 18}
514:
355:
184:
137:
2056:
1977:
1933:
1907:
1805:
1779:
1747:
1673:
1638:
1511:
1461:
1429:
1363:
665:. Then the incidence of the point
74:
14:
2488:10.1090/s0002-9947-1902-1500595-3
2778:, 9.18 for a sketch of the proof
545:A projective plane is called a
1919:{\displaystyle \dim \Sigma =39}
1791:{\displaystyle \dim \Sigma =40}
522:{\displaystyle {\mathfrak {L}}}
192:{\displaystyle {\mathfrak {L}}}
105:Definition and basic properties
2250:{\displaystyle {\mathcal {P}}}
2167:{\displaystyle {\mathcal {P}}}
2036:{\displaystyle {\mathcal {P}}}
2012:{\displaystyle {\mathcal {P}}}
1963:{\displaystyle {\mathcal {P}}}
1727:{\displaystyle {\mathcal {P}}}
1703:{\displaystyle {\mathcal {P}}}
1618:{\displaystyle {\mathcal {P}}}
1594:{\displaystyle {\mathcal {P}}}
1565:{\displaystyle {\mathcal {C}}}
1541:{\displaystyle {\mathcal {C}}}
1497:{\displaystyle {\mathcal {P}}}
1409:{\displaystyle {\mathcal {P}}}
1341:{\displaystyle {\mathcal {P}}}
1217:{\displaystyle {\mathcal {P}}}
1184:{\displaystyle {\mathcal {E}}}
1160:{\displaystyle {\mathcal {P}}}
1094:{\displaystyle {\mathcal {P}}}
1023:{\displaystyle {\mathcal {P}}}
990:{\displaystyle {\mathcal {E}}}
898:. These planes are self-dual.
870:
862:
808:
799:
788:
779:
728:
710:
690:
672:
618:{\displaystyle {\mathcal {P}}}
418:{\displaystyle {\mathcal {P}}}
390:{\displaystyle {\mathcal {P}}}
360:
344:
281:{\displaystyle {\mathcal {P}}}
142:
126:
48:{\displaystyle {\mathcal {E}}}
1:
235:{\displaystyle 1\leq m\leq 4}
1167:is the classical real plane
645:, say, by vectors of length
576:on the set of points not on
82:{\displaystyle =C^{\infty }}
59:(infinitely differentiable
2915:
891:{\displaystyle |t|<1/9}
155:consists of a point space
109:A smooth projective plane
2851:Compact Projective Planes
1548:invariant and induces on
2114:is the largest value of
22:smooth projective planes
2746:10.1023/A:1005020223604
2475:Trans. Amer. Math. Soc.
2261:Complex analytic planes
2178:projective plane, then
1983:{\displaystyle \Sigma }
1939:{\displaystyle \Sigma }
1864:, and the affine plane
1811:{\displaystyle \Sigma }
1679:{\displaystyle \Sigma }
1517:{\displaystyle \Sigma }
1348:have been discussed in
734:{\displaystyle (a,b,c)}
696:{\displaystyle (x,y,z)}
639:homogeneous coordinates
635:non-Desarguesian planes
2634:10.1215/ijm/1258488168
2251:
2227:
2168:
2144:
2108:
2069:
2037:
2013:
1984:
1964:
1940:
1920:
1888:
1858:
1857:{\displaystyle v\in W}
1832:
1812:
1792:
1760:
1728:
1704:
1680:
1651:
1619:
1595:
1566:
1542:
1518:
1498:
1474:
1442:
1410:
1386:
1342:
1320:Compact 8-dimensional
1305:
1254:
1218:
1185:
1161:
1137:
1095:
1066:
1024:
991:
967:
947:
923:
892:
848:
828:
735:
697:
659:
619:
590:
563:
523:
499:
479:
449:
419:
391:
367:
312:
282:
236:
193:
169:
149:
83:
49:
2858:Salzmann, H. (2014),
2662:10.1007/s000130050022
2461:10.1515/advg.2003.011
2383:10.1515/form.10.6.751
2302:J. Reine Angew. Math.
2252:
2228:
2169:
2145:
2109:
2107:{\displaystyle c_{m}}
2070:
2038:
2014:
1985:
1965:
1941:
1921:
1889:
1859:
1833:
1813:
1793:
1761:
1729:
1705:
1681:
1662:16-dimensional planes
1652:
1620:
1596:
1567:
1543:
1519:
1499:
1475:
1443:
1411:
1387:
1350:Salzmann et al. (1995
1343:
1306:
1255:
1219:
1186:
1162:
1138:
1096:
1067:
1025:
992:
968:
948:
924:
893:
849:
829:
736:
698:
660:
620:
591:
564:
524:
500:
480:
450:
420:
392:
368:
313:
283:
237:
194:
170:
150:
84:
50:
30:real projective plane
2822:Salzmann et al. 1995
2704:Salzmann et al. 1995
2607:Salzmann et al. 1995
2571:Salzmann et al. 1995
2559:Salzmann et al. 1995
2501:Salzmann et al. 1995
2319:Salzmann et al. 1995
2288:Salzmann et al. 1995
2237:
2182:
2154:
2118:
2091:
2047:
2023:
1999:
1974:
1950:
1930:
1898:
1868:
1842:
1822:
1802:
1770:
1738:
1714:
1690:
1670:
1629:
1605:
1581:
1552:
1528:
1508:
1484:
1452:
1420:
1396:
1360:
1328:
1316:8-dimensional planes
1273:
1228:
1204:
1196:4-dimensional planes
1171:
1147:
1105:
1081:
1034:
1010:
999:Salzmann et al. 1995
977:
957:
937:
913:
902:2-dimensional planes
858:
838:
745:
707:
669:
649:
605:
580:
574:sharply transitively
553:
509:
489:
459:
429:
405:
377:
331:
292:
268:
214:
179:
159:
113:
63:
35:
2894:Projective geometry
2878:2014arXiv1402.0304S
1200:All compact planes
320:uniform convergence
2802:10.1007/bf01111021
2527:10.1007/bf01222580
2427:10.1007/bf01265639
2345:10.1007/bf01196303
2247:
2223:
2164:
2140:
2104:
2065:
2033:
2009:
1980:
1960:
1936:
1916:
1884:
1854:
1828:
1808:
1788:
1756:
1724:
1700:
1676:
1647:
1615:
1591:
1562:
1538:
1514:
1494:
1470:
1438:
1406:
1382:
1338:
1301:
1250:
1214:
1181:
1157:
1133:
1091:
1062:
1020:
987:
963:
943:
919:
888:
844:
824:
731:
693:
655:
615:
586:
559:
541:Translation planes
519:
495:
475:
445:
415:
387:
363:
308:
278:
232:
189:
165:
145:
79:
45:
2837:Bödi, R. (1996),
2728:Bödi, R. (1998),
2621:Illinois J. Math.
2365:Bödi, R. (1998),
1831:{\displaystyle W}
966:{\displaystyle S}
946:{\displaystyle S}
929:, each line is a
922:{\displaystyle S}
847:{\displaystyle t}
658:{\displaystyle 1}
589:{\displaystyle W}
562:{\displaystyle W}
547:translation plane
535:simple Lie groups
498:{\displaystyle P}
175:and a line space
168:{\displaystyle P}
26:projective planes
2906:
2880:
2871:
2854:
2845:
2843:Thesis, TĂĽbingen
2825:
2819:
2813:
2812:
2785:
2779:
2773:
2767:
2766:
2757:
2725:
2719:
2713:
2707:
2701:
2695:
2689:
2683:
2682:
2673:
2645:
2639:
2637:
2636:
2627:(1–2): 485–513,
2616:
2610:
2604:
2598:
2592:
2586:
2580:
2574:
2568:
2562:
2556:
2550:
2544:
2538:
2537:
2510:
2504:
2498:
2492:
2491:
2490:
2470:
2464:
2463:
2444:
2438:
2437:
2410:
2404:
2403:
2394:
2362:
2356:
2355:
2328:
2322:
2316:
2310:
2309:
2297:
2291:
2285:
2257:is even smooth.
2256:
2254:
2253:
2248:
2246:
2245:
2232:
2230:
2229:
2224:
2216:
2215:
2203:
2202:
2173:
2171:
2170:
2165:
2163:
2162:
2149:
2147:
2146:
2141:
2139:
2138:
2113:
2111:
2110:
2105:
2103:
2102:
2074:
2072:
2071:
2066:
2042:
2040:
2039:
2034:
2032:
2031:
2018:
2016:
2015:
2010:
2008:
2007:
1989:
1987:
1986:
1981:
1969:
1967:
1966:
1961:
1959:
1958:
1945:
1943:
1942:
1937:
1925:
1923:
1922:
1917:
1893:
1891:
1890:
1885:
1877:
1876:
1863:
1861:
1860:
1855:
1837:
1835:
1834:
1829:
1817:
1815:
1814:
1809:
1797:
1795:
1794:
1789:
1765:
1763:
1762:
1757:
1733:
1731:
1730:
1725:
1723:
1722:
1709:
1707:
1706:
1701:
1699:
1698:
1685:
1683:
1682:
1677:
1656:
1654:
1653:
1648:
1624:
1622:
1621:
1616:
1614:
1613:
1600:
1598:
1597:
1592:
1590:
1589:
1571:
1569:
1568:
1563:
1561:
1560:
1547:
1545:
1544:
1539:
1537:
1536:
1523:
1521:
1520:
1515:
1503:
1501:
1500:
1495:
1493:
1492:
1479:
1477:
1476:
1471:
1447:
1445:
1444:
1439:
1415:
1413:
1412:
1407:
1405:
1404:
1391:
1389:
1388:
1383:
1381:
1380:
1347:
1345:
1344:
1339:
1337:
1336:
1310:
1308:
1307:
1302:
1294:
1293:
1259:
1257:
1256:
1251:
1243:
1242:
1223:
1221:
1220:
1215:
1213:
1212:
1190:
1188:
1187:
1182:
1180:
1179:
1166:
1164:
1163:
1158:
1156:
1155:
1142:
1140:
1139:
1134:
1126:
1125:
1100:
1098:
1097:
1092:
1090:
1089:
1071:
1069:
1068:
1063:
1055:
1054:
1029:
1027:
1026:
1021:
1019:
1018:
996:
994:
993:
988:
986:
985:
972:
970:
969:
964:
952:
950:
949:
944:
928:
926:
925:
920:
897:
895:
894:
889:
884:
873:
865:
853:
851:
850:
845:
833:
831:
830:
825:
817:
816:
811:
802:
797:
796:
791:
782:
740:
738:
737:
732:
702:
700:
699:
694:
664:
662:
661:
656:
624:
622:
621:
616:
614:
613:
595:
593:
592:
587:
568:
566:
565:
560:
528:
526:
525:
520:
518:
517:
504:
502:
501:
496:
484:
482:
481:
476:
474:
473:
454:
452:
451:
446:
444:
443:
424:
422:
421:
416:
414:
413:
396:
394:
393:
388:
386:
385:
372:
370:
369:
364:
359:
358:
340:
339:
317:
315:
314:
309:
307:
306:
287:
285:
284:
279:
277:
276:
241:
239:
238:
233:
199:that are smooth
198:
196:
195:
190:
188:
187:
174:
172:
171:
166:
154:
152:
151:
146:
141:
140:
122:
121:
88:
86:
85:
80:
78:
77:
54:
52:
51:
46:
44:
43:
2914:
2913:
2909:
2908:
2907:
2905:
2904:
2903:
2884:
2883:
2857:
2853:, W. de Gruyter
2848:
2836:
2833:
2828:
2820:
2816:
2787:
2786:
2782:
2774:
2770:
2727:
2726:
2722:
2714:
2710:
2702:
2698:
2690:
2686:
2647:
2646:
2642:
2618:
2617:
2613:
2605:
2601:
2593:
2589:
2581:
2577:
2569:
2565:
2557:
2553:
2545:
2541:
2512:
2511:
2507:
2499:
2495:
2472:
2471:
2467:
2446:
2445:
2441:
2412:
2411:
2407:
2364:
2363:
2359:
2330:
2329:
2325:
2317:
2313:
2299:
2298:
2294:
2286:
2282:
2278:
2263:
2235:
2234:
2207:
2180:
2179:
2152:
2151:
2116:
2115:
2094:
2089:
2088:
2082:
2045:
2044:
2021:
2020:
1997:
1996:
1972:
1971:
1948:
1947:
1928:
1927:
1896:
1895:
1866:
1865:
1840:
1839:
1820:
1819:
1800:
1799:
1768:
1767:
1736:
1735:
1712:
1711:
1688:
1687:
1668:
1667:
1664:
1627:
1626:
1603:
1602:
1579:
1578:
1550:
1549:
1526:
1525:
1506:
1505:
1482:
1481:
1450:
1449:
1418:
1417:
1394:
1393:
1358:
1357:
1354:Salzmann (2014)
1326:
1325:
1318:
1271:
1270:
1226:
1225:
1202:
1201:
1198:
1169:
1168:
1145:
1144:
1103:
1102:
1079:
1078:
1032:
1031:
1008:
1007:
975:
974:
955:
954:
935:
934:
911:
910:
904:
856:
855:
836:
835:
806:
786:
743:
742:
705:
704:
667:
666:
647:
646:
603:
602:
578:
577:
551:
550:
543:
507:
506:
487:
486:
457:
456:
427:
426:
425:coincides with
403:
402:
375:
374:
329:
328:
290:
289:
288:form the group
266:
265:
255:
212:
211:
177:
176:
157:
156:
111:
110:
107:
91:complex numbers
69:
61:
60:
33:
32:
12:
11:
5:
2912:
2910:
2902:
2901:
2896:
2886:
2885:
2882:
2881:
2855:
2846:
2832:
2829:
2827:
2826:
2814:
2796:(5): 429–432,
2780:
2768:
2740:(3): 283–298,
2734:Geom. Dedicata
2720:
2708:
2696:
2684:
2640:
2611:
2599:
2587:
2575:
2563:
2551:
2539:
2505:
2493:
2481:(2): 192–195,
2465:
2455:(2): 163–176,
2439:
2421:(2): 203–212,
2415:Geom. Dedicata
2405:
2377:(6): 751–773,
2357:
2323:
2311:
2292:
2279:
2277:
2274:
2262:
2259:
2244:
2222:
2219:
2214:
2210:
2206:
2201:
2196:
2193:
2190:
2187:
2161:
2137:
2132:
2129:
2126:
2123:
2101:
2097:
2081:
2078:
2064:
2061:
2058:
2055:
2052:
2030:
2006:
1979:
1957:
1935:
1915:
1912:
1909:
1906:
1903:
1883:
1880:
1875:
1853:
1850:
1847:
1827:
1807:
1787:
1784:
1781:
1778:
1775:
1755:
1752:
1749:
1746:
1743:
1721:
1697:
1675:
1663:
1660:
1646:
1643:
1640:
1637:
1634:
1612:
1588:
1559:
1535:
1513:
1491:
1469:
1466:
1463:
1460:
1457:
1437:
1434:
1431:
1428:
1425:
1403:
1379:
1374:
1371:
1368:
1365:
1335:
1317:
1314:
1300:
1297:
1292:
1287:
1284:
1281:
1278:
1249:
1246:
1241:
1236:
1233:
1211:
1197:
1194:
1178:
1154:
1132:
1129:
1124:
1119:
1116:
1113:
1110:
1088:
1061:
1058:
1053:
1048:
1045:
1042:
1039:
1017:
984:
962:
942:
918:
903:
900:
887:
883:
879:
876:
872:
868:
864:
843:
823:
820:
815:
810:
805:
801:
795:
790:
785:
781:
777:
774:
771:
768:
765:
762:
759:
756:
753:
750:
741:is defined by
730:
727:
724:
721:
718:
715:
712:
692:
689:
686:
683:
680:
677:
674:
654:
612:
585:
558:
542:
539:
516:
494:
472:
467:
464:
442:
437:
434:
412:
384:
362:
357:
352:
349:
346:
343:
338:
305:
300:
297:
275:
254:
251:
231:
228:
225:
222:
219:
186:
164:
144:
139:
134:
131:
128:
125:
120:
106:
103:
76:
72:
68:
42:
13:
10:
9:
6:
4:
3:
2:
2911:
2900:
2897:
2895:
2892:
2891:
2889:
2879:
2875:
2870:
2865:
2861:
2856:
2852:
2847:
2844:
2840:
2835:
2834:
2830:
2823:
2818:
2815:
2811:
2807:
2803:
2799:
2795:
2791:
2784:
2781:
2777:
2776:Salzmann 2014
2772:
2769:
2765:
2761:
2756:
2751:
2747:
2743:
2739:
2735:
2731:
2724:
2721:
2717:
2712:
2709:
2705:
2700:
2697:
2693:
2692:Salzmann 2014
2688:
2685:
2681:
2677:
2672:
2667:
2663:
2659:
2655:
2651:
2644:
2641:
2635:
2630:
2626:
2622:
2615:
2612:
2608:
2603:
2600:
2596:
2595:Salzmann 2014
2591:
2588:
2584:
2579:
2576:
2572:
2567:
2564:
2560:
2555:
2552:
2548:
2543:
2540:
2536:
2532:
2528:
2524:
2520:
2516:
2509:
2506:
2502:
2497:
2494:
2489:
2484:
2480:
2476:
2469:
2466:
2462:
2458:
2454:
2450:
2443:
2440:
2436:
2432:
2428:
2424:
2420:
2416:
2409:
2406:
2402:
2398:
2393:
2388:
2384:
2380:
2376:
2372:
2368:
2361:
2358:
2354:
2350:
2346:
2342:
2338:
2334:
2327:
2324:
2320:
2315:
2312:
2307:
2303:
2296:
2293:
2289:
2284:
2281:
2275:
2273:
2271:
2266:
2260:
2258:
2220:
2217:
2212:
2208:
2204:
2194:
2191:
2188:
2185:
2177:
2130:
2127:
2124:
2121:
2099:
2095:
2085:
2079:
2077:
2075:
2062:
2059:
2053:
2050:
1991:
1913:
1910:
1904:
1901:
1881:
1878:
1851:
1848:
1845:
1825:
1818:fixes a line
1785:
1782:
1776:
1773:
1753:
1750:
1744:
1741:
1661:
1659:
1657:
1644:
1641:
1635:
1632:
1573:
1467:
1464:
1458:
1455:
1435:
1432:
1426:
1423:
1372:
1369:
1366:
1355:
1351:
1323:
1315:
1313:
1311:
1298:
1295:
1285:
1282:
1279:
1276:
1265:
1263:
1247:
1244:
1234:
1231:
1195:
1193:
1191:
1130:
1127:
1117:
1114:
1111:
1108:
1073:
1059:
1056:
1046:
1043:
1040:
1037:
1004:
1000:
960:
940:
932:
916:
909:
901:
899:
885:
881:
877:
874:
866:
841:
821:
818:
813:
803:
793:
783:
775:
772:
769:
766:
763:
760:
757:
754:
751:
748:
725:
722:
719:
716:
713:
703:and the line
687:
684:
681:
678:
675:
652:
644:
640:
636:
633:
632:real analytic
628:
626:
597:
583:
575:
572:
556:
548:
540:
538:
536:
531:
529:
492:
465:
462:
435:
432:
398:
350:
347:
341:
323:
321:
298:
295:
263:
259:
258:Automorphisms
253:Automorphisms
252:
250:
248:
243:
229:
226:
223:
220:
217:
209:
204:
202:
162:
132:
129:
123:
104:
102:
100:
96:
92:
70:
66:
58:
31:
27:
23:
19:
2859:
2850:
2842:
2817:
2793:
2789:
2783:
2771:
2737:
2733:
2723:
2711:
2699:
2687:
2653:
2649:
2643:
2624:
2620:
2614:
2602:
2590:
2578:
2566:
2554:
2542:
2518:
2514:
2508:
2496:
2478:
2474:
2468:
2452:
2448:
2442:
2418:
2414:
2408:
2374:
2370:
2360:
2336:
2332:
2326:
2314:
2305:
2301:
2295:
2283:
2269:
2267:
2264:
2175:
2086:
2083:
2080:Main theorem
1994:
1992:
1838:and a point
1665:
1576:
1574:
1321:
1319:
1268:
1266:
1199:
1076:
1074:
931:Jordan curve
905:
629:
600:
598:
544:
532:
455:. Moreover,
400:
326:
324:
262:collineation
256:
246:
244:
207:
205:
108:
56:
24:are special
21:
15:
2650:Arch. Math.
2521:: 304–309,
2515:Arch. Math.
2371:Forum Math.
2333:Arch. Math.
2176:topological
1322:topological
643:quaternions
322:. We have:
95:quaternions
2888:Categories
2831:References
2755:11475/3238
2718:, Chap. 12
2671:11475/3229
2449:Adv. Geom.
2392:11475/3260
1262:Bödi (1996
1030:such that
97:, and the
2869:1402.0304
2810:120984088
2716:Bödi 1996
2680:120222293
2656:: 73–80,
2585:, (10.11)
2583:Bödi 1996
2547:Bödi 1996
2535:119885473
2435:120238728
2339:: 85–91,
2308:: 108–122
2268:Theorem.
2233:whenever
2218:−
2205:≤
2195:
2189:
2131:
2125:
2060:≤
2057:Σ
2054:
1993:Theorem.
1978:Σ
1934:Σ
1908:Σ
1905:
1879:∖
1849:∈
1806:Σ
1780:Σ
1777:
1751:≤
1748:Σ
1745:
1710:. Either
1674:Σ
1642:≤
1639:Σ
1636:
1575:Theorem.
1512:Σ
1465:≥
1462:Σ
1459:
1433:≤
1430:Σ
1427:
1392:. Either
1373:
1364:Σ
1296:≤
1286:
1280:
1267:Theorem.
1245:≥
1235:
1128:≥
1118:
1112:
1075:Theorem.
1057:≥
1047:
1041:
599:Theorem.
466:
436:
397:is smooth
325:Theorem.
299:
245:Theorem.
227:≤
221:≤
201:manifolds
99:octonions
75:∞
2899:Surfaces
2790:Math. Z.
2764:56094550
2401:54504153
2353:15480568
2150:, where
834:, where
18:geometry
2874:Bibcode
2561:, 74.27
2549:, (9.1)
2321:, 54.11
1926:, then
1798:, then
1480:, then
1324:planes
1143:, then
1003:Moulton
908:surface
505:and of
208:compact
2824:, 75.1
2808:
2762:
2706:, 87.7
2694:, 9.17
2678:
2597:, 1.10
2533:
2433:
2399:
2351:
2290:, 42.4
1356:. Put
93:, the
57:smooth
2864:arXiv
2806:S2CID
2760:S2CID
2676:S2CID
2609:, §86
2573:, §74
2531:S2CID
2503:, §34
2431:S2CID
2397:S2CID
2349:S2CID
2276:Notes
1766:. If
1448:. If
2638:3.19
1970:nor
1666:Let
875:<
571:acts
569:and
2798:doi
2750:hdl
2742:doi
2666:hdl
2658:doi
2629:doi
2523:doi
2483:doi
2457:doi
2423:doi
2387:hdl
2379:doi
2341:doi
2306:343
2192:Aut
2186:dim
2128:Aut
2122:dim
2087:If
2051:dim
1995:If
1902:dim
1774:dim
1742:dim
1633:dim
1577:If
1456:dim
1424:dim
1370:Aut
1283:Aut
1277:dim
1232:Aut
1115:Aut
1109:dim
1077:If
1044:Aut
1038:dim
933:in
463:Aut
433:Aut
327:If
296:Aut
16:In
2890::
2872:,
2862:,
2841:,
2804:,
2794:99
2792:,
2758:,
2748:,
2738:72
2736:,
2732:,
2674:,
2664:,
2654:73
2652:,
2625:47
2623:,
2529:,
2519:22
2517:,
2477:,
2451:,
2429:,
2419:58
2417:,
2395:,
2385:,
2375:10
2373:,
2369:,
2347:,
2337:63
2335:,
2304:,
2272:.
2076:.
2063:38
1914:39
1786:40
1754:40
1658:.
1645:16
1468:17
1436:18
1312:.
1192:.
627:.
596:.
530:.
399:;
249:.
20:,
2876::
2866::
2800::
2752::
2744::
2668::
2660::
2631::
2525::
2485::
2479:3
2459::
2453:3
2425::
2389::
2381::
2343::
2243:P
2221:2
2213:m
2209:c
2200:P
2160:P
2136:P
2100:m
2096:c
2029:P
2005:P
1956:P
1911:=
1882:W
1874:P
1852:W
1846:v
1826:W
1783:=
1720:P
1696:P
1611:P
1587:P
1558:C
1534:C
1490:P
1402:P
1378:P
1367:=
1334:P
1299:6
1291:P
1248:7
1240:P
1210:P
1177:E
1153:P
1131:3
1123:P
1087:P
1060:3
1052:P
1016:P
983:E
961:S
941:S
917:S
886:9
882:/
878:1
871:|
867:t
863:|
842:t
822:z
819:c
814:2
809:|
804:z
800:|
794:2
789:|
784:c
780:|
776:t
773:=
770:z
767:c
764:+
761:y
758:b
755:+
752:x
749:a
729:)
726:c
723:,
720:b
717:,
714:a
711:(
691:)
688:z
685:,
682:y
679:,
676:x
673:(
653:1
611:P
584:W
557:W
515:L
493:P
471:P
441:P
411:P
383:P
361:)
356:L
351:,
348:P
345:(
342:=
337:P
304:P
274:P
230:4
224:m
218:1
185:L
163:P
143:)
138:L
133:,
130:P
127:(
124:=
119:P
71:C
67:=
41:E
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