909:
space. How many dimensions of
Euclidean space does it take to isometrically embed hyperbolic 1-space? Or what I'm really looking for is how many dimensions of Euclidean space does it take for hyperbolic 1-space to be embedded in and be as much itself, if you know what I mean, as the circle is in the Euclidean plane and the n-sphere is in Euclidean n+1-space. That may be eqivilent in all cases to a manifold being able to be isometrically embedded in a certain space, but I'm not sure. The number of dimensions it takes to isometrically embed hyperbolic 1-space could shed some insight into the number of dimensions it takes to isometrically embed hyperbolic 2-space, which I believe has been narrowed down to 4 or 5 now but I'm not sure if it's been proven that it doesn't take more than 5. Any answers to these questions would be appreciated.
1057:
if you look at the bowl from the inside it is concave in all directions so principal curvatures are all negative giving again a positive
Gaussian curvature as the product of two negative numbers. It is the same for the hyperbolic model. From the outside it has positive curvature in all directions but from the inside it has negative curvature in all directions. So it has positive Gaussian curvature however you look at it. Of course there is also the hyperbolic surface in one sheet which does have saddle point structure with negative Gaussian curvature but that is not a model of hyperbolic space.
1734:
the term (since Klein), and as far as i know it is still the primary meaning today. One ambiguity is that a priori there is no reason for the curvature to be normalised to -1, scaling the metric results in a space with the same geometric properties (up to constants in metric equalities, etc.), this is done only for the sake of clarity, so that "n-hyperbolic space" refers only to one object rather than an infinite family; but of course every simply-connected manifold of constant negative curvature is as deserving of the name "hyperbolic space" in principle.
1740:
of ancient euclidean geometry. At last this seems to be the assumption under which part of the "hyperbolic geometry" article is written. I agree with you that this is nonsensical, there is no serious reason to distinguish between dimensions in this article, i think in the current version there is no such discrimination. (Another reason is that in colloquial meaning "space" refers to 3-dimensional space, and so "hyperbolic space" may in context refer to actual hyperbolic 3-space ; but i don't think it makes sense with respect to this article either).
84:
74:
53:
22:
1711:? Reading the comments here, it seems that some people regards hyperbolic geometry to be a broader term than hyperbolic space. So can it be said that hyperbolic space is a form of hyperbolic geometry? And if so, what kinds of hyperbolic geometries are there that do not also count as hyperbolic spaces?
1739:
I think the reason some people wanted a distinction between 2- and higher-dimensional spaces is that the hyperbolic plane can be characterised by axioms on points, lines, circles, etc. in the same way as the
Euclidean plane, so "hyperbolic geometry" is the study of this axiomatic geometry in the vein
1733:
Hyperbolic geometry is the study of the geometric properties of hyperbolic space, like
Euclidean geometry is the study of properties of Euclidean space. As to what hyperbolic space is: by default it should be used to denote the spaces described in this article, since this is the historical meaning of
1056:
Everywhere you find stated that hyperbolic space has negative curvature with a saddle point structure. But is this actually true? Consider a hemispherical bowl. From the outside it is convex in all directions so its principal curvatures are positive and so the
Gaussian curvature is also positive. But
927:
Thanks
Fropuff. I now get why negative curvature doesn't exist (and why, if it did exist by the absence of the absolute value function in the measurement of curvature for curves, a curve or section thereof with non-zero curvature would have both positive and negative curvature of the same magnitude
252:
It would be good to explain the relation between the hyperboloid model and the other models in some detail. For example, stereographic projection of the hyperboloid model gives the
Poincaré disk model. These relations between the models should be given as explicit as possible. Perhaps as many models
248:
The hyperboloid model is perhaps the least known model of hyperbolic space. It has some advantages: it is easy to describe the group of isometries for all dimensions (just as SO(1,n)); it is rather easy to describe the geodesics (as any non-empty intersection between a planes passing the origin and
214:
is meant to be for those with more mathematical training. That's what I've observed in watching these articles develop. I think also that some editors had the notion that the geometry article should cover the plane, and higher dimensions in general should be covered here. Unfortunately, strictly
1803:
like "the sphere is the unique simply connected, 2-dimensional, orientable
Riemannian manifold of constant sectional curvature equal to 1" I feel like there would be significant pushback. Emphasizing that hyperbolic space is "unique" (i.e. every hyperbolic space is isomorphic) doesn't seem like an
1098:
Yes you are right and the article does say so. I had verified it some time ago by transforming to the hyperbolic analogue of spherical coordinates. What I don’t understand is how so many people (Helmholtz, Killing, Poincare etc) previous to
Minkowski's introduction of his space in 1908 are said to
917:
The hyperbolic line doesn't actually exist. The reason is that
Riemannian geometry is trivial in dimension 1; meaning that all 1-dimensional Riemannian manifolds are locally isometric. Therefore, the curvature of everything is 0. Negative curvature spaces exists only in dimension two or higher. --
1798:
It might be better to define "a hyperbolic space" in such a way that arbitrary constant negative sectional curvature is allowed, instead of trying to define "the hyperbolic space". After all, if someone comes across such a space they are still likely to call it "hyperbolic space", in the same way
908:
2, hyperbolic 1-space, the "maximally symmetric, simply connected ," 1-dimensional "Riemannian manifold with constant sectional curvature −1," is not often talked about, and I am curious about it. What is it called? The hyperbolic line? I can tell that hyperbolas are not examples of hyperbolic-1
1687:
There's also a lot of results on minimal surfaces and such which i am not familiar with, and probably a lot more i'm not even aware of. The article is also very much lacking in references, i'll add some (eg. to Ratcliffe's book or Milnor's article on "the first 150 years") when have the time to
1121:
I am not familiar with the history or those old papers, so whatever I say here is speculation. However, it would not surprise me at all if Minkowski's space was already well known before Minkowski. Mathematically it is very simple, and thus probably very old. Minkowski's innovation was probably
193:
We could move some of the material to hyperboloid model. But if we only talk about general properties, what distinguishes this article from the one on hyperbolic geometry? A general discussion of models might make sense, on the grounds that a hyperbolic space, concretely, means some model.
219:
cover a lot more of the general stuff and applications than hyperbolic space, since the term refers to more than hyperbolic space itself. Hyperbolic space could cover, as you say, a general discussion of the standard models and various other representations, including say, combinatorial
1666:
I started editing this page as it seemed a bit bare-bones and not super clear about what it wanted to be about. I removed most of the details on the specific models as they have their own articles. Maybe some details of the isomorphisms between the various models should be added back.
784:
176:
Yeah, I noticed that. I think this article should be more about the general properties of hyperbolic space, mentioning only a model when it makes it particularly easy to see something. Also some general stuff about applications and so forth.
1718:). Why? This seems like a rather arbitrary constraint to me. What would be the purpose of discriminating between different spaces in this way? To me, this only seems to add a layer of linguistic complexity that isn't really justified. —
478:
1498:
1633:
1404:
1313:
696:
1229:
223:
In any case, clearly a reorganization of some sort is needed, which takes into account these issues. BTW, your work on these hyperbolic geometry related pages is greatly appreciated and needed. --
140:
1000:
which does mention the hyperbolic metric. It is not possible to give a complete analogy with Riemann space because the quadratic differential form does not define a metric in the same way.
546:
885:
330:
688:
1520:
829:
347:
1421:
1834:
130:
1829:
1149:
I want to make this page the main page for higher dimensional (3 dimensional and higher) hyperbolic geometry. For me this is mostly to "unburden" the
1540:
1024:
March 2006 issue; written by David Samuels; photography by Richard Barnes). Could be useful to somewhat analogically explain hyperbolic space.
106:
1775:
By H×E×S, do you mean one dimension each? H¹×E¹ is indistinguishable from E²; E²×S¹ is a flat 3-space that is periodic in one direction.
1760:
Has anyone thought about HxExS geometry, or H3xS3? It just seems like they would be really cool for a game Also how would HxExS work?
981:
779:{\displaystyle {\begin{pmatrix}1&0&\ldots &0\\0&&&\\\vdots &&A&\\0&&&\\\end{pmatrix}}}
1099:
have used the 'well known hyperbolic model' without using Minkowski space. I did look at Killing's article but saw nothing about it.
97:
58:
1335:
1240:
1761:
1799:
that a (hyper)sphere of radius other than 1 is still going to be called a "(hyper)sphere". If we tried to make a definition at
1040:
210:(because of the popularity of the term in various books, articles, etc.) is meant to be quite wide, while the audience for
1173:
271:
33:
1714:
Also, some people seem to consider hyperbolic spaces to strictly be of dimension 3 or greater (i.e., not including the
599:+1)/2, the maximal dimension of the isometry group of a Riemannian manifold. Therefore, hyperbolic space is said to be
974:
distance function) typically seems to be given, it would also be highly relevant to give the infinitesimal version.
1745:
1693:
1639:
929:
21:
513:
997:
985:
1765:
860:
952:
39:
83:
285:
1652:
1028:
977:
335:
240:
195:
167:
649:
1812:
1741:
1708:
1689:
1161:
1154:
1150:
1127:
1085:
1036:
933:
910:
888:
207:
162:
This strikes me as a dubious article for hyperbolic space, since it is largely a discussion of the
105:
on Knowledge. If you would like to participate, please visit the project page, where you can join
612:
89:
73:
52:
1122:
merely to notice that it was applicable to the physical world described by special relativity.
1783:
1503:
967:
831:
808:
635:
236:
163:
1805:
1715:
948:
211:
1648:
1104:
1077:
1073:
1062:
1005:
257:
473:{\displaystyle \langle x,y\rangle =-x_{0}y_{0}+x_{1}y_{1}+x_{2}y_{2}+\cdots +x_{n}y_{n}.}
239:, changing notation to fit that introduced there. The original section is the one below.
256:
I also would like to thank the authors up to now for their effort writing this article.
1809:
1523:
1493:{\displaystyle \Omega _{n}={\frac {2\pi ^{n/2}}{\Gamma \left({\frac {n}{2}}\right)}}\,}
1123:
1081:
1032:
843:
794:
643:
584:
1823:
1723:
500:
339:
227:
181:
1644:
But i am wondering where to put it on this page, also some cross linking is needed.
945:
of course any connected 1-dimensional manifold, even the circle, is simply connected
1804:
essential part of the definition, but is worth discussing, as is currently done at
1779:
1021:
971:
919:
623:
560:
1157:
page that still needs to be included here are the formulas for spheres and balls:
907:
For all the talk about the hyperbolic plane and even hyperbolic n-space of n : -->
966:
There appears to be no proper article or section of an article on the hyperbolic
1411:
835:
332:
102:
1628:{\displaystyle \Omega _{n}R^{n-1}\int _{0}^{r}\sinh ^{n-1}{\frac {s}{R}}ds\,.}
1100:
1058:
1001:
79:
279:
1815:
1787:
1769:
1749:
1727:
1697:
1656:
1131:
1108:
1089:
1066:
1044:
1009:
989:
955:
936:
922:
260:
243:
230:
198:
184:
170:
1719:
1319:
488:
224:
206:
Sure, a general discussion would be good. I think that the audience of
178:
1680:
discrete subgroups of isometries (in the "hyperbolic manifold" section);
1800:
1670:
There's plenty of stuff that could (should?) be added, for instance:
1778:
H³×S³ is a six-dimensional space; how would you use that in a game?
220:
representations and its relevance to delta-hyperbolic spaces.
1072:
You are ignoring the fact that the hyperbolic model is embedded in
1052:
Does the hyperbolic model really have negative Gaussian curvature?
1399:{\displaystyle \Omega _{n}R^{n-1}\sinh ^{n-1}{\frac {r}{R}}\,}
15:
1308:{\displaystyle \pi R^{3}\sinh {\frac {2r}{R}}-2\pi R^{2}r\,.}
1703:
Difference between hyperbolic space and hyperbolic geometry?
1640:
Talk:Hyperbolic geometry#Removed spheres and balls sections
932:). Your reply was definately helpful, as was Tom Ruen's.
575:. This group clearly preserves the hyperbolic metric on
495:
fixing the origin. This group is sometimes called the (
1153:
page form the higher dimensional "stuff". Bits of the
1020:
Just happened to bump into this old Discover article (
705:
1543:
1506:
1424:
1338:
1243:
1224:{\displaystyle 4\pi R^{2}\sinh ^{2}{\frac {r}{R}}\,.}
1176:
863:
811:
699:
652:
516:
350:
288:
1707:
What is the difference between hyperbolic space and
603:. The group of orientation preserving isometries of
101:, a collaborative effort to improve the coverage of
1808:. The comments there could plausibly be extended. –
1627:
1514:
1492:
1398:
1329:dimensional space the corresponding expression is
1307:
1223:
879:
823:
778:
682:
540:
472:
324:
1794:Hyperbolic space with other sectional curvature
1683:the boundary at infinity and harmonic analysis;
928:as Tomruen said the circle had in his reply in
253:as possible should be included in the article?
1674:hyperbolic isometries and the isometry group;
618:The orientation preserving isometry group SO(
8:
874:
864:
529:
517:
363:
351:
947:" is false. For example, the circle is not
503:. The subgroup which preserves the sign of
19:
1016:An Unusual Explanation Of Hyperbolic Space
970:. Although its "integrated" version (the
47:
1604:
1589:
1579:
1574:
1558:
1548:
1542:
1505:
1471:
1452:
1448:
1438:
1429:
1423:
1384:
1369:
1353:
1343:
1337:
1291:
1263:
1251:
1242:
1206:
1197:
1187:
1175:
862:
810:
700:
698:
651:
515:
461:
451:
432:
422:
409:
399:
386:
376:
349:
287:
541:{\displaystyle \langle x,x\rangle <0}
1620:
1510:
1488:
1394:
1300:
1216:
872:
867:
49:
1677:hyperplanes and hyperbolic polyhedra;
880:{\displaystyle \langle \,,\,\rangle }
249:the upper part of the hyperboloid).
7:
591:. This isometry group has dimension
95:This article is within the scope of
1234:The volume of the enclosed ball is
895:,1)-invariant quadratic form on SO(
235:I've moved the symmetry section to
38:It is of interest to the following
1545:
1507:
1464:
1426:
1340:
1142:Goal of this Hyperbolic space page
903:Questions about hyperbolic 1-space
14:
1835:Mid-priority mathematics articles
1080:. That makes all the difference.
325:{\displaystyle (n+1)\times (n+1)}
115:Knowledge:WikiProject Mathematics
1830:Start-Class mathematics articles
1647:can we work together to do this?
1167:The surface area of a sphere is
118:Template:WikiProject Mathematics
82:
72:
51:
20:
842:is therefore isomorphic to the
683:{\displaystyle (1,0,\ldots ,0)}
135:This article has been rated as
1132:09:17, 14 September 2013 (UTC)
1109:05:55, 14 September 2013 (UTC)
1090:22:52, 12 September 2013 (UTC)
1067:19:15, 12 September 2013 (UTC)
1010:19:46, 12 September 2013 (UTC)
677:
653:
319:
307:
301:
289:
215:speaking, hyperbolic geometry
1:
1788:03:48, 27 November 2023 (UTC)
1698:16:39, 20 December 2021 (UTC)
109:and see a list of open tasks.
1770:19:09, 9 February 2022 (UTC)
1530:The measure of the enclosed
1160:The text I removed from the
1750:10:49, 5 January 2022 (UTC)
1728:16:28, 4 January 2022 (UTC)
1662:Modifications and TODO list
990:21:07, 9 October 2009 (UTC)
956:18:02, 31 August 2006 (UTC)
624:transitively and faithfully
615:of the full Lorentz group.
550:orthochronous Lorentz group
1851:
1816:01:18, 13 March 2024 (UTC)
1657:11:18, 12 April 2015 (UTC)
913:17:34, 29 May 2006 (UTC)
571:restricts to an action on
261:23:18, 26 April 2006 (UTC)
244:08:21, 19 April 2006 (UTC)
231:22:46, 17 April 2006 (UTC)
199:20:31, 17 April 2006 (UTC)
185:08:48, 17 April 2006 (UTC)
171:08:25, 17 April 2006 (UTC)
1515:{\displaystyle \Gamma \,}
824:{\displaystyle n\times n}
134:
67:
46:
1045:19:08, 17 May 2010 (UTC)
937:20:07, 5 June 2006 (UTC)
930:Talk:Hyperbolic geometry
923:05:47, 4 June 2006 (UTC)
690:is a matrix of the form
141:project's priority scale
1688:locate them precisely.
998:pseudo-Riemannian space
630:. Which is to say that
98:WikiProject Mathematics
1806:§ Hyperbolic manifolds
1629:
1516:
1494:
1400:
1318:For the measure of an
1309:
1225:
881:
825:
780:
684:
542:
474:
326:
28:This article is rated
1630:
1517:
1495:
1401:
1310:
1226:
996:There is the article
882:
838:+1. Hyperbolic space
826:
781:
685:
638:for the action of SO(
543:
475:
327:
1541:
1504:
1422:
1336:
1241:
1174:
1162:hyperbolic geometry
861:
809:
697:
650:
514:
487:,1) is the group of
348:
286:
121:mathematics articles
1709:hyperbolic geometry
1584:
1155:hyperbolic geometry
1151:hyperbolic geometry
889:Cartan-Killing form
793:is a matrix in the
601:maximally symmetric
491:of Minkowski space
208:hyperbolic geometry
1625:
1621:
1570:
1512:
1511:
1490:
1489:
1396:
1395:
1305:
1301:
1221:
1217:
877:
873:
868:
857:The bilinear form
821:
776:
770:
680:
613:identity component
611:,1), which is the
538:
470:
338:that preserve the
322:
90:Mathematics portal
34:content assessment
1612:
1486:
1479:
1392:
1276:
1214:
1048:
1031:comment added by
980:comment added by
968:Riemannian metric
962:Hyperbolic metric
832:orthogonal matrix
636:homogeneous space
237:hyperboloid model
164:hyperboloid model
155:
154:
151:
150:
147:
146:
1842:
1716:hyperbolic plane
1634:
1632:
1631:
1626:
1613:
1605:
1600:
1599:
1583:
1578:
1569:
1568:
1553:
1552:
1521:
1519:
1518:
1513:
1499:
1497:
1496:
1491:
1487:
1485:
1484:
1480:
1472:
1462:
1461:
1460:
1456:
1439:
1434:
1433:
1405:
1403:
1402:
1397:
1393:
1385:
1380:
1379:
1364:
1363:
1348:
1347:
1314:
1312:
1311:
1306:
1296:
1295:
1277:
1272:
1264:
1256:
1255:
1230:
1228:
1227:
1222:
1215:
1207:
1202:
1201:
1192:
1191:
1047:
1025:
992:
949:simply connected
943:Also note that "
891:, the unique SO(
886:
884:
883:
878:
830:
828:
827:
822:
785:
783:
782:
777:
775:
774:
689:
687:
686:
681:
607:is the group SO(
583:,1) is the full
548:) is called the
547:
545:
544:
539:
499:+1)-dimensional
479:
477:
476:
471:
466:
465:
456:
455:
437:
436:
427:
426:
414:
413:
404:
403:
391:
390:
381:
380:
331:
329:
328:
323:
212:hyperbolic space
123:
122:
119:
116:
113:
92:
87:
86:
76:
69:
68:
63:
55:
48:
31:
25:
24:
16:
1850:
1849:
1845:
1844:
1843:
1841:
1840:
1839:
1820:
1819:
1796:
1758:
1705:
1664:
1585:
1554:
1544:
1539:
1538:
1502:
1501:
1467:
1463:
1444:
1440:
1425:
1420:
1419:
1406:where the full
1365:
1349:
1339:
1334:
1333:
1287:
1265:
1247:
1239:
1238:
1193:
1183:
1172:
1171:
1144:
1078:Euclidean space
1074:Minkowski space
1054:
1026:
1018:
975:
964:
905:
859:
858:
807:
806:
769:
768:
766:
764:
762:
756:
755:
753:
748:
746:
740:
739:
737:
735:
733:
727:
726:
721:
716:
711:
701:
695:
694:
648:
647:
512:
511:
509:
457:
447:
428:
418:
405:
395:
382:
372:
346:
345:
284:
283:
268:
241:Gene Ward Smith
196:Gene Ward Smith
168:Gene Ward Smith
160:
120:
117:
114:
111:
110:
88:
81:
61:
32:on Knowledge's
29:
12:
11:
5:
1848:
1846:
1838:
1837:
1832:
1822:
1821:
1795:
1792:
1791:
1790:
1776:
1757:
1754:
1753:
1752:
1736:
1735:
1704:
1701:
1685:
1684:
1681:
1678:
1675:
1663:
1660:
1636:
1635:
1624:
1619:
1616:
1611:
1608:
1603:
1598:
1595:
1592:
1588:
1582:
1577:
1573:
1567:
1564:
1561:
1557:
1551:
1547:
1528:
1527:
1524:Gamma function
1509:
1483:
1478:
1475:
1470:
1466:
1459:
1455:
1451:
1447:
1443:
1437:
1432:
1428:
1416:
1415:
1391:
1388:
1383:
1378:
1375:
1372:
1368:
1362:
1359:
1356:
1352:
1346:
1342:
1316:
1315:
1304:
1299:
1294:
1290:
1286:
1283:
1280:
1275:
1271:
1268:
1262:
1259:
1254:
1250:
1246:
1232:
1231:
1220:
1213:
1210:
1205:
1200:
1196:
1190:
1186:
1182:
1179:
1143:
1140:
1139:
1138:
1137:
1136:
1135:
1134:
1114:
1113:
1112:
1111:
1093:
1092:
1053:
1050:
1017:
1014:
1013:
1012:
963:
960:
959:
958:
941:
940:
939:
934:Kevin Lamoreau
911:Kevin Lamoreau
904:
901:
876:
871:
866:
844:quotient space
820:
817:
814:
795:rotation group
787:
786:
773:
767:
765:
763:
761:
758:
757:
754:
752:
749:
747:
745:
742:
741:
738:
736:
734:
732:
729:
728:
725:
722:
720:
717:
715:
712:
710:
707:
706:
704:
679:
676:
673:
670:
667:
664:
661:
658:
655:
646:of the vector
644:isotropy group
585:isometry group
579:. In fact, O(
537:
534:
531:
528:
525:
522:
519:
507:
481:
480:
469:
464:
460:
454:
450:
446:
443:
440:
435:
431:
425:
421:
417:
412:
408:
402:
398:
394:
389:
385:
379:
375:
371:
368:
365:
362:
359:
356:
353:
321:
318:
315:
312:
309:
306:
303:
300:
297:
294:
291:
267:
264:
204:
203:
202:
201:
188:
187:
159:
156:
153:
152:
149:
148:
145:
144:
133:
127:
126:
124:
107:the discussion
94:
93:
77:
65:
64:
56:
44:
43:
37:
26:
13:
10:
9:
6:
4:
3:
2:
1847:
1836:
1833:
1831:
1828:
1827:
1825:
1818:
1817:
1814:
1811:
1807:
1802:
1793:
1789:
1785:
1781:
1777:
1774:
1773:
1772:
1771:
1767:
1763:
1755:
1751:
1747:
1743:
1738:
1737:
1732:
1731:
1730:
1729:
1725:
1721:
1717:
1712:
1710:
1702:
1700:
1699:
1695:
1691:
1682:
1679:
1676:
1673:
1672:
1671:
1668:
1661:
1659:
1658:
1654:
1650:
1645:
1642:
1641:
1622:
1617:
1614:
1609:
1606:
1601:
1596:
1593:
1590:
1586:
1580:
1575:
1571:
1565:
1562:
1559:
1555:
1549:
1537:
1536:
1535:
1533:
1525:
1481:
1476:
1473:
1468:
1457:
1453:
1449:
1445:
1441:
1435:
1430:
1418:
1417:
1413:
1410:-dimensional
1409:
1389:
1386:
1381:
1376:
1373:
1370:
1366:
1360:
1357:
1354:
1350:
1344:
1332:
1331:
1330:
1328:
1324:
1322:
1302:
1297:
1292:
1288:
1284:
1281:
1278:
1273:
1269:
1266:
1260:
1257:
1252:
1248:
1244:
1237:
1236:
1235:
1218:
1211:
1208:
1203:
1198:
1194:
1188:
1184:
1180:
1177:
1170:
1169:
1168:
1165:
1163:
1158:
1156:
1152:
1147:
1141:
1133:
1129:
1125:
1120:
1119:
1118:
1117:
1116:
1115:
1110:
1106:
1102:
1097:
1096:
1095:
1094:
1091:
1087:
1083:
1079:
1075:
1071:
1070:
1069:
1068:
1064:
1060:
1051:
1049:
1046:
1042:
1038:
1034:
1030:
1023:
1015:
1011:
1007:
1003:
999:
995:
994:
993:
991:
987:
983:
979:
973:
969:
961:
957:
954:
953:68.100.203.44
950:
946:
942:
938:
935:
931:
926:
925:
924:
921:
916:
915:
914:
912:
902:
900:
898:
894:
890:
869:
855:
853:
849:
845:
841:
837:
833:
818:
815:
812:
804:
800:
796:
792:
771:
759:
750:
743:
730:
723:
718:
713:
708:
702:
693:
692:
691:
674:
671:
668:
665:
662:
659:
656:
645:
641:
637:
633:
629:
625:
621:
616:
614:
610:
606:
602:
598:
594:
590:
586:
582:
578:
574:
570:
566:
562:
557:
555:
551:
535:
532:
526:
523:
520:
506:
502:
501:Lorentz group
498:
494:
490:
486:
467:
462:
458:
452:
448:
444:
441:
438:
433:
429:
423:
419:
415:
410:
406:
400:
396:
392:
387:
383:
377:
373:
369:
366:
360:
357:
354:
344:
343:
342:
341:
340:bilinear form
337:
334:
316:
313:
310:
304:
298:
295:
292:
281:
277:
275:
265:
263:
262:
259:
254:
250:
246:
245:
242:
238:
233:
232:
229:
226:
221:
218:
213:
209:
200:
197:
192:
191:
190:
189:
186:
183:
180:
175:
174:
173:
172:
169:
165:
157:
142:
138:
132:
129:
128:
125:
108:
104:
100:
99:
91:
85:
80:
78:
75:
71:
70:
66:
60:
57:
54:
50:
45:
41:
35:
27:
23:
18:
17:
1797:
1759:
1713:
1706:
1686:
1669:
1665:
1646:
1643:
1638:See further
1637:
1531:
1529:
1407:
1326:
1320:
1317:
1233:
1166:
1159:
1148:
1145:
1076:rather than
1055:
1019:
982:72.95.242.63
972:metric space
965:
944:
906:
896:
892:
856:
851:
847:
839:
802:
801:); that is,
798:
790:
788:
639:
631:
627:
619:
617:
608:
604:
600:
596:
592:
588:
580:
576:
572:
568:
564:
558:
553:
552:, denoted O(
549:
504:
496:
492:
484:
482:
273:
269:
255:
251:
247:
234:
222:
216:
205:
161:
137:Mid-priority
136:
96:
62:Mid‑priority
40:WikiProjects
1412:solid angle
1027:—Preceding
1022:Knit Theory
976:—Preceding
836:determinant
483:That is, O(
112:Mathematics
103:mathematics
59:Mathematics
30:Start-class
1824:Categories
1762:92.9.33.11
1649:WillemienH
1164:page was:
489:isometries
270:The group
258:Pierreback
1810:jacobolus
1534:ball is:
1323:-1 sphere
1124:JRSpriggs
1082:JRSpriggs
1033:Komitsuki
642:,1). The
622:,1) acts
280:Lie group
1780:—Tamfang
1742:jraimbau
1690:jraimbau
1041:contribs
1029:unsigned
978:unsigned
336:matrices
266:Symmetry
1756:Hmmmmmm
1522:is the
920:Fropuff
887:is the
850:,1)/SO(
567:,1) on
278:is the
158:Content
139:on the
1801:sphere
805:is an
789:where
561:action
228:(Talk)
217:should
182:(Talk)
36:scale.
1101:JFB80
1059:JFB80
1002:JFB80
899:,1).
834:with
634:is a
563:of O(
556:,1).
1784:talk
1766:talk
1746:talk
1724:talk
1694:talk
1653:talk
1587:sinh
1500:and
1367:sinh
1258:sinh
1195:sinh
1146:Hi
1128:talk
1105:talk
1086:talk
1063:talk
1037:talk
1006:talk
986:talk
559:The
533:<
510:(if
333:real
282:of
1813:(t)
1720:Kri
1325:in
854:).
846:SO(
797:SO(
626:on
587:of
276:,1)
225:C S
179:C S
131:Mid
1826::
1786:)
1768:)
1748:)
1726:)
1696:)
1655:)
1602:
1594:−
1572:∫
1563:−
1546:Ω
1508:Γ
1465:Γ
1446:π
1427:Ω
1414:is
1382:
1374:−
1358:−
1341:Ω
1285:π
1279:−
1261:
1245:π
1204:
1181:π
1130:)
1107:)
1088:)
1065:)
1043:)
1039:•
1008:)
988:)
951:.
875:⟩
865:⟨
816:×
744:⋮
719:…
669:…
530:⟩
518:⟨
442:⋯
370:−
364:⟩
352:⟨
305:×
272:O(
177:--
166:.
1782:(
1764:(
1744:(
1722:(
1692:(
1651:(
1623:.
1618:s
1615:d
1610:R
1607:s
1597:1
1591:n
1581:r
1576:0
1566:1
1560:n
1556:R
1550:n
1532:n
1526:.
1482:)
1477:2
1474:n
1469:(
1458:2
1454:/
1450:n
1442:2
1436:=
1431:n
1408:n
1390:R
1387:r
1377:1
1371:n
1361:1
1355:n
1351:R
1345:n
1327:n
1321:n
1303:.
1298:r
1293:2
1289:R
1282:2
1274:R
1270:r
1267:2
1253:3
1249:R
1219:.
1212:R
1209:r
1199:2
1189:2
1185:R
1178:4
1126:(
1103:(
1084:(
1061:(
1035:(
1004:(
984:(
897:n
893:n
870:,
852:n
848:n
840:H
819:n
813:n
803:A
799:n
791:A
772:)
760:0
751:A
731:0
724:0
714:0
709:1
703:(
678:)
675:0
672:,
666:,
663:0
660:,
657:1
654:(
640:n
632:H
628:H
620:n
609:n
605:H
597:n
595:(
593:n
589:H
581:n
577:H
573:H
569:R
565:n
554:n
536:0
527:x
524:,
521:x
508:0
505:x
497:n
493:R
485:n
468:.
463:n
459:y
453:n
449:x
445:+
439:+
434:2
430:y
424:2
420:x
416:+
411:1
407:y
401:1
397:x
393:+
388:0
384:y
378:0
374:x
367:=
361:y
358:,
355:x
320:)
317:1
314:+
311:n
308:(
302:)
299:1
296:+
293:n
290:(
274:n
143:.
42::
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
