953:
961:
4652:
4347:
3080:
3541:
3304:
4647:{\displaystyle \mathbf {g} _{-1}=\left\{\left.{\begin{pmatrix}0&^{t}p&0\\0&0&J^{-1}p\\0&0&0\end{pmatrix}}\right|p\in \mathbb {R} ^{n}\right\},\quad \mathbf {g} _{-1}=\left\{\left.{\begin{pmatrix}0&0&0\\^{t}q&0&0\\0&qJ^{-1}&0\end{pmatrix}}\right|q\in (\mathbb {R} ^{n})^{*}\right\}}
4818:
2900:
206:
is a locally conformally flat manifold that is not globally conformally flat in this sense, whereas a
Euclidean space, a torus, or any conformal manifold that is covered by an open subset of Euclidean space is (globally) conformally flat in this sense. A locally conformally flat manifold is locally
3323:
1391:
In two dimensions, the group of conformal automorphisms of a space can be quite large (as in the case of
Lorentzian signature) or variable (as with the case of Euclidean signature). The comparative lack of rigidity of the two-dimensional case with that of higher dimensions owes to the analytical
3095:
1410:
The general theory of conformal geometry is similar, although with some differences, in the cases of
Euclidean and pseudo-Euclidean signature. In either case, there are a number of ways of introducing the model space of conformally flat geometry. Unless otherwise clear from the context, this
534:
2480:
240:
Conformal geometry has a number of features which distinguish it from (pseudo-)Riemannian geometry. The first is that although in (pseudo-)Riemannian geometry one has a well-defined metric at each point, in conformal geometry one only has a class of metrics. Thus the length of a
2234:
1392:
fact that the asymptotic developments of the infinitesimal automorphisms of the structure are relatively unconstrained. In
Lorentzian signature, the freedom is in a pair of real valued functions. In Euclidean, the freedom is in a single holomorphic function.
914:
182:. Thus, a conformal metric may be regarded as a metric that is only defined "up to scale". Often conformal metrics are treated by selecting a metric in the conformal class, and applying only "conformally invariant" constructions to the chosen metric.
4660:
1871:
1641:
4200:
4336:
3689:
3075:{\displaystyle \mathbf {y} \in \mathbf {R} ^{n}\mapsto \left({\frac {2\mathbf {y} }{\left|\mathbf {y} \right|^{2}+1}},{\frac {\left|\mathbf {y} \right|^{2}-1}{\left|\mathbf {y} \right|^{2}+1}}\right)\in S\subset \mathbf {R} ^{n+1}.}
3536:{\displaystyle x_{0}=t{\sqrt {2}}{\frac {\left|\mathbf {y} \right|^{2}}{1+\left|\mathbf {y} \right|^{2}}},x_{i}=t{\frac {y_{i}}{\left|\mathbf {y} \right|^{2}+1}},x_{n+1}=t{\sqrt {2}}{\frac {1}{\left|\mathbf {y} \right|^{2}+1}}.}
3299:{\displaystyle x_{0}={\sqrt {2}}{\frac {\left|\mathbf {y} \right|^{2}}{1+\left|\mathbf {y} \right|^{2}}},x_{i}={\frac {y_{i}}{\left|\mathbf {y} \right|^{2}+1}},x_{n+1}={\sqrt {2}}{\frac {1}{\left|\mathbf {y} \right|^{2}+1}}.}
3839:
1281:
411:
2297:
2622:
2813:
2721:
2065:
195:
vanishes. It may only be possible to find a metric in the conformal class that is flat in an open neighborhood of each point. When it is necessary to distinguish these cases, the latter is called
3971:
2082:
364:. Formally, its group of conformal transformations is infinite-dimensional. By contrast, the group of conformal transformations of the compactified Euclidean plane is only 6-dimensional.
1368:
1192:
1031:
808:
4813:{\displaystyle \mathbf {g} _{0}=\left\{\left.{\begin{pmatrix}-a&0&0\\0&A&0\\0&0&a\end{pmatrix}}\right|A\in {\mathfrak {so}}(p,q),a\in \mathbb {R} \right\}.}
1128:
1716:
583:
157:
1395:
In the case of higher dimensions, the asymptotic developments of infinitesimal symmetries are at most quadratic polynomials. In particular, they form a finite-dimensional
632:
356:
At an abstract level, the
Euclidean and pseudo-Euclidean spaces can be handled in much the same way, except in the case of dimension two. The compactified two-dimensional
1474:
293:, although only being defined once a particular representative of the conformal structure has been singled out, do satisfy certain transformation laws involving the
1436:, any angle-preserving local (conformal) transformation is of this form. From this perspective, the transformation properties of flat conformal space are those of
4113:
4271:
3560:
1433:
802:, there is no obstruction to integrating the infinitesimal symmetries, and so the group of conformal transformations is the infinite-dimensional Lie group
3718:
529:{\displaystyle \operatorname {CSO} (1,1)=\left\{\left.{\begin{pmatrix}e^{a}&0\\0&e^{b}\end{pmatrix}}\right|a,b\in \mathbb {R} \right\},}
2475:{\displaystyle x_{0}={\frac {z+1}{\kappa (x){\sqrt {2}}}},\,x_{1}=x_{1},\,\ldots ,\,x_{n}=x_{n},\,x_{n+1}={\frac {(z-1)\kappa (x)}{\sqrt {2}}}}
1305:
The conformal isometries of a domain therefore consist of holomorphic self-maps. In particular, on the conformal compactification – the
1198:
935:
2522:
2751:
2239:
It is readily seen that the image of the sphere under this transformation is null in the
Minkowski space, and so it lies on the cone
5060:
5030:
2633:
1977:
1399:. The pointwise infinitesimal conformal symmetries of a manifold can be integrated precisely when the manifold is a certain model
1962:
215:
from the manifold into a Möbius geometry. In two dimensions, every conformal metric is locally conformally flat. In dimension
5114:
346:
2509:
on the sphere is a metric which is proportional to the standard sphere metric. This gives a realization of the sphere as a
5088:
4989:
952:
2229:{\displaystyle x_{0}={\frac {z+1}{\sqrt {2}}},\,x_{1}=x_{1},\,\ldots ,\,x_{n}=x_{n},\,x_{n+1}={\frac {z-1}{\sqrt {2}}}.}
960:
1291:
5109:
5083:
4984:
4883:
4089:
is again taken as the stabilizer of a null line. Note that both the
Euclidean and pseudo-Euclidean model spaces are
3896:
5094:
4893:
1318:
1139:
98:
70:
974:
309:. Moreover, even though there is no Levi-Civita connection on a conformal manifold, one can instead work with a
187:
4939:
Ecole d'Eté de
Physique Théorique: Champs, cordes et phénomÚnes critiques/Fields, strings and critical phenomena
909:{\displaystyle (\mathbb {Z} \rtimes \mathrm {Diff} (S^{1}))\times (\mathbb {Z} \rtimes \mathrm {Diff} (S^{1})),}
297:
and its derivatives when a different representative is chosen. In particular, (in dimension higher than 3) the
2877:
192:
1310:
3089:
in
Minkowski space. Using the embedding given above, the representative metric section of the null cone is
1674:
1965:.) The Euclidean sphere can be mapped to the conformal sphere in a canonical manner, but not vice versa.
1926:
1866:{\displaystyle N=\left\{(x_{0},\ldots ,x_{n+1})\mid -2x_{0}x_{n+1}+x_{1}^{2}+\cdots +x_{n}^{2}=0\right\}.}
782:
246:
3085:
In terms of these stereographic coordinates, it is possible to give a coordinate system on the null cone
5078:
4979:
1085:
559:
123:
4263:
2853:
1411:
article treats the case of
Euclidean conformal geometry with the understanding that it also applies,
1295:
1052:
924:
310:
245:
cannot be defined, but the angle between two vectors still can. Another feature is that there is no
212:
1636:{\displaystyle q(x_{0},x_{1},\ldots ,x_{n+1})=-2x_{0}x_{n+1}+x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}.}
1944:
591:
66:
289:
Despite these differences, conformal geometry is still tractable. The Levi-Civita connection and
4930:
4052:
2844:: to give a section of this bundle is tantamount to specifying a metric in the conformal class .
1884:
be the future part of the null cone (with the origin deleted). Then the tautological projection
1437:
261:
4101:
To describe the groups and algebras involved in the flat model space, fix the following form on
1425:
The inversive model of conformal geometry consists of the group of local transformations on the
208:
5056:
5026:
4888:
4006:
2859:
2506:
314:
102:
50:
4195:{\displaystyle Q={\begin{pmatrix}0&0&-1\\0&J&0\\-1&0&0\end{pmatrix}}}
4898:
4850:
4331:{\displaystyle \mathbf {g} =\mathbf {g} _{-1}\oplus \mathbf {g} _{0}\oplus \mathbf {g} _{1}}
4056:
3986:
3684:{\displaystyle \rho ={\frac {-2x_{0}x_{n+1}+x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}}{t^{2}}}.}
1947:
of a sphere. Conformal symmetries of a sphere are generated by the inversion in all of its
1693:
1453:
1413:
942:
788:
The conformal compactification of the Minkowski plane is a Cartesian product of two circles
46:
49:. In space higher than two dimensions, conformal geometry may refer either to the study of
1669:
is the projective (or Möbius) model of conformal geometry. A conformal transformation on
1426:
799:
378:
357:
338:
334:
318:
167:
54:
4903:
4823:
Alternatively, this decomposition agrees with a natural Lie algebra structure defined on
5000:
5041:
4044:
4002:
1461:
1306:
290:
242:
82:
4849:
The stabilizer of the null ray pointing up the last coordinate vector is given by the
1935:, since these are homogeneous linear transformations preserving the future null cone.
769:
Conversely, given any such pair of real-valued functions, there exists a vector field
5103:
4941:(Les Houches), ed. by E. Brézin and J. Zinn-Justin, Elsevier Science Publishers B.V.
4922:
4090:
1076:
400:
234:
106:
74:
39:
17:
4997:
Extension au case des trois dimensions de la question du tracé géographique, Note VI
3981:
Consider first the case of the flat conformal geometry in Euclidean signature. The
1299:
778:
45:
In a real two dimensional space, conformal geometry is precisely the geometry of
5095:
http://www.euclideanspace.com/maths/geometry/space/nonEuclid/conformal/index.htm
1948:
1914:
1396:
774:
540:
298:
223:
31:
191:
if there is a metric representing it that is flat, in the usual sense that the
77:
that are defined up to scale. Study of the flat structures is sometimes termed
4048:
1943:
Intuitively, the conformally flat geometry of a sphere is less rigid than the
637:
A 1-parameter group of conformal transformations gives rise to a vector field
3834:{\displaystyle t^{2}g_{ij}(y)\,dy^{i}\,dy^{j}+2\rho \,dt^{2}+2t\,dt\,d\rho ,}
2513:. The standard sphere metric is the restriction of the Euclidean metric on
1697:
403:
342:
4070:, the model flat geometry is defined analogously as the homogeneous space
282:
would involve derivatives of the function λ whereas those associated with
1957:
1952:
1276:{\displaystyle \mathbf {L} _{X}\,d{\bar {z}}=f({\bar {z}})\,d{\bar {z}},}
361:
350:
200:
58:
4934:
3886:. This yields the following representative of the conformal metric on
2858:
Another way to realize the representative metrics is through a special
1449:
199:, although often in the literature no distinction is maintained. The
1448:
The projective model identifies the conformal sphere with a certain
2243:. Consequently, it determines a cross-section of the line bundle
1404:
959:
951:
222:
a conformal metric is locally conformally flat if and only if its
2617:{\displaystyle g=dz^{2}+dx_{1}^{2}+dx_{2}^{2}+\cdots +dx_{n}^{2}}
2808:{\displaystyle =\left\{\lambda ^{2}g\mid \lambda >0\right\}.}
27:
Study of angle-preserving transformations of a geometric space
5046:. Research Lecture Notes, University of Vienna (Dissertation).
2822:, as in the previous section, determines a conformal scale on
2741:
is a positive function on the sphere. The conformal class of
2716:{\displaystyle z^{2}+x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}=1.}
2060:{\displaystyle z^{2}+x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}=1.}
682:
In particular, using the above description of the Lie algebra
260:
are two representatives of the conformal structure, then the
777:
of infinitesimal symmetries of the conformal structure, the
4685:
4517:
4375:
2745:, denoted , is the collection of all such representatives:
445:
3860:
consists of a specification of the value of the variable
353:
is concerned with the implications of preserving angles.
968:
The group of conformal symmetries of the quadratic form
1407:
taking universal covers and discrete group quotients).
1309:– the conformal transformations are given by the
4692:
4524:
4382:
4128:
4051:. Thus the conformally flat models are the spaces of
452:
4663:
4350:
4274:
4116:
3899:
3721:
3563:
3326:
3098:
2903:
2840:
is identified with the bundle of conformal scales on
2830:
is given by such an embedding. Thus the line bundle
2754:
2636:
2525:
2300:
2085:
1980:
1719:
1477:
1321:
1201:
1142:
1088:
977:
811:
594:
562:
414:
174:. An equivalence class of such metrics is known as a
126:
1876:
This is the affine cone over the projective quadric
317:
modelled on the associated Möbius geometry, or as a
4812:
4646:
4330:
4194:
3965:
3833:
3683:
3535:
3298:
3074:
2807:
2715:
2616:
2474:
2228:
2059:
1865:
1635:
1362:
1275:
1186:
1122:
1025:
956:A coordinate grid prior to a Möbius transformation
908:
626:
577:
528:
151:
2256:Nevertheless, there was an arbitrary choice. If
325:and other invariants of the conformal structure.
211:, meaning that there exists an angle preserving
3966:{\displaystyle t(y)^{2}g_{ij}\,dy^{i}\,dy^{j}.}
1133:The infinitesimal conformal symmetries satisfy
934:and its Lie algebra are of current interest in
5023:Transformation Groups in Differential Geometry
345:added at infinity". That is, the setting is a
4959:Due to a general theorem of Sternberg (1962).
1363:{\displaystyle z\mapsto {\frac {az+b}{cz+d}}}
1187:{\displaystyle \mathbf {L} _{X}\,dz=f(z)\,dz}
641:with the property that the Lie derivative of
38:is the study of the set of angle-preserving (
8:
3317:, so that the null cone is coordinatized by
1706:, which is equipped with the quadratic form
1055:of the complex numbers. Its Lie algebra is
1955:of a sphere are generated by inversions in
1026:{\displaystyle q(z,{\bar {z}})=z{\bar {z}}}
964:The same grid after a Möbius transformation
313:, which can be handled either as a type of
5043:Invariant Operators on Conformal Manifolds
2818:An embedding of the Euclidean sphere into
2070:This can be mapped to the Minkowski space
1968:The Euclidean unit sphere is the locus in
170:defined on the manifold and is called the
53:of what are called "flat spaces" (such as
4798:
4797:
4764:
4763:
4687:
4670:
4665:
4662:
4633:
4623:
4619:
4618:
4582:
4547:
4519:
4499:
4494:
4478:
4474:
4473:
4425:
4393:
4377:
4357:
4352:
4349:
4322:
4317:
4307:
4302:
4289:
4284:
4275:
4273:
4123:
4115:
3954:
3946:
3940:
3932:
3923:
3913:
3898:
3821:
3814:
3799:
3791:
3776:
3768:
3762:
3754:
3736:
3726:
3720:
3670:
3659:
3654:
3635:
3630:
3617:
3612:
3593:
3583:
3570:
3562:
3515:
3506:
3495:
3488:
3470:
3448:
3439:
3427:
3421:
3409:
3393:
3384:
3366:
3357:
3350:
3343:
3331:
3325:
3278:
3269:
3258:
3251:
3236:
3214:
3205:
3193:
3187:
3178:
3162:
3153:
3135:
3126:
3119:
3112:
3103:
3097:
3057:
3052:
3022:
3013:
2994:
2985:
2977:
2959:
2950:
2938:
2932:
2918:
2913:
2904:
2902:
2880:. This consists of the following map of
2776:
2753:
2701:
2696:
2677:
2672:
2659:
2654:
2641:
2635:
2608:
2603:
2581:
2576:
2560:
2555:
2539:
2524:
2434:
2419:
2414:
2405:
2392:
2387:
2380:
2371:
2358:
2353:
2340:
2314:
2305:
2299:
2203:
2188:
2183:
2174:
2161:
2156:
2149:
2140:
2127:
2122:
2099:
2090:
2084:
2045:
2040:
2021:
2016:
2003:
1998:
1985:
1979:
1843:
1838:
1819:
1814:
1795:
1785:
1757:
1738:
1718:
1624:
1619:
1600:
1595:
1582:
1577:
1558:
1548:
1520:
1501:
1488:
1476:
1328:
1320:
1259:
1258:
1254:
1240:
1239:
1219:
1218:
1214:
1208:
1203:
1200:
1177:
1155:
1149:
1144:
1141:
1106:
1105:
1101:
1087:
1012:
1011:
991:
990:
976:
891:
870:
863:
862:
844:
823:
816:
815:
810:
611:
604:
593:
569:
565:
564:
561:
514:
513:
483:
459:
447:
413:
137:
125:
3856:In these terms, a section of the bundle
2510:
275:would not agree. Those associated with
5002:Application de l'Analyse à la géometrie
4915:
3712:, the Minkowski metric takes the form:
1710:as above. The null cone is defined by
1688:In a related construction, the quadric
1432:generated by inversion in spheres. By
337:with a point added at infinity", or a "
3550:be the following defining function of
2826:. Conversely, any conformal scale on
936:two-dimensional conformal field theory
5005:. Bachelier, Paris. pp. 609â615.
1927:orthochronous Lorentz transformations
1685:) that leaves the quadric invariant.
1417:, to the pseudo-Euclidean situation.
339:Minkowski (or pseudo-Euclidean) space
7:
4768:
4765:
1123:{\displaystyle g=dz\,d{\bar {z}}.}
880:
877:
874:
871:
833:
830:
827:
824:
556:Consider now the Minkowski plane,
25:
5053:Lectures on differential geometry
4209:is a quadratic form of signature
1951:. On the other hand, Riemannian
381:for the Minkowski quadratic form
333:Möbius geometry is the study of "
4666:
4495:
4353:
4318:
4303:
4285:
4276:
3507:
3440:
3385:
3358:
3270:
3206:
3154:
3127:
3053:
3014:
2986:
2951:
2939:
2914:
2905:
1921:. Conformal transformations on
1675:projective linear transformation
1204:
1145:
773:satisfying 1. and 2. Hence the
578:{\displaystyle \mathbb {R} ^{2}}
549:consisting of all real diagonal
152:{\displaystyle h=\lambda ^{2}g,}
4492:
2878:stereographic coordinate system
750:for some real-valued functions
4927:Applied Conformal Field Theory
4785:
4773:
4630:
4614:
4035:-dimensional Lorentzian space
3997:-dimensional Lorentzian space
3910:
3903:
3751:
3745:
3313:corresponding to dilations up
2924:
2868:. Suppose that the Euclidean
2761:
2755:
2726:A conformal representative of
2461:
2455:
2449:
2437:
2337:
2331:
2264:) is any positive function of
1769:
1731:
1532:
1481:
1325:
1264:
1251:
1245:
1236:
1224:
1174:
1168:
1111:
1017:
1002:
996:
981:
900:
897:
884:
859:
853:
850:
837:
812:
433:
421:
117:are equivalent if and only if
42:) transformations on a space.
1:
5021:Kobayashi, Shoshichi (1970).
3853:is the metric on the sphere.
627:{\displaystyle g=2\,dx\,dy~.}
360:exhibits extensive conformal
321:. This allows one to define
5025:(First ed.). Springer.
4262:. The Lie algebra admits a
758:depending, respectively, on
5084:Encyclopedia of Mathematics
4985:Encyclopedia of Mathematics
4884:Conformal geometric algebra
4055:. For pseudo-Euclidean of
4047:of a fixed null ray in the
301:turns out not to depend on
71:pseudo-Riemannian manifolds
5131:
5051:Sternberg, Shlomo (1983).
4894:Conformal Killing equation
4097:The conformal Lie algebras
3985:-dimensional model is the
2851:
2493:is an arbitrary choice of
2485:also gives a mapping into
1899:restricts to a projection
940:
99:pseudo-Riemannian manifold
5077:G.V. Bushmanova (2001) ,
3309:Introduce a new variable
1079:equipped with the metric
1075:Consider the (Euclidean)
585:equipped with the metric
349:of a familiar space; the
51:conformal transformations
2730:is a metric of the form
1963:CartanâDieudonnĂ© theorem
1646:In the projective space
197:locally conformally flat
193:Riemann curvature tensor
4978:S.A. Stepanov (2001) ,
4001:. Here the model is a
1700:in the Minkowski space
1298:over its domain. (See
1292:CauchyâRiemann equation
226:vanishes; in dimension
109:, in which two metrics
4814:
4648:
4332:
4196:
3967:
3835:
3685:
3537:
3300:
3076:
2809:
2717:
2618:
2501:Representative metrics
2476:
2291:, then the assignment
2230:
2061:
1961:hyperspheres (see the
1867:
1637:
1460:denote the Lorentzian
1364:
1311:Möbius transformations
1277:
1188:
1124:
1027:
965:
957:
910:
628:
579:
530:
247:Levi-Civita connection
185:A conformal metric is
153:
61:), or to the study of
5115:Differential geometry
5055:. New York: Chelsea.
4815:
4649:
4333:
4248:matrices stabilizing
4197:
3968:
3875:as a function of the
3836:
3686:
3538:
3301:
3077:
2810:
2718:
2619:
2477:
2231:
2062:
1868:
1692:is thought of as the
1638:
1365:
1278:
1189:
1125:
1028:
963:
955:
911:
629:
580:
531:
233:, if and only if the
154:
18:Conformal equivalence
5079:"Conformal geometry"
5040:SlovĂĄk, Jan (1993).
4999:(by J. Liouville)".
4980:"Liouville theorems"
4661:
4348:
4272:
4264:Cartan decomposition
4114:
3897:
3879:along the null cone
3719:
3561:
3324:
3096:
2901:
2854:Ambient construction
2848:Ambient metric model
2752:
2634:
2523:
2298:
2083:
1978:
1939:The Euclidean sphere
1717:
1475:
1444:The projective model
1319:
1199:
1140:
1086:
1053:multiplicative group
975:
930:The conformal group
925:diffeomorphism group
809:
783:infinite-dimensional
689:, this implies that
592:
560:
412:
399:in the plane is the
311:conformal connection
213:local diffeomorphism
124:
3664:
3640:
3622:
2706:
2682:
2664:
2613:
2586:
2565:
2050:
2026:
2008:
1945:Riemannian geometry
1925:are induced by the
1913:the structure of a
1848:
1824:
1696:at infinity of the
1629:
1605:
1587:
1434:Liouville's theorem
1421:The inversive model
649:is proportional to
323:conformal curvature
307:conformal invariant
262:Christoffel symbols
89:Conformal manifolds
81:, and is a type of
63:conformal manifolds
5110:Conformal geometry
4995:G. Monge (1850). "
4810:
4747:
4644:
4598:
4456:
4328:
4192:
4186:
4053:inversive geometry
3977:The Kleinian model
3963:
3831:
3681:
3650:
3626:
3608:
3533:
3296:
3072:
2805:
2713:
2692:
2668:
2650:
2614:
2599:
2572:
2551:
2511:conformal manifold
2472:
2226:
2057:
2036:
2012:
1994:
1863:
1834:
1810:
1633:
1615:
1591:
1573:
1438:inversive geometry
1360:
1273:
1184:
1120:
1023:
966:
958:
906:
624:
575:
526:
491:
149:
95:conformal manifold
36:conformal geometry
4950:Kobayashi (1972).
4889:Conformal gravity
4007:homogeneous space
3676:
3528:
3493:
3461:
3400:
3348:
3291:
3256:
3227:
3169:
3117:
3035:
2972:
2860:coordinate system
2507:Riemannian metric
2505:A representative
2470:
2469:
2348:
2345:
2221:
2220:
2117:
2116:
1387:Higher dimensions
1358:
1267:
1248:
1227:
1114:
1020:
999:
653:. Symbolically,
620:
315:Cartan connection
305:, and so it is a
166:is a real-valued
103:equivalence class
101:equipped with an
16:(Redirected from
5122:
5091:
5066:
5047:
5036:
5008:
5006:
4992:
4975:
4969:
4966:
4960:
4957:
4951:
4948:
4942:
4920:
4899:Erlangen program
4851:Borel subalgebra
4845:
4819:
4817:
4816:
4811:
4806:
4802:
4801:
4772:
4771:
4756:
4752:
4751:
4675:
4674:
4669:
4653:
4651:
4650:
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4628:
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4622:
4607:
4603:
4602:
4590:
4589:
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4507:
4506:
4498:
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4483:
4482:
4477:
4465:
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4460:
4433:
4432:
4398:
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4365:
4364:
4356:
4337:
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4326:
4321:
4312:
4311:
4306:
4297:
4296:
4288:
4279:
4261:
4247:
4235:
4220:
4201:
4199:
4198:
4193:
4191:
4190:
4106:
4084:
4069:
4057:metric signature
4034:
4026:
3996:
3987:celestial sphere
3972:
3970:
3969:
3964:
3959:
3958:
3945:
3944:
3931:
3930:
3918:
3917:
3885:
3874:
3840:
3838:
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3832:
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3781:
3780:
3767:
3766:
3744:
3743:
3731:
3730:
3711:
3690:
3688:
3687:
3682:
3677:
3675:
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3665:
3663:
3658:
3639:
3634:
3621:
3616:
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3588:
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3571:
3542:
3540:
3539:
3534:
3529:
3527:
3520:
3519:
3514:
3510:
3496:
3494:
3489:
3481:
3480:
3462:
3460:
3453:
3452:
3447:
3443:
3432:
3431:
3422:
3414:
3413:
3401:
3399:
3398:
3397:
3392:
3388:
3371:
3370:
3365:
3361:
3351:
3349:
3344:
3336:
3335:
3305:
3303:
3302:
3297:
3292:
3290:
3283:
3282:
3277:
3273:
3259:
3257:
3252:
3247:
3246:
3228:
3226:
3219:
3218:
3213:
3209:
3198:
3197:
3188:
3183:
3182:
3170:
3168:
3167:
3166:
3161:
3157:
3140:
3139:
3134:
3130:
3120:
3118:
3113:
3108:
3107:
3081:
3079:
3078:
3073:
3068:
3067:
3056:
3041:
3037:
3036:
3034:
3027:
3026:
3021:
3017:
3006:
2999:
2998:
2993:
2989:
2978:
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2971:
2964:
2963:
2958:
2954:
2943:
2942:
2933:
2923:
2922:
2917:
2908:
2893:
2867:
2839:
2814:
2812:
2811:
2806:
2801:
2797:
2781:
2780:
2722:
2720:
2719:
2714:
2705:
2700:
2681:
2676:
2663:
2658:
2646:
2645:
2623:
2621:
2620:
2615:
2612:
2607:
2585:
2580:
2564:
2559:
2544:
2543:
2489:. The function
2481:
2479:
2478:
2473:
2471:
2465:
2464:
2435:
2430:
2429:
2410:
2409:
2397:
2396:
2376:
2375:
2363:
2362:
2349:
2347:
2346:
2341:
2326:
2315:
2310:
2309:
2290:
2252:
2235:
2233:
2232:
2227:
2222:
2216:
2215:
2204:
2199:
2198:
2179:
2178:
2166:
2165:
2145:
2144:
2132:
2131:
2118:
2112:
2111:
2100:
2095:
2094:
2075:
2066:
2064:
2063:
2058:
2049:
2044:
2025:
2020:
2007:
2002:
1990:
1989:
1934:
1908:
1898:
1872:
1870:
1869:
1864:
1859:
1855:
1847:
1842:
1823:
1818:
1806:
1805:
1790:
1789:
1768:
1767:
1743:
1742:
1705:
1694:celestial sphere
1664:
1658:be the locus of
1642:
1640:
1639:
1634:
1628:
1623:
1604:
1599:
1586:
1581:
1569:
1568:
1553:
1552:
1531:
1530:
1506:
1505:
1493:
1492:
1454:projective space
1414:mutatis mutandis
1401:conformally flat
1382:
1369:
1367:
1366:
1361:
1359:
1357:
1343:
1329:
1282:
1280:
1279:
1274:
1269:
1268:
1260:
1250:
1249:
1241:
1229:
1228:
1220:
1213:
1212:
1207:
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1190:
1185:
1154:
1153:
1148:
1129:
1127:
1126:
1121:
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1115:
1107:
1071:
1050:
1032:
1030:
1029:
1024:
1022:
1021:
1013:
1001:
1000:
992:
943:Virasoro algebra
933:
915:
913:
912:
907:
896:
895:
883:
866:
849:
848:
836:
819:
797:
746:
718:
688:
674: for some
673:
633:
631:
630:
625:
618:
584:
582:
581:
576:
574:
573:
568:
552:
548:
535:
533:
532:
527:
522:
518:
517:
500:
496:
495:
488:
487:
464:
463:
398:
347:compactification
291:curvature tensor
232:
221:
188:conformally flat
176:conformal metric
172:conformal factor
158:
156:
155:
150:
142:
141:
73:with a class of
55:Euclidean spaces
47:Riemann surfaces
21:
5130:
5129:
5125:
5124:
5123:
5121:
5120:
5119:
5100:
5099:
5076:
5073:
5063:
5050:
5039:
5033:
5020:
5017:
5012:
5011:
4994:
4977:
4976:
4972:
4967:
4963:
4958:
4954:
4949:
4945:
4937:. Published in
4921:
4917:
4912:
4880:
4872:
4865:
4824:
4746:
4745:
4740:
4735:
4729:
4728:
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4718:
4712:
4711:
4706:
4701:
4688:
4684:
4683:
4679:
4664:
4659:
4658:
4629:
4617:
4597:
4596:
4591:
4578:
4573:
4567:
4566:
4561:
4556:
4544:
4541:
4540:
4535:
4530:
4520:
4516:
4515:
4511:
4493:
4472:
4455:
4454:
4449:
4444:
4438:
4437:
4421:
4419:
4414:
4408:
4407:
4402:
4390:
4388:
4378:
4374:
4373:
4369:
4351:
4346:
4345:
4316:
4301:
4283:
4270:
4269:
4249:
4237:
4222:
4210:
4185:
4184:
4179:
4174:
4165:
4164:
4159:
4154:
4148:
4147:
4139:
4134:
4124:
4112:
4111:
4102:
4099:
4071:
4059:
4028:
4017:
3990:
3979:
3950:
3936:
3919:
3909:
3895:
3894:
3880:
3861:
3852:
3795:
3772:
3758:
3732:
3722:
3717:
3716:
3707:
3706:coordinates on
3666:
3589:
3579:
3572:
3559:
3558:
3502:
3501:
3500:
3466:
3435:
3434:
3433:
3423:
3405:
3380:
3379:
3372:
3353:
3352:
3327:
3322:
3321:
3265:
3264:
3263:
3232:
3201:
3200:
3199:
3189:
3174:
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3141:
3122:
3121:
3099:
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3051:
3009:
3008:
3007:
2981:
2980:
2979:
2946:
2945:
2944:
2934:
2931:
2927:
2912:
2899:
2898:
2881:
2863:
2856:
2850:
2831:
2772:
2771:
2767:
2750:
2749:
2637:
2632:
2631:
2535:
2521:
2520:
2503:
2495:conformal scale
2436:
2415:
2401:
2388:
2367:
2354:
2327:
2316:
2301:
2296:
2295:
2288:
2279:
2265:
2244:
2205:
2184:
2170:
2157:
2136:
2123:
2101:
2086:
2081:
2080:
2071:
1981:
1976:
1975:
1941:
1930:
1900:
1885:
1791:
1781:
1753:
1734:
1730:
1726:
1715:
1714:
1701:
1659:
1554:
1544:
1516:
1497:
1484:
1473:
1472:
1446:
1427:Euclidean space
1423:
1389:
1374:
1344:
1330:
1317:
1316:
1202:
1197:
1196:
1143:
1138:
1137:
1084:
1083:
1062:
1056:
1041:
1037:
973:
972:
950:
948:Euclidean space
945:
931:
927:of the circle.
887:
840:
807:
806:
800:universal cover
789:
730:
728:
702:
700:
683:
665:
657:
590:
589:
563:
558:
557:
550:
543:
490:
489:
479:
477:
471:
470:
465:
455:
448:
444:
443:
439:
410:
409:
382:
379:conformal group
375:
373:Minkowski plane
370:
358:Minkowski plane
335:Euclidean space
331:
329:Möbius geometry
319:Weyl connection
227:
216:
209:Möbius geometry
207:conformal to a
180:conformal class
168:smooth function
133:
122:
121:
91:
79:Möbius geometry
28:
23:
22:
15:
12:
11:
5:
5128:
5126:
5118:
5117:
5112:
5102:
5101:
5098:
5097:
5092:
5072:
5071:External links
5069:
5068:
5067:
5061:
5048:
5037:
5031:
5016:
5013:
5010:
5009:
4970:
4968:Slovak (1993).
4961:
4952:
4943:
4935:hep-th/9108028
4914:
4913:
4911:
4908:
4907:
4906:
4901:
4896:
4891:
4886:
4879:
4876:
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4140:
4138:
4135:
4133:
4130:
4129:
4127:
4122:
4119:
4098:
4095:
4045:isotropy group
4027:acting on the
4003:Klein geometry
3978:
3975:
3974:
3973:
3962:
3957:
3953:
3949:
3943:
3939:
3935:
3929:
3926:
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3908:
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3842:
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3830:
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3820:
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3813:
3810:
3807:
3802:
3798:
3794:
3790:
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3784:
3779:
3775:
3771:
3765:
3761:
3757:
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3750:
3747:
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3729:
3725:
3692:
3691:
3680:
3673:
3669:
3662:
3657:
3653:
3649:
3646:
3643:
3638:
3633:
3629:
3625:
3620:
3615:
3611:
3607:
3602:
3599:
3596:
3592:
3586:
3582:
3578:
3575:
3569:
3566:
3544:
3543:
3532:
3526:
3523:
3518:
3513:
3509:
3505:
3499:
3492:
3487:
3484:
3479:
3476:
3473:
3469:
3465:
3459:
3456:
3451:
3446:
3442:
3438:
3430:
3426:
3420:
3417:
3412:
3408:
3404:
3396:
3391:
3387:
3383:
3378:
3375:
3369:
3364:
3360:
3356:
3347:
3342:
3339:
3334:
3330:
3307:
3306:
3295:
3289:
3286:
3281:
3276:
3272:
3268:
3262:
3255:
3250:
3245:
3242:
3239:
3235:
3231:
3225:
3222:
3217:
3212:
3208:
3204:
3196:
3192:
3186:
3181:
3177:
3173:
3165:
3160:
3156:
3152:
3147:
3144:
3138:
3133:
3129:
3125:
3116:
3111:
3106:
3102:
3083:
3082:
3071:
3066:
3063:
3060:
3055:
3050:
3047:
3044:
3040:
3033:
3030:
3025:
3020:
3016:
3012:
3005:
3002:
2997:
2992:
2988:
2984:
2976:
2970:
2967:
2962:
2957:
2953:
2949:
2941:
2937:
2930:
2926:
2921:
2916:
2911:
2907:
2849:
2846:
2816:
2815:
2804:
2800:
2796:
2793:
2790:
2787:
2784:
2779:
2775:
2770:
2766:
2763:
2760:
2757:
2724:
2723:
2712:
2709:
2704:
2699:
2695:
2691:
2688:
2685:
2680:
2675:
2671:
2667:
2662:
2657:
2653:
2649:
2644:
2640:
2627:to the sphere
2625:
2624:
2611:
2606:
2602:
2598:
2595:
2592:
2589:
2584:
2579:
2575:
2571:
2568:
2563:
2558:
2554:
2550:
2547:
2542:
2538:
2534:
2531:
2528:
2502:
2499:
2483:
2482:
2468:
2463:
2460:
2457:
2454:
2451:
2448:
2445:
2442:
2439:
2433:
2428:
2425:
2422:
2418:
2413:
2408:
2404:
2400:
2395:
2391:
2386:
2383:
2379:
2374:
2370:
2366:
2361:
2357:
2352:
2344:
2339:
2336:
2333:
2330:
2325:
2322:
2319:
2313:
2308:
2304:
2284:
2277:
2237:
2236:
2225:
2219:
2214:
2211:
2208:
2202:
2197:
2194:
2191:
2187:
2182:
2177:
2173:
2169:
2164:
2160:
2155:
2152:
2148:
2143:
2139:
2135:
2130:
2126:
2121:
2115:
2110:
2107:
2104:
2098:
2093:
2089:
2068:
2067:
2056:
2053:
2048:
2043:
2039:
2035:
2032:
2029:
2024:
2019:
2015:
2011:
2006:
2001:
1997:
1993:
1988:
1984:
1940:
1937:
1909:. This gives
1874:
1873:
1862:
1858:
1854:
1851:
1846:
1841:
1837:
1833:
1830:
1827:
1822:
1817:
1813:
1809:
1804:
1801:
1798:
1794:
1788:
1784:
1780:
1777:
1774:
1771:
1766:
1763:
1760:
1756:
1752:
1749:
1746:
1741:
1737:
1733:
1729:
1725:
1722:
1644:
1643:
1632:
1627:
1622:
1618:
1614:
1611:
1608:
1603:
1598:
1594:
1590:
1585:
1580:
1576:
1572:
1567:
1564:
1561:
1557:
1551:
1547:
1543:
1540:
1537:
1534:
1529:
1526:
1523:
1519:
1515:
1512:
1509:
1504:
1500:
1496:
1491:
1487:
1483:
1480:
1462:quadratic form
1445:
1442:
1422:
1419:
1388:
1385:
1371:
1370:
1356:
1353:
1350:
1347:
1342:
1339:
1336:
1333:
1327:
1324:
1307:Riemann sphere
1290:satisfies the
1284:
1283:
1272:
1266:
1263:
1257:
1253:
1247:
1244:
1238:
1235:
1232:
1226:
1223:
1217:
1211:
1206:
1194:
1183:
1180:
1176:
1173:
1170:
1167:
1164:
1161:
1158:
1152:
1147:
1131:
1130:
1119:
1113:
1110:
1104:
1100:
1097:
1094:
1091:
1060:
1039:
1034:
1033:
1019:
1016:
1010:
1007:
1004:
998:
995:
989:
986:
983:
980:
949:
946:
917:
916:
905:
902:
899:
894:
890:
886:
882:
879:
876:
873:
869:
865:
861:
858:
855:
852:
847:
843:
839:
835:
832:
829:
826:
822:
818:
814:
748:
747:
724:
719:
696:
680:
679:
661:
635:
634:
623:
617:
614:
610:
607:
603:
600:
597:
572:
567:
537:
536:
525:
521:
516:
512:
509:
506:
503:
499:
494:
486:
482:
478:
476:
473:
472:
469:
466:
462:
458:
454:
453:
451:
446:
442:
438:
435:
432:
429:
426:
423:
420:
417:
374:
371:
369:
368:Two dimensions
366:
330:
327:
243:tangent vector
160:
159:
148:
145:
140:
136:
132:
129:
107:metric tensors
90:
87:
83:Klein geometry
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
5127:
5116:
5113:
5111:
5108:
5107:
5105:
5096:
5093:
5090:
5086:
5085:
5080:
5075:
5074:
5070:
5064:
5062:0-8284-0316-3
5058:
5054:
5049:
5045:
5044:
5038:
5034:
5032:3-540-05848-6
5028:
5024:
5019:
5018:
5014:
5004:
5003:
4998:
4991:
4987:
4986:
4981:
4974:
4971:
4965:
4962:
4956:
4953:
4947:
4944:
4940:
4936:
4932:
4928:
4924:
4923:Paul Ginsparg
4919:
4916:
4909:
4905:
4902:
4900:
4897:
4895:
4892:
4890:
4887:
4885:
4882:
4881:
4877:
4869:
4862:
4858:
4855:
4854:
4853:
4852:
4847:
4843:
4839:
4835:
4831:
4827:
4807:
4803:
4794:
4791:
4788:
4782:
4779:
4776:
4760:
4757:
4753:
4748:
4742:
4737:
4732:
4725:
4720:
4715:
4708:
4703:
4698:
4695:
4689:
4680:
4676:
4671:
4657:
4656:
4640:
4634:
4624:
4611:
4608:
4604:
4599:
4593:
4586:
4583:
4579:
4575:
4570:
4563:
4558:
4553:
4548:
4545:
4537:
4532:
4527:
4521:
4512:
4508:
4503:
4500:
4489:
4485:
4479:
4469:
4466:
4462:
4457:
4451:
4446:
4441:
4434:
4429:
4426:
4422:
4416:
4411:
4404:
4399:
4394:
4391:
4385:
4379:
4370:
4366:
4361:
4358:
4344:
4343:
4342:
4323:
4313:
4308:
4298:
4293:
4290:
4280:
4268:
4267:
4266:
4265:
4260:
4256:
4252:
4245:
4241:
4233:
4229:
4225:
4218:
4214:
4208:
4187:
4181:
4176:
4171:
4168:
4161:
4156:
4151:
4144:
4141:
4136:
4131:
4125:
4120:
4117:
4110:
4109:
4108:
4105:
4096:
4094:
4092:
4088:
4083:
4079:
4075:
4067:
4063:
4058:
4054:
4050:
4046:
4042:
4038:
4032:
4024:
4020:
4015:
4011:
4008:
4004:
4000:
3994:
3988:
3984:
3976:
3960:
3955:
3951:
3947:
3941:
3937:
3933:
3927:
3924:
3920:
3914:
3906:
3900:
3893:
3892:
3891:
3889:
3883:
3878:
3872:
3868:
3864:
3859:
3854:
3851:
3847:
3828:
3825:
3822:
3818:
3815:
3811:
3808:
3805:
3800:
3796:
3792:
3788:
3785:
3782:
3777:
3773:
3769:
3763:
3759:
3755:
3748:
3740:
3737:
3733:
3727:
3723:
3715:
3714:
3713:
3710:
3705:
3701:
3697:
3678:
3671:
3667:
3660:
3655:
3651:
3647:
3644:
3641:
3636:
3631:
3627:
3623:
3618:
3613:
3609:
3605:
3600:
3597:
3594:
3590:
3584:
3580:
3576:
3573:
3567:
3564:
3557:
3556:
3555:
3553:
3549:
3546:Finally, let
3530:
3524:
3521:
3516:
3511:
3503:
3497:
3490:
3485:
3482:
3477:
3474:
3471:
3467:
3463:
3457:
3454:
3449:
3444:
3436:
3428:
3424:
3418:
3415:
3410:
3406:
3402:
3394:
3389:
3381:
3376:
3373:
3367:
3362:
3354:
3345:
3340:
3337:
3332:
3328:
3320:
3319:
3318:
3316:
3312:
3293:
3287:
3284:
3279:
3274:
3266:
3260:
3253:
3248:
3243:
3240:
3237:
3233:
3229:
3223:
3220:
3215:
3210:
3202:
3194:
3190:
3184:
3179:
3175:
3171:
3163:
3158:
3150:
3145:
3142:
3136:
3131:
3123:
3114:
3109:
3104:
3100:
3092:
3091:
3090:
3088:
3069:
3064:
3061:
3058:
3048:
3045:
3042:
3038:
3031:
3028:
3023:
3018:
3010:
3003:
3000:
2995:
2990:
2982:
2974:
2968:
2965:
2960:
2955:
2947:
2935:
2928:
2919:
2909:
2897:
2896:
2895:
2892:
2888:
2884:
2879:
2875:
2871:
2866:
2861:
2855:
2847:
2845:
2843:
2838:
2834:
2829:
2825:
2821:
2802:
2798:
2794:
2791:
2788:
2785:
2782:
2777:
2773:
2768:
2764:
2758:
2748:
2747:
2746:
2744:
2740:
2736:
2733:
2729:
2710:
2707:
2702:
2697:
2693:
2689:
2686:
2683:
2678:
2673:
2669:
2665:
2660:
2655:
2651:
2647:
2642:
2638:
2630:
2629:
2628:
2609:
2604:
2600:
2596:
2593:
2590:
2587:
2582:
2577:
2573:
2569:
2566:
2561:
2556:
2552:
2548:
2545:
2540:
2536:
2532:
2529:
2526:
2519:
2518:
2517:
2516:
2512:
2508:
2500:
2498:
2496:
2492:
2488:
2466:
2458:
2452:
2446:
2443:
2440:
2431:
2426:
2423:
2420:
2416:
2411:
2406:
2402:
2398:
2393:
2389:
2384:
2381:
2377:
2372:
2368:
2364:
2359:
2355:
2350:
2342:
2334:
2328:
2323:
2320:
2317:
2311:
2306:
2302:
2294:
2293:
2292:
2287:
2283:
2276:
2272:
2268:
2263:
2259:
2254:
2251:
2247:
2242:
2223:
2217:
2212:
2209:
2206:
2200:
2195:
2192:
2189:
2185:
2180:
2175:
2171:
2167:
2162:
2158:
2153:
2150:
2146:
2141:
2137:
2133:
2128:
2124:
2119:
2113:
2108:
2105:
2102:
2096:
2091:
2087:
2079:
2078:
2077:
2074:
2054:
2051:
2046:
2041:
2037:
2033:
2030:
2027:
2022:
2017:
2013:
2009:
2004:
1999:
1995:
1991:
1986:
1982:
1974:
1973:
1972:
1971:
1966:
1964:
1960:
1959:
1954:
1950:
1946:
1938:
1936:
1933:
1928:
1924:
1920:
1916:
1912:
1907:
1903:
1896:
1892:
1888:
1883:
1879:
1860:
1856:
1852:
1849:
1844:
1839:
1835:
1831:
1828:
1825:
1820:
1815:
1811:
1807:
1802:
1799:
1796:
1792:
1786:
1782:
1778:
1775:
1772:
1764:
1761:
1758:
1754:
1750:
1747:
1744:
1739:
1735:
1727:
1723:
1720:
1713:
1712:
1711:
1709:
1704:
1699:
1695:
1691:
1686:
1684:
1680:
1676:
1672:
1668:
1662:
1657:
1653:
1649:
1630:
1625:
1620:
1616:
1612:
1609:
1606:
1601:
1596:
1592:
1588:
1583:
1578:
1574:
1570:
1565:
1562:
1559:
1555:
1549:
1545:
1541:
1538:
1535:
1527:
1524:
1521:
1517:
1513:
1510:
1507:
1502:
1498:
1494:
1489:
1485:
1478:
1471:
1470:
1469:
1467:
1463:
1459:
1455:
1451:
1443:
1441:
1439:
1435:
1431:
1428:
1420:
1418:
1416:
1415:
1408:
1406:
1402:
1398:
1393:
1386:
1384:
1381:
1377:
1354:
1351:
1348:
1345:
1340:
1337:
1334:
1331:
1322:
1315:
1314:
1313:
1312:
1308:
1303:
1301:
1297:
1293:
1289:
1270:
1261:
1255:
1242:
1233:
1230:
1221:
1215:
1209:
1195:
1181:
1178:
1171:
1165:
1162:
1159:
1156:
1150:
1136:
1135:
1134:
1117:
1108:
1102:
1098:
1095:
1092:
1089:
1082:
1081:
1080:
1078:
1077:complex plane
1073:
1070:
1066:
1059:
1054:
1049:
1045:
1036:is the group
1014:
1008:
1005:
993:
987:
984:
978:
971:
970:
969:
962:
954:
947:
944:
939:
937:
928:
926:
922:
903:
892:
888:
867:
856:
845:
841:
820:
805:
804:
803:
801:
796:
792:
786:
784:
780:
776:
772:
767:
765:
761:
757:
753:
745:
741:
737:
733:
727:
723:
720:
717:
713:
709:
705:
699:
695:
692:
691:
690:
686:
677:
672:
668:
664:
660:
656:
655:
654:
652:
648:
644:
640:
621:
615:
612:
608:
605:
601:
598:
595:
588:
587:
586:
570:
554:
546:
542:
523:
519:
510:
507:
504:
501:
497:
492:
484:
480:
474:
467:
460:
456:
449:
440:
436:
430:
427:
424:
418:
415:
408:
407:
406:
405:
402:
397:
393:
389:
385:
380:
372:
367:
365:
363:
359:
354:
352:
348:
344:
340:
336:
328:
326:
324:
320:
316:
312:
308:
304:
300:
296:
292:
287:
285:
281:
278:
274:
271:
267:
263:
259:
256:
252:
248:
244:
238:
236:
235:Cotton tensor
230:
225:
219:
214:
210:
205:
203:
198:
194:
190:
189:
183:
181:
177:
173:
169:
165:
146:
143:
138:
134:
130:
127:
120:
119:
118:
116:
112:
108:
104:
100:
96:
88:
86:
84:
80:
76:
72:
68:
64:
60:
56:
52:
48:
43:
41:
37:
33:
19:
5082:
5052:
5042:
5022:
5001:
4996:
4983:
4973:
4964:
4955:
4946:
4938:
4926:
4918:
4904:Möbius plane
4867:
4860:
4856:
4848:
4841:
4837:
4833:
4829:
4825:
4822:
4340:
4258:
4254:
4250:
4243:
4239:
4236:consists of
4231:
4227:
4223:
4216:
4212:
4206:
4204:
4103:
4100:
4086:
4081:
4077:
4073:
4065:
4061:
4040:
4036:
4030:
4022:
4018:
4013:
4009:
3998:
3992:
3982:
3980:
3887:
3881:
3876:
3870:
3866:
3862:
3857:
3855:
3849:
3845:
3843:
3708:
3703:
3699:
3695:
3693:
3551:
3547:
3545:
3314:
3310:
3308:
3086:
3084:
2890:
2886:
2882:
2873:
2869:
2864:
2857:
2841:
2836:
2832:
2827:
2823:
2819:
2817:
2742:
2738:
2734:
2731:
2727:
2725:
2626:
2514:
2504:
2494:
2490:
2486:
2484:
2285:
2281:
2274:
2270:
2266:
2261:
2257:
2255:
2249:
2245:
2240:
2238:
2072:
2069:
1969:
1967:
1956:
1949:hyperspheres
1942:
1931:
1922:
1918:
1910:
1905:
1901:
1894:
1890:
1886:
1881:
1877:
1875:
1707:
1702:
1689:
1687:
1682:
1678:
1670:
1666:
1660:
1655:
1651:
1647:
1645:
1465:
1457:
1447:
1429:
1424:
1412:
1409:
1400:
1394:
1390:
1383:is nonzero.
1379:
1375:
1372:
1304:
1300:Witt algebra
1294:, and so is
1287:
1285:
1132:
1074:
1068:
1064:
1057:
1047:
1043:
1035:
967:
929:
920:
918:
794:
790:
787:
779:Witt algebra
770:
768:
763:
759:
755:
751:
749:
743:
739:
735:
731:
725:
721:
715:
711:
707:
703:
697:
693:
684:
681:
675:
670:
666:
662:
658:
650:
646:
642:
638:
636:
555:
544:
538:
395:
391:
387:
383:
376:
355:
332:
322:
306:
302:
294:
288:
283:
279:
276:
272:
269:
265:
257:
254:
250:
239:
228:
217:
201:
196:
186:
184:
179:
175:
171:
163:
161:
114:
110:
94:
92:
78:
62:
44:
35:
29:
2076:by letting
1915:line bundle
1468:defined by
1397:Lie algebra
1296:holomorphic
919:where Diff(
775:Lie algebra
541:Lie algebra
299:Weyl tensor
286:would not.
249:because if
224:Weyl tensor
32:mathematics
5104:Categories
5015:References
4049:light cone
2876:carries a
2852:See also:
1953:isometries
941:See also:
798:. On the
553:matrices.
237:vanishes.
67:Riemannian
65:which are
5089:EMS Press
4990:EMS Press
4795:∈
4761:∈
4696:−
4635:∗
4612:∈
4584:−
4501:−
4470:∈
4427:−
4359:−
4314:⊕
4299:⊕
4291:−
4169:−
4142:−
3826:ρ
3789:ρ
3645:⋯
3574:−
3565:ρ
3049:⊂
3043:∈
3001:−
2925:↦
2910:∈
2789:λ
2786:∣
2774:λ
2687:⋯
2591:⋯
2453:κ
2444:−
2382:…
2329:κ
2210:−
2151:…
2031:⋯
1829:⋯
1776:−
1773:∣
1748:…
1698:null cone
1610:⋯
1539:−
1511:…
1326:↦
1265:¯
1246:¯
1225:¯
1112:¯
1018:¯
997:¯
932:CSO(1, 1)
923:) is the
868:⋊
857:×
821:⋊
511:∈
419:
404:Lie group
343:null cone
135:λ
40:conformal
4925:(1989),
4878:See also
4253: :
4242:+ 2) Ă (
4221:. Then
4085:, where
2872:-sphere
2737:, where
1958:geodesic
1889:\ {0} â
1665:. Then
1378:−
362:symmetry
351:geometry
4091:compact
4043:is the
4025:+ 1, 1)
3989:of the
3694:In the
2280:, ...,
1880:. Let
1654:), let
1456:. Let
1450:quadric
1403:space (
729:
701:
401:abelian
341:with a
204:-sphere
75:metrics
59:spheres
5059:
5029:
4341:where
4205:where
4016:where
3844:where
1373:where
1286:where
1051:, the
781:, is
687:(1, 1)
645:along
619:
547:(1, 1)
220:> 3
162:where
4931:arXiv
4910:Notes
4840:) â (
4230:+ 1,
4080:+ 1)/
4076:+ 1,
4021:= SO(
1917:over
1673:is a
1452:in a
1405:up to
551:2 Ă 2
539:with
394:) = 2
97:is a
5057:ISBN
5027:ISBN
4246:+ 2)
4234:+ 1)
4226:= O(
4039:and
4033:+ 2)
4005:: a
3995:+ 2)
2792:>
1067:) =
1046:) =
762:and
754:and
377:The
268:and
253:and
113:and
4993:.
4830:cso
4255:MQM
3884:= 0
2862:on
2269:= (
1929:of
1677:of
1663:= 0
1464:on
1302:.)
685:cso
545:cso
416:CSO
264:of
231:= 3
178:or
105:of
69:or
57:or
30:In
5106::
5087:,
5081:,
4988:,
4982:,
4929:.
4866:â
4859:=
4846:.
4836:,
4828:â
4257:=
4215:,
4107::
4093:.
4072:O(
4064:,
3890::
3865:=
3850:ij
3702:,
3698:,
3554::
2894::
2889:â
2885:â
2835:â
2711:1.
2497:.
2273:,
2253:.
2248:â
2055:1.
1904:â
1440:.
1380:bc
1376:ad
1072:.
1058:gl
1038:GL
938:.
793:Ă
785:.
766:.
744:dy
742:)
734:=
732:dy
716:dx
714:)
706:=
704:dx
671:λg
669:=
396:xy
390:,
93:A
85:.
34:,
5065:.
5035:.
5007:.
4933::
4873:.
4871:1
4868:g
4864:0
4861:g
4857:h
4844:)
4842:R
4838:q
4834:p
4832:(
4826:R
4808:.
4804:}
4799:R
4792:a
4789:,
4786:)
4783:q
4780:,
4777:p
4774:(
4769:o
4766:s
4758:A
4754:|
4749:)
4743:a
4738:0
4733:0
4726:0
4721:A
4716:0
4709:0
4704:0
4699:a
4690:(
4681:{
4677:=
4672:0
4667:g
4641:}
4631:)
4625:n
4620:R
4615:(
4609:q
4605:|
4600:)
4594:0
4587:1
4580:J
4576:q
4571:0
4564:0
4559:0
4554:q
4549:t
4538:0
4533:0
4528:0
4522:(
4513:{
4509:=
4504:1
4496:g
4490:,
4486:}
4480:n
4475:R
4467:p
4463:|
4458:)
4452:0
4447:0
4442:0
4435:p
4430:1
4423:J
4417:0
4412:0
4405:0
4400:p
4395:t
4386:0
4380:(
4371:{
4367:=
4362:1
4354:g
4324:1
4319:g
4309:0
4304:g
4294:1
4286:g
4281:=
4277:g
4259:Q
4251:Q
4244:n
4240:n
4238:(
4232:q
4228:p
4224:G
4219:)
4217:q
4213:p
4211:(
4207:J
4188:)
4182:0
4177:0
4172:1
4162:0
4157:J
4152:0
4145:1
4137:0
4132:0
4126:(
4121:=
4118:Q
4104:R
4087:H
4082:H
4078:q
4074:p
4068:)
4066:q
4062:p
4060:(
4041:H
4037:R
4031:n
4029:(
4023:n
4019:G
4014:H
4012:/
4010:G
3999:R
3993:n
3991:(
3983:n
3961:.
3956:j
3952:y
3948:d
3942:i
3938:y
3934:d
3928:j
3925:i
3921:g
3915:2
3911:)
3907:y
3904:(
3901:t
3888:S
3882:Ï
3877:y
3873:)
3871:y
3869:(
3867:t
3863:t
3858:N
3846:g
3829:,
3823:d
3819:t
3816:d
3812:t
3809:2
3806:+
3801:2
3797:t
3793:d
3786:2
3783:+
3778:j
3774:y
3770:d
3764:i
3760:y
3756:d
3752:)
3749:y
3746:(
3741:j
3738:i
3734:g
3728:2
3724:t
3709:R
3704:y
3700:Ï
3696:t
3679:.
3672:2
3668:t
3661:2
3656:n
3652:x
3648:+
3642:+
3637:2
3632:2
3628:x
3624:+
3619:2
3614:1
3610:x
3606:+
3601:1
3598:+
3595:n
3591:x
3585:0
3581:x
3577:2
3568:=
3552:N
3548:Ï
3531:.
3525:1
3522:+
3517:2
3512:|
3508:y
3504:|
3498:1
3491:2
3486:t
3483:=
3478:1
3475:+
3472:n
3468:x
3464:,
3458:1
3455:+
3450:2
3445:|
3441:y
3437:|
3429:i
3425:y
3419:t
3416:=
3411:i
3407:x
3403:,
3395:2
3390:|
3386:y
3382:|
3377:+
3374:1
3368:2
3363:|
3359:y
3355:|
3346:2
3341:t
3338:=
3333:0
3329:x
3315:N
3311:t
3294:.
3288:1
3285:+
3280:2
3275:|
3271:y
3267:|
3261:1
3254:2
3249:=
3244:1
3241:+
3238:n
3234:x
3230:,
3224:1
3221:+
3216:2
3211:|
3207:y
3203:|
3195:i
3191:y
3185:=
3180:i
3176:x
3172:,
3164:2
3159:|
3155:y
3151:|
3146:+
3143:1
3137:2
3132:|
3128:y
3124:|
3115:2
3110:=
3105:0
3101:x
3087:N
3070:.
3065:1
3062:+
3059:n
3054:R
3046:S
3039:)
3032:1
3029:+
3024:2
3019:|
3015:y
3011:|
3004:1
2996:2
2991:|
2987:y
2983:|
2975:,
2969:1
2966:+
2961:2
2956:|
2952:y
2948:|
2940:y
2936:2
2929:(
2920:n
2915:R
2906:y
2891:R
2887:S
2883:R
2874:S
2870:n
2865:R
2842:S
2837:S
2833:N
2828:S
2824:S
2820:N
2803:.
2799:}
2795:0
2783:g
2778:2
2769:{
2765:=
2762:]
2759:g
2756:[
2743:g
2739:λ
2735:g
2732:λ
2728:g
2708:=
2703:2
2698:n
2694:x
2690:+
2684:+
2679:2
2674:2
2670:x
2666:+
2661:2
2656:1
2652:x
2648:+
2643:2
2639:z
2610:2
2605:n
2601:x
2597:d
2594:+
2588:+
2583:2
2578:2
2574:x
2570:d
2567:+
2562:2
2557:1
2553:x
2549:d
2546:+
2541:2
2537:z
2533:d
2530:=
2527:g
2515:R
2491:Îș
2487:N
2467:2
2462:)
2459:x
2456:(
2450:)
2447:1
2441:z
2438:(
2432:=
2427:1
2424:+
2421:n
2417:x
2412:,
2407:n
2403:x
2399:=
2394:n
2390:x
2385:,
2378:,
2373:1
2369:x
2365:=
2360:1
2356:x
2351:,
2343:2
2338:)
2335:x
2332:(
2324:1
2321:+
2318:z
2312:=
2307:0
2303:x
2289:)
2286:n
2282:x
2278:0
2275:x
2271:z
2267:x
2262:x
2260:(
2258:Îș
2250:S
2246:N
2241:N
2224:.
2218:2
2213:1
2207:z
2201:=
2196:1
2193:+
2190:n
2186:x
2181:,
2176:n
2172:x
2168:=
2163:n
2159:x
2154:,
2147:,
2142:1
2138:x
2134:=
2129:1
2125:x
2120:,
2114:2
2109:1
2106:+
2103:z
2097:=
2092:0
2088:x
2073:R
2052:=
2047:2
2042:n
2038:x
2034:+
2028:+
2023:2
2018:2
2014:x
2010:+
2005:2
2000:1
1996:x
1992:+
1987:2
1983:z
1970:R
1932:R
1923:S
1919:S
1911:N
1906:S
1902:N
1897:)
1895:R
1893:(
1891:P
1887:R
1882:N
1878:S
1861:.
1857:}
1853:0
1850:=
1845:2
1840:n
1836:x
1832:+
1826:+
1821:2
1816:1
1812:x
1808:+
1803:1
1800:+
1797:n
1793:x
1787:0
1783:x
1779:2
1770:)
1765:1
1762:+
1759:n
1755:x
1751:,
1745:,
1740:0
1736:x
1732:(
1728:{
1724:=
1721:N
1708:q
1703:R
1690:S
1683:R
1681:(
1679:P
1671:S
1667:S
1661:q
1656:S
1652:R
1650:(
1648:P
1631:.
1626:2
1621:n
1617:x
1613:+
1607:+
1602:2
1597:2
1593:x
1589:+
1584:2
1579:1
1575:x
1571:+
1566:1
1563:+
1560:n
1556:x
1550:0
1546:x
1542:2
1536:=
1533:)
1528:1
1525:+
1522:n
1518:x
1514:,
1508:,
1503:1
1499:x
1495:,
1490:0
1486:x
1482:(
1479:q
1466:R
1458:q
1430:E
1355:d
1352:+
1349:z
1346:c
1341:b
1338:+
1335:z
1332:a
1323:z
1288:f
1271:,
1262:z
1256:d
1252:)
1243:z
1237:(
1234:f
1231:=
1222:z
1216:d
1210:X
1205:L
1182:z
1179:d
1175:)
1172:z
1169:(
1166:f
1163:=
1160:z
1157:d
1151:X
1146:L
1118:.
1109:z
1103:d
1099:z
1096:d
1093:=
1090:g
1069:C
1065:C
1063:(
1061:1
1048:C
1044:C
1042:(
1040:1
1015:z
1009:z
1006:=
1003:)
994:z
988:,
985:z
982:(
979:q
921:S
904:,
901:)
898:)
893:1
889:S
885:(
881:f
878:f
875:i
872:D
864:Z
860:(
854:)
851:)
846:1
842:S
838:(
834:f
831:f
828:i
825:D
817:Z
813:(
795:S
791:S
771:X
764:y
760:x
756:b
752:a
740:y
738:(
736:b
726:X
722:L
712:x
710:(
708:a
698:X
694:L
678:.
676:λ
667:g
663:X
659:L
651:g
647:X
643:g
639:X
622:.
616:y
613:d
609:x
606:d
602:2
599:=
596:g
571:2
566:R
524:,
520:}
515:R
508:b
505:,
502:a
498:|
493:)
485:b
481:e
475:0
468:0
461:a
457:e
450:(
441:{
437:=
434:)
431:1
428:,
425:1
422:(
392:y
388:x
386:(
384:q
303:λ
295:λ
284:g
280:g
277:λ
273:g
270:λ
266:g
258:g
255:λ
251:g
229:n
218:n
202:n
164:λ
147:,
144:g
139:2
131:=
128:h
115:h
111:g
20:)
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