31:
1748:. For three dimensions, this gives a total of six linearly independent Killing vector fields; homogeneous 3-spaces have the property that one may use linear combinations of these to find three everywhere non-vanishing Killing vector fields
1626:
The definition is more restrictive than it initially appears: such spaces have remarkable properties, and there is a classification of irreducible prehomogeneous vector spaces, up to a transformation known as "castling".
801:
with norm 1. If we consider one of these vectors as a base vector, then any other vector can be constructed using an orthogonal transformation. If we consider the span of this vector as a one dimensional subspace of
1882:
1570:
given by the minors are 6 in number, this means that the latter are not independent of each other. In fact, a single quadratic relation holds between the six minors, as was known to nineteenth-century geometers.
362:
1674:
these are the maximally symmetric lorentzian spacetimes. There are also homogeneous spaces of relevance in physics that are non-lorentzian, for example
Galilean, Carrollian or Aristotelian spacetimes.
1701:
1347:
1119:
A further classical example is the space of lines in projective space of three dimensions (equivalently, the space of two-dimensional subspaces of a four-dimensional
1220:
with a distinguished point, namely the coset of the identity. Thus a homogeneous space can be thought of as a coset space without a choice of origin.
2106:
1768:
2195:
1578:, other than a projective space. There are many further homogeneous spaces of the classical linear groups in common use in mathematics.
2161:
309:
2070:
2171:
1135:
of the 4Ă2 matrix with columns two basis vectors for the subspace. The geometry of the resulting homogeneous space is the
1291:
1681:
2223:
2213:
1587:
1236:
897:
851:
1704:
may be represented by subsets of the
Bianchi I (flat), V (open), VII (flat or open) and IX (closed) types, while the
2154:
1605:
492:
178:
114:
111:
67:
1169:
66:
is, very informally, a space that looks the same everywhere, as you move through it, with movement given by the
2167:
1906:
in its lower two indices (on the left-hand side, the brackets denote antisymmetrisation and ";" represents the
1559:
by looking for the stabilizer of the subspace spanned by the first two standard basis vectors. That shows that
1505:
1039:
830:
1489:
1567:
1453:
459:
1897:
1685:
1132:
1105:
245:
174:
2181:
162:
161:
looks locally the same at each point, either in the sense of isometry (rigid geometry), diffeomorphism (
1045:
There are other interesting homogeneous spaces, in particular with relevance in physics: This includes
1907:
1903:
1693:
1671:
1521:
1181:
980:
932:
198:
146:
2218:
2186:
1983:
1082:
969:
444:
150:
87:
71:
177:(non-identity elements act non-trivially), although the present article does not. Thus there is a
2035:
1978:
1956:
1705:
1697:
1677:
1276:
134:
2157:
2146:
2102:
2053:
1636:
1140:
901:
416:
101:
83:
495:
are an important class of homogeneous spaces, and include many of the examples listed below.
2177:
2045:
1968:
1620:
1475:
1113:
1097:
1070:
775:
2121:
2094:
1745:
1689:
1667:
1663:
1609:
1467:
1429:
1421:
1089:
1046:
874:
79:
2190:
2078:
1973:
1413:
1101:
463:
273:
142:
51:
39:
2207:
1643:
1136:
420:
43:
814:-dimensional vector space that is invariant under an orthogonal transformation from
1598:
1575:
1482:
1449:
1120:
1093:
1006:
976:
965:
368:
300:
253:
17:
2142:
2049:
59:
2090:
1911:
1591:
47:
2057:
1441:
1109:
94:
75:
2023:
1208:
corresponds to the coset of the identity. Conversely, given a coset space
1709:
1074:
166:
97:
1623:(and so, dense). An example is GL(1) acting on a one-dimensional space.
1877:{\displaystyle \xi _{}^{(a)}=C_{\ bc}^{a}\xi _{i}^{(b)}\xi _{k}^{(c)},}
1900:
1384:. Note that the inner automorphism (1) does not depend on which such
264:
preserve the structure associated with the category (for example, if
2099:
Course of
Theoretical Physics vol. 2: The Classical Theory of Fields
399:) is a transitive group of symmetries of the underlying set of
2040:
189:
that can be thought of as preserving some "geometric structure" on
1650:
1189:
35:
1100:
are all in natural ways homogeneous spaces for their respective
30:
1081:. This was true of essentially all geometries proposed before
793:. This is true because of the following observations: First,
1073:, one may understand that "all points are the same", in the
50:
is homogeneous under its diffeomorphism, homeomorphism, and
357:{\displaystyle \rho :G\to \mathrm {Aut} _{\mathbf {C} }(X)}
2005:. The distinction is only important in the description of
2022:
Figueroa-OâFarrill, JosĂ©; Prohazka, Stefan (2019-01-31).
1516:
For example, in the line geometry case, we can identify
1127:
acts transitively on those. We can parameterize them by
1688:
system. Homogeneous spaces in relativity represent the
900:, point stabilizer orthogonal group, corresponding to
2168:
1926:(type I), but in the case of a closed FLRW universe,
1771:
1524:, GL(4), defined by conditions on the matrix entries
1294:
312:
1520:as a 12-dimensional subgroup of the 16-dimensional
1876:
1341:
356:
240:acts by automorphisms (bijections) on the set. If
2199:, volume 2, chapter X, (Wiley Classics Library)
1574:This example was the first known example of a
1123:). It is simple linear algebra to show that GL
74:. Homogeneous spaces occur in the theories of
38:. The standard torus is homogeneous under its
1246:-torsors are often described intuitively as "
8:
2024:"Spatially isotropic homogeneous spacetimes"
419:, then group elements are assumed to act as
86:. More precisely, a homogeneous space for a
1085:, in the middle of the nineteenth century.
256:in the same category. That is, the maps on
228:-space if it is equipped with an action of
129:. A special case of this is when the group
1702:FriedmannâLemaĂźtreâRobertsonâWalker metric
1104:. The same is true of the models found of
169:). Some authors insist that the action of
2039:
1859:
1854:
1838:
1833:
1823:
1812:
1793:
1776:
1770:
1330:
1320:
1299:
1293:
1253:In general, a different choice of origin
877:, point stabilizer is orthogonal group):
338:
337:
326:
311:
500:
29:
1994:
1700:; for example, the three cases of the
485:into the diffeomorphism group of
1500:, where Î is a discrete subgroup (of
822:. This shows us why we can construct
391:defines a homogeneous space provided
272:then the action is required to be by
141:â here "automorphism group" can mean
7:
2196:Foundations of Differential Geometry
2017:
2015:
2001:We assume that the action is on the
1896:, the "structure constants", form a
1285:
1712:example of a Bianchi IX cosmology.
1342:{\displaystyle H_{o'}=gH_{o}g^{-1}}
1478:compatible with the group action.
1147:Homogeneous spaces as coset spaces
1060:or Galilean and Carrollian spaces.
333:
330:
327:
25:
1615:, such that there is an orbit of
27:Topological space in group theory
1388:is selected; it depends only on
1257:will lead to a quotient of
1216:, it is a homogeneous space for
339:
1908:covariant differential operator
502:Examples of homogeneous spaces
470:-space is a group homomorphism
431:-space is a group homomorphism
2028:Journal of High Energy Physics
1866:
1860:
1845:
1839:
1800:
1794:
1789:
1777:
1069:From the point of view of the
462:, then the group elements are
351:
345:
322:
157:is homogeneous if intuitively
1:
2172:Kyungpook National University
1631:Homogeneous spaces in physics
291:is an object of the category
1682:general theory of relativity
1582:Prehomogeneous vector spaces
806:, then the complement is an
276:). A homogeneous space is a
244:in addition belongs to some
1719:dimensions admits a set of
1597:It is a finite-dimensional
1588:prehomogeneous vector space
918:Oriented hyperbolic space:
898:orthochronous Lorentz group
852:projective orthogonal group
498:Concrete examples include:
493:Riemannian symmetric spaces
2240:
2155:Princeton University Press
1250:with forgotten identity".
1227:is the identity subgroup {
1155:is a homogeneous space of
1042:(in the sense of topology)
295:, then the structure of a
236:. Note that automatically
2126:Gravitation and Cosmology
2101:, Butterworth-Heinemann,
2075:Lie Groups for Physicists
1040:Topological vector spaces
797:is the set of vectors in
1506:properly discontinuously
1261:by a different subgroup
831:special orthogonal group
669:anti-de Sitter space AdS
260:coming from elements of
2050:10.1007/JHEP01(2019)229
1912:flat isotropic universe
1715:A homogeneous space of
1568:homogeneous coordinates
1204:, and the marked point
1188:correspond to the left
870:Flat (zero curvature):
826:as a homogeneous space.
460:differentiable manifold
248:, then the elements of
216:be a non-empty set and
2151:Characteristic Classes
1878:
1686:Bianchi classification
1481:One can go further to
1343:
1106:non-Euclidean geometry
358:
252:are assumed to act as
55:
2182:Heidelberg University
2128:, John Wiley and Sons
1914:, one possibility is
1879:
1642:and its subgroup the
1619:that is open for the
1344:
1242:, which explains why
1172:of some marked point
466:. The structure of a
427:. The structure of a
359:
165:), or homeomorphism (
163:differential geometry
33:
1910:). In the case of a
1769:
1672:Anti-de Sitter space
1522:general linear group
1490:CliffordâKlein forms
1436:. In particular, if
1292:
1131:: these are the 2Ă2
981:general linear group
933:Anti-de Sitter space
893:Negative curvature:
771:Positive curvature:
310:
147:diffeomorphism group
2187:Shoshichi Kobayashi
1984:Homogeneous variety
1870:
1849:
1828:
1804:
1698:cosmological models
1268:that is related to
1088:Thus, for example,
1083:Riemannian geometry
979:, point stabilizer
503:
445:homeomorphism group
439: â Homeo(
284:acts transitively.
151:homeomorphism group
133:in question is the
117:. The elements of
18:Inhomogeneous space
2224:Homogeneous spaces
2214:Topological groups
2178:Homogeneous Spaces
1979:Heap (mathematics)
1957:Levi-Civita symbol
1901:order-three tensor
1874:
1850:
1829:
1808:
1772:
1706:Mixmaster universe
1678:Physical cosmology
1590:was introduced by
1369:is any element of
1339:
1277:inner automorphism
896:Hyperbolic space (
850:Projective space (
631:hyperbolic space H
571:projective space P
501:
367:into the group of
354:
135:automorphism group
84:topological groups
56:
2176:Menelaos Zikidis
2147:James D. Stasheff
2108:978-0-7506-2768-9
2009:as a coset space.
1887:where the object
1815:
1684:makes use of the
1563:has dimension 4.
1474:carries a unique
1404:If the action of
1363:
1362:
1283:. Specifically,
1184:), the points of
1129:line co-ordinates
902:hyperboloid model
873:Euclidean space (
829:Oriented sphere (
763:
762:
593:Euclidean space E
417:topological space
208:Formal definition
102:topological space
64:homogeneous space
16:(Redirected from
2231:
2130:
2129:
2118:
2112:
2111:
2087:
2081:
2068:
2062:
2061:
2043:
2019:
2010:
1999:
1969:Erlangen program
1945:
1925:
1883:
1881:
1880:
1875:
1869:
1858:
1848:
1837:
1827:
1822:
1813:
1803:
1792:
1761:
1760:
1744:
1735:
1733:
1732:
1729:
1726:
1666:. Together with
1661:
1621:Zariski topology
1488:spaces, notably
1476:smooth structure
1465:
1454:Cartan's theorem
1383:
1357:
1348:
1346:
1345:
1340:
1338:
1337:
1325:
1324:
1309:
1308:
1307:
1286:
1223:For example, if
1114:hyperbolic space
1098:projective space
1071:Erlangen program
1059:
1036:
1002:
953:
929:
915:
888:
865:
847:
821:
813:
792:
776:orthogonal group
694:Grassmannian Gr(
527:spherical space
504:
484:
411:For example, if
390:
375:in the category
363:
361:
360:
355:
344:
343:
342:
336:
280:-space on which
268:is an object in
153:. In this case,
80:algebraic groups
46:groups, and the
21:
2239:
2238:
2234:
2233:
2232:
2230:
2229:
2228:
2204:
2203:
2202:
2138:
2133:
2122:Steven Weinberg
2120:
2119:
2115:
2109:
2095:Evgeny Lifshitz
2089:
2088:
2084:
2069:
2065:
2021:
2020:
2013:
2000:
1996:
1992:
1965:
1954:
1944:
1935:
1927:
1923:
1915:
1895:
1767:
1766:
1759:
1754:
1753:
1752:
1746:Killing vectors
1730:
1727:
1724:
1723:
1721:
1720:
1668:de Sitter space
1664:Minkowski space
1653:
1649:, the space of
1633:
1610:algebraic group
1584:
1554:
1547:
1540:
1533:
1514:
1468:smooth manifold
1457:
1430:closed subgroup
1400:
1374:
1355:
1326:
1316:
1300:
1295:
1290:
1289:
1273:
1266:
1203:
1167:
1151:In general, if
1149:
1126:
1102:symmetry groups
1090:Euclidean space
1067:
1049:
1047:Minkowski space
1010:
984:
943:
936:
919:
905:
878:
875:Euclidean group
855:
834:
815:
807:
779:
766:Isometry groups
729:affine space A(
675:
471:
464:diffeomorphisms
409:
380:
325:
308:
307:
274:diffeomorphisms
210:
121:are called the
52:isometry groups
28:
23:
22:
15:
12:
11:
5:
2237:
2235:
2227:
2226:
2221:
2216:
2206:
2205:
2201:
2200:
2191:Katsumi Nomizu
2184:
2174:
2164:
2139:
2137:
2134:
2132:
2131:
2113:
2107:
2082:
2079:W. A. Benjamin
2071:Robert Hermann
2063:
2011:
1993:
1991:
1988:
1987:
1986:
1981:
1976:
1974:Klein geometry
1971:
1964:
1961:
1950:
1940:
1931:
1919:
1891:
1885:
1884:
1873:
1868:
1865:
1862:
1857:
1853:
1847:
1844:
1841:
1836:
1832:
1826:
1821:
1818:
1811:
1807:
1802:
1799:
1796:
1791:
1788:
1785:
1782:
1779:
1775:
1755:
1708:represents an
1692:of background
1637:Poincaré group
1632:
1629:
1586:The idea of a
1583:
1580:
1557:
1556:
1552:
1545:
1538:
1531:
1513:
1510:
1396:
1361:
1360:
1351:
1349:
1336:
1333:
1329:
1323:
1319:
1315:
1312:
1306:
1303:
1298:
1271:
1264:
1199:
1163:
1148:
1145:
1141:Julius PlĂŒcker
1124:
1066:
1063:
1062:
1061:
1043:
1037:
1004:
962:
961:
957:
956:
955:
954:
938:
930:
916:
891:
890:
889:
868:
867:
866:
848:
827:
768:
767:
761:
760:
749:
738:
726:
725:
710:
703:
691:
690:
683:
676:
670:
666:
665:
658:
651:
647:
646:
639:
632:
628:
627:
620:
613:
609:
608:
601:
594:
590:
589:
582:
575:
568:
567:
560:
553:
546:
545:
538:
531:
524:
523:
517:
511:
454:Similarly, if
421:homeomorphisms
408:
405:
371:of the object
365:
364:
353:
350:
347:
341:
335:
332:
329:
324:
321:
318:
315:
220:a group. Then
209:
206:
197:into a single
143:isometry group
40:diffeomorphism
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2236:
2225:
2222:
2220:
2217:
2215:
2212:
2211:
2209:
2198:
2197:
2192:
2188:
2185:
2183:
2179:
2175:
2173:
2169:
2166:Takashi Koda
2165:
2163:
2162:0-691-08122-0
2159:
2156:
2152:
2148:
2144:
2141:
2140:
2135:
2127:
2123:
2117:
2114:
2110:
2104:
2100:
2096:
2092:
2086:
2083:
2080:
2076:
2072:
2067:
2064:
2059:
2055:
2051:
2047:
2042:
2037:
2033:
2029:
2025:
2018:
2016:
2012:
2008:
2004:
1998:
1995:
1989:
1985:
1982:
1980:
1977:
1975:
1972:
1970:
1967:
1966:
1962:
1960:
1958:
1953:
1949:
1943:
1939:
1934:
1930:
1922:
1918:
1913:
1909:
1905:
1904:antisymmetric
1902:
1899:
1894:
1890:
1871:
1863:
1855:
1851:
1842:
1834:
1830:
1824:
1819:
1816:
1809:
1805:
1797:
1786:
1783:
1780:
1773:
1765:
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1763:
1758:
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1747:
1742:
1738:
1718:
1713:
1711:
1707:
1703:
1699:
1695:
1691:
1687:
1683:
1679:
1675:
1673:
1669:
1665:
1660:
1656:
1652:
1648:
1645:
1644:Lorentz group
1641:
1638:
1630:
1628:
1624:
1622:
1618:
1614:
1611:
1607:
1603:
1600:
1595:
1593:
1589:
1581:
1579:
1577:
1572:
1569:
1564:
1562:
1551:
1544:
1537:
1530:
1527:
1526:
1525:
1523:
1519:
1511:
1509:
1507:
1503:
1499:
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1491:
1487:
1485:
1479:
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1473:
1469:
1464:
1460:
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1451:
1447:
1443:
1439:
1435:
1431:
1427:
1423:
1419:
1415:
1411:
1407:
1402:
1399:
1395:
1391:
1387:
1381:
1377:
1372:
1368:
1359:
1352:
1350:
1334:
1331:
1327:
1321:
1317:
1313:
1310:
1304:
1301:
1296:
1288:
1287:
1284:
1282:
1278:
1274:
1267:
1260:
1256:
1251:
1249:
1245:
1241:
1239:
1234:
1230:
1226:
1221:
1219:
1215:
1211:
1207:
1202:
1198:
1194:
1191:
1187:
1183:
1180:(a choice of
1179:
1175:
1171:
1166:
1162:
1158:
1154:
1146:
1144:
1142:
1138:
1137:line geometry
1134:
1130:
1122:
1117:
1115:
1111:
1107:
1103:
1099:
1095:
1091:
1086:
1084:
1080:
1076:
1072:
1064:
1057:
1053:
1048:
1044:
1041:
1038:
1034:
1030:
1026:
1022:
1018:
1014:
1008:
1005:
1000:
996:
992:
988:
982:
978:
974:
971:
967:
964:
963:
959:
958:
951:
947:
941:
934:
931:
927:
923:
917:
913:
909:
903:
899:
895:
894:
892:
886:
882:
876:
872:
871:
869:
863:
859:
853:
849:
845:
841:
837:
832:
828:
825:
819:
811:
805:
800:
796:
790:
786:
782:
777:
773:
772:
770:
769:
765:
764:
758:
754:
750:
747:
743:
739:
736:
732:
728:
727:
723:
719:
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708:
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697:
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688:
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681:
677:
673:
668:
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583:
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369:automorphisms
348:
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137:of the space
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44:homeomorphism
41:
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32:
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2194:
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2098:
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1606:group action
1601:
1599:vector space
1596:
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1576:Grassmannian
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1471:
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1450:Lie subgroup
1445:
1437:
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1393:
1392:modulo
1389:
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1121:vector space
1118:
1108:of constant
1094:affine space
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1078:
1068:
1055:
1051:
1032:
1028:
1024:
1020:
1016:
1012:
1007:Grassmannian
998:
994:
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977:affine group
972:
966:Affine space
949:
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366:
301:homomorphism
299:-space is a
296:
292:
288:
286:
281:
277:
269:
265:
261:
257:
249:
241:
237:
233:
229:
225:
224:is called a
221:
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186:
182:
179:group action
170:
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115:transitively
108:
104:
90:
63:
57:
2143:John Milnor
1710:anisotropic
519:stabilizer
443:) into the
379:. The pair
60:mathematics
2219:Lie groups
2208:Categories
2136:References
2091:Lev Landau
2077:, page 4,
2041:1809.01224
2034:(1): 229.
1690:space part
1680:using the
1635:Given the
1592:Mikio Sato
1566:Since the
1414:continuous
1373:for which
1170:stabilizer
1112:, such as
650:oriented H
612:oriented E
123:symmetries
76:Lie groups
48:flat torus
2058:1029-8479
1852:ξ
1831:ξ
1774:ξ
1696:for some
1504:) acting
1456:. Hence
1442:Lie group
1422:Hausdorff
1332:−
1110:curvature
948:) / O(1,
906:H â
O(1,
549:oriented
479:â Diffeo(
323:→
314:ρ
107:on which
95:non-empty
2124:(1972),
2097:(1980),
1963:See also
1946:, where
1898:constant
1305:′
1279:of
1231:}, then
1075:geometry
1065:Geometry
1050:M â
ISO(
985:A = Aff(
774:Sphere (
475: :
447:of
407:Examples
246:category
175:faithful
167:topology
98:manifold
2193:(1969)
2149:(1974)
2073:(1966)
1955:is the
1734:
1722:
1694:metrics
1662:is the
1604:with a
1512:Example
1470:and so
1444:, then
1424:, then
1240:-torsor
1235:is the
1168:is the
1054:) / SO(
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856:P â
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1651:cosets
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1365:where
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1190:cosets
1182:origin
1159:, and
1133:minors
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960:Others
920:SO(1,
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879:E â
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716:) Ă O(
653:SO(1,
513:group
507:space
202:-orbit
68:action
2180:from
2170:from
2036:arXiv
1990:Notes
1486:coset
1466:is a
1448:is a
1440:is a
1428:is a
1052:n-1,1
975:(for
970:field
968:over
838:â
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685:O(1,
678:O(2,
634:O(1,
588:â 1)
566:â 1)
544:â 1)
458:is a
415:is a
149:, or
93:is a
88:group
72:group
70:of a
36:torus
2158:ISBN
2103:ISBN
2093:and
2054:ISSN
2032:2019
2003:left
1743:+ 1)
1670:and
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1416:and
1096:and
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270:Diff
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112:acts
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