5946:
6008:
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31:
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Intuitively, the transition maps indicate how to glue two planes together to form the
Riemann sphere. The planes are glued in an "inside-out" manner, so that they overlap almost everywhere, with each plane contributing just one point (its origin) missing from the other plane. In other words, (almost)
4927:
of automorphisms, meaning the maps from the object to itself that preserve the essential structure of the object. In the case of the
Riemann sphere, an automorphism is an invertible conformal map (i.e. biholomorphic map) from the Riemann sphere to itself. It turns out that the only such maps are the
527:
5725:. This construction is helpful in the study of holomorphic and meromorphic functions. For example, on a compact Riemann surface there are no non-constant holomorphic maps to the complex numbers, but holomorphic maps to the complex projective line are abundant.
4096:. The Riemann surface's conformal structure does, however, determine a class of metrics: all those whose subordinate conformal structure is the given one. In more detail: The complex structure of the Riemann surface does uniquely determine a metric up to
3465:
3688:
4263:
4108:
uniquely determines a complex structure, which depends on the metric only up to conformal equivalence. Complex structures on an oriented surface are therefore in one-to-one correspondence with conformal classes of metrics on that surface.
4604:
980:
5368:
4798:. All such metrics determine the same conformal geometry. The round metric is therefore not intrinsic to the Riemann sphere, since "roundness" is not an invariant of conformal geometry. The Riemann sphere is only a
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795:
5471:
5423:
921:
430:
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593:
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is the simplest and most common choice). That is because only a round metric on the
Riemann sphere has its isometry group be a 3-dimensional group. (Namely, the group known as
4112:
Within a given conformal class, one can use conformal symmetry to find a representative metric with convenient properties. In particular, there is always a complete metric with
3308:
3364:
1037:
678:
643:
5569:
4891:
4209:
3584:
3126:
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559:
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5504:
4430:{\displaystyle ds^{2}=\left({\frac {2}{1+|\zeta |^{2}}}\right)^{2}\,|d\zeta |^{2}={\frac {4}{\left(1+\zeta {\overline {\zeta }}\right)^{2}}}\,d\zeta \,d{\overline {\zeta }}.}
1704:
1645:
1127:
847:
821:
5539:
4858:
4721:
4652:
1199:
4687:
4180:
3994:
3911:
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2350:
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2260:
2186:
2136:
2094:
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2010:
1952:
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1611:
452:
376:
5121:
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4056:
3232:
1071:
189:
6226:
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1552:
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and
Jonathan Rogness (a video by two University of Minnesota professors explaining and illustrating Möbius transformations using stereographic projection from a sphere)
1926:
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1503:
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100:
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2424:
1099:
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5645:
5625:
5083:
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5043:
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4792:
4772:
4744:
4144:
2919:
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1972:
1306:
1003:
763:
701:
144:
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929:
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5156:
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The inverses of these two stereographic projections are maps from the complex plane to the sphere. The first inverse covers the sphere except the point
6649:
6884:
5432:, then not all Möbius transformations are isometries; for example, the dilations and translations are not. The isometries form a proper subgroup of
4806:. However, if one needs to do Riemannian geometry on the Riemann sphere, the round metric is a natural choice (with any fixed radius, though radius
4620:
metric on the sphere whose group of orientation-preserving isometries is 3-dimensional (and none is more than 3-dimensional); that group is called
5377:. Since they act on projective coordinates, two matrices yield the same Möbius transformation if and only if they differ by a nonzero factor. The
6378:
5798:
of strings are
Riemann surfaces, and the Riemann sphere, being the simplest Riemann surface, plays a significant role. It is also important in
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6124:
4902:
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6999:
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6807:
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6160:
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3012:
This treatment of the
Riemann sphere connects most readily to projective geometry. For example, any line (or smooth conic) in the
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6212:
5728:
The
Riemann sphere has many uses in physics. In quantum mechanics, points on the complex projective line are natural values for
5846:
4938:
2556:
of a plane into the sphere. However, the
Riemann sphere is not merely a topological sphere. It is a sphere with a well-defined
7044:
7039:
7014:
6869:
6837:
6654:
6565:
4147:
2553:
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6502:
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2613:). Hence the two-dimensional sphere admits a unique complex structure turning it into a one-dimensional complex manifold.
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6472:
5579:
In complex analysis, a meromorphic function on the complex plane (or on any
Riemann surface, for that matter) is a ratio
522:{\displaystyle {\widehat {\mathbf {C} }},\quad {\overline {\mathbf {C} }},\quad {\text{or}}\quad \mathbf {C} _{\infty }.}
7095:
6979:
6368:
1719:
7004:
6731:
6695:
5892:
6394:
6307:
6182:
7029:
7009:
6016:
5960:
5954:
246:
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6343:
768:
5435:
5387:
892:
401:
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6830:
6751:
3799:
2729:
564:
7019:
6664:
6644:
6580:
6497:
6358:
5971:
4655:. In this sense, this is by far the most symmetric metric on the sphere. (The group of all isometries, known as
4120:
6909:
6874:
6399:
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3129:
3013:
234:
35:
6363:
5429:
4929:
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2785:
713:
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6555:
5382:
4747:
203:
6944:
6348:
5143:
5132:
3460:{\displaystyle \zeta ={\frac {x+iy}{1-z}}={\cot }{\bigl (}{\tfrac {1}{2}}\theta {\bigr )}\,e^{i\varphi }.}
2846:
2610:
2590:
886:
882:
34:
The
Riemann sphere can be visualized as the complex number plane wrapped around a sphere (by some form of
6726:
6462:
3845:-coordinates are obtained by composing one projection with the inverse of the other. They turn out to be
3683:{\displaystyle \xi ={\frac {x-iy}{1+z}}={\tan }{\bigl (}{\tfrac {1}{2}}\theta {\bigr )}\,e^{-i\varphi }.}
3043:
2194:
6969:
6262:
4097:
3772:. The two complex planes, that are the domains of these maps, are identified differently with the plane
3281:
3276:
3235:
6424:
6373:
1008:
651:
620:
5545:
5135:, and complex inversion. In fact, any Möbius transformation can be written as a composition of these.
4867:
4185:
3102:
2636:
1800:
535:
294:
6802:
6674:
6585:
6333:
5507:
5476:
4827:
4751:
2263:
1678:
1648:
1619:
1104:
826:
800:
207:
199:
6639:
6177:
6022:
5511:
4830:
4693:
4624:
4100:. (Two metrics are said to be conformally equivalent if they differ by multiplication by a positive
1143:
6994:
6904:
6889:
6517:
6482:
6439:
6419:
5841:
5732:
5378:
4924:
4803:
4659:
4153:
3963:
3882:
3848:
2321:
1555:
1367:
854:
341:
230:
2567:
2243:
2169:
2119:
2077:
2035:
1993:
1931:
1861:
1590:
435:
359:
6914:
6769:
6353:
5811:
5783:
5743:
5124:
5088:
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4443:
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3016:
is biholomorphic to the complex projective line. It is also convenient for studying the sphere's
1042:
850:
329:
160:
55:
6560:
6540:
6512:
5821:
3543:
1528:
2270:. As a complex manifold of 1 complex dimension (i.e. 2 real dimensions), this is also called a
1896:
332:. It also finds utility in other disciplines that depend on analysis and geometry, such as the
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1472:
1436:
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1138:
337:
195:
71:
5373:
Thus the Möbius transformations can be described as two-by-two complex matrices with nonzero
4599:{\displaystyle ds^{2}={\frac {4}{\left(1+u^{2}+v^{2}\right)^{2}}}\left(du^{2}+dv^{2}\right).}
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381:
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85:
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2015:
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51:
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129:
79:
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321:
67:
975:{\displaystyle {\frac {z}{0}}=\infty \quad {\text{and}}\quad {\frac {z}{\infty }}=0}
6934:
6736:
6710:
6700:
6690:
6492:
6312:
5856:
5779:
5363:{\displaystyle {\begin{pmatrix}a&c\\b&d\end{pmatrix}}\ =\ \ =\ \left\ =\ .}
3017:
333:
288:
191:
4901:
6959:
6611:
6449:
5851:
5831:
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5374:
2601:, or the Riemann sphere. Of these, the Riemann sphere is the only one that is a
2593:, a central result in the classification of Riemann surfaces, states that every
43:
2560:, so that around every point on the sphere there is a neighborhood that can be
6606:
5795:
5139:
17:
6204:
5571:(which, when restricted to the sphere, become the isometries of the sphere).
3960:-chart, and the equator of the unit sphere are all identified. The unit disk
561:
has also seen use, but as this notation is also used for the punctured plane
6467:
5888:
4754:). By the uniformization theorem there exists a unique complex structure on
3028:
2262:—the so-called charts—glueing them together. Since the transition maps are
6194:
30:
5128:
4923:
The study of any mathematical object is aided by an understanding of its
4613:
on complex projective space (of which the Riemann sphere is an example).
3802:-reversal is necessary to maintain consistent orientation on the sphere.
2549:
853:. Unlike the complex numbers, the extended complex numbers do not form a
614:
598:
Geometrically, the set of extended complex numbers is referred to as the
218:
211:
103:
6779:
6764:
3355:
6759:
5729:
4861:, a continuous ("Lie") group that is topologically the 3-dimensional
3331:
6822:
1616:
The set of complex rational functions—whose mathematical symbol is
206:
of the rational function mapping to infinity. More generally, any
4900:
4621:
29:
1974:
becomes a continuous function from the Riemann sphere to itself.
5740:
4656:
6826:
6208:
5699:
to the complex projective line that is well-defined even where
432:, and is often denoted by adding some decoration to the letter
324:
Riemann surface, the sphere may also be viewed as a projective
157:
in some circumstances, in a way that makes expressions such as
6135:
6028:
You can help by providing page numbers for existing citations.
6001:
5939:
4609:
Up to a constant factor, this metric agrees with the standard
2625:. The points of the complex projective line can be defined as
4092:
A Riemann surface does not come equipped with any particular
5647:. As a map to the complex numbers, it is undefined wherever
4123:
implies that a constant-curvature metric must have positive
27:
Model of the extended complex plane plus a point at infinity
4774:
up to conformal equivalence. It follows that any metric on
2597:
Riemann surface is biholomorphic to the complex plane, the
1651:
from the Riemann sphere to itself, when it is viewed as a
5542:, which is the group of symmetries of the unit sphere in
4995:{\displaystyle f(\zeta )={\frac {a\zeta +b}{c\zeta +d}},}
398:. The set of extended complex numbers may be written as
3040:
The Riemann sphere can be visualized as the unit sphere
3009:, which is in a chart for the Riemann sphere manifold.
5516:
5481:
5440:
5392:
5274:
5183:
4835:
4698:
4664:
4629:
3639:
3419:
221:, the Riemann sphere is the prototypical example of a
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5653:
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5613:
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5438:
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5031:
5011:
4941:
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4833:
4812:
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4627:
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4217:
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4156:
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4002:
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3926:
3885:
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identified with another copy of the complex plane by
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3284:
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3205:
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3138:
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2246:
2197:
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2144:
2122:
2102:
2080:
2060:
2038:
2018:
1996:
1990:, both with domain equal to the complex number plane
1960:
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991:
932:
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863:
829:
803:
771:
751:
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689:
654:
623:
567:
538:
463:
438:
404:
384:
362:
297:
249:
163:
132:
112:
88:
3731:, and the second covers the sphere except the point
1655:, except for the constant function taking the value
617:
of complex numbers may be extended by defining, for
6788:
6750:
6719:
6683:
6632:
6625:
6599:
6531:
6448:
6412:
6387:
6321:
6290:
6281:
6243:
6081:. Oxford : New York: Oxford University Press.
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5017:
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4249:
4223:
4203:
4174:
4138:
4076:
4050:
4014:
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3932:
3920:Under this diffeomorphism, the unit circle in the
3905:
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3532:
3503:
3459:
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3001:
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2004:
1966:
1946:
1920:
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1787:{\displaystyle f(z)={\frac {6z^{2}+1}{2z^{2}-50}}}
1786:
1698:
1667:
1639:
1605:
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1497:
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734:
695:
672:
637:
587:
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521:
446:
424:
390:
370:
312:
275:
210:can be thought of as a holomorphic function whose
183:
138:
118:
94:
126:is near to very large numbers, just as the point
6096:Griffiths, Phillip & Harris, Joseph (1978).
6140:Journal fĂĽr die reine und angewandte Mathematik
5667:is zero. However, it induces a holomorphic map
2813:In this case, the equivalence class is written
5782:). The Riemann sphere has been suggested as a
4231:-chart on the Riemann sphere, the metric with
3913:, as described above. Thus the unit sphere is
2621:The Riemann sphere can also be defined as the
1397:is a complex number such that the denominator
276:{\displaystyle \mathbf {P} ^{1}(\mathbf {C} )}
6838:
6220:
5774:, and 2-state particles in general (see also
5123:. Examples of Möbius transformations include
3655:
3633:
3435:
3413:
3032:Stereographic projection of a complex number
2278:every point in the Riemann sphere has both a
2266:, they define a complex manifold, called the
1986:, the Riemann sphere can be described by two
1708:the field of rational functions on the sphere
790:{\displaystyle \infty \times \infty =\infty }
8:
6880:Grothendieck–Hirzebruch–Riemann–Roch theorem
5466:{\displaystyle {\mbox{PGL}}(2,\mathbf {C} )}
5418:{\displaystyle {\mbox{PGL}}(2,\mathbf {C} )}
3199:which we identify with the complex plane by
1366:have no common factor) can be extended to a
916:{\displaystyle \mathbf {C} \cup \{\infty \}}
910:
904:
582:
576:
425:{\displaystyle \mathbf {C} \cup \{\infty \}}
419:
413:
6058:Brown, James & Churchill, Ruel (1989).
4058:is identified with the northern hemisphere
3996:is identified with the southern hemisphere
2775:{\displaystyle (w,z)=(\lambda u,\lambda v)}
588:{\displaystyle \mathbf {C} \setminus \{0\}}
149:The extended complex numbers are useful in
6845:
6831:
6823:
6629:
6287:
6227:
6213:
6205:
5428:If one endows the Riemann sphere with the
4909:acting on the sphere, and on the plane by
198:on the complex plane can be extended to a
7025:Riemann–Roch theorem for smooth manifolds
6044:Learn how and when to remove this message
5990:Learn how and when to remove this message
5756:
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3594:
3586:
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3516:
3475:
3470:Similarly, stereographic projection from
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2986:
2952:
2926:
2906:
2886:
2854:
2818:
2795:
2787:
2731:
2699:
2667:
2646:
2641:
2638:
2571:
2569:
2531:
2511:
2491:
2486:-chart. Symmetrically, the origin of the
2471:
2451:
2431:
2408:
2403:
2383:
2357:
2334:
2323:
2318:value, and the two values are related by
2303:
2283:
2247:
2245:
2213:
2196:
2173:
2171:
2148:
2143:
2123:
2121:
2101:
2081:
2079:
2059:
2039:
2037:
2017:
1997:
1995:
1959:
1933:
1898:
1863:
1840:
1802:
1769:
1748:
1738:
1721:
1682:
1680:
1660:
1623:
1621:
1592:
1563:
1530:
1510:
1486:
1474:
1450:
1438:
1414:
1402:
1381:
1375:
1370:on the Riemann sphere. Specifically, if
1342:
1313:
1293:
1264:
1235:
1206:
1174:
1145:
1111:
1106:
1083:
1078:
1049:
1044:
1015:
1010:
990:
956:
950:
933:
931:
896:
894:
885:. Nonetheless, it is customary to define
862:
828:
802:
770:
750:
715:
688:
653:
630:
622:
568:
566:
545:
540:
537:
510:
505:
498:
484:
482:
467:
465:
464:
462:
439:
437:
405:
403:
383:
363:
361:
304:
299:
296:
265:
256:
251:
248:
167:
162:
131:
111:
87:
5953:This article includes a list of general
3027:
2803:{\displaystyle \lambda \in \mathbf {C} }
6138:[Theory of Abelian functions].
5925:
5913:
5875:
5868:
4119:In the case of the Riemann sphere, the
2881:in the complex projective line, one of
2096:. Identify each nonzero complex number
2074:be a complex number in another copy of
735:{\displaystyle z\times \infty =\infty }
573:
6650:Clifford's theorem on special divisors
6119:. London: National Geographic Books.
5842:Parallel (operator) § Properties
5506:. This subgroup is isomorphic to the
4211:via stereographic projection. In the
4146:. It follows that the metric must be
3132:from the unit sphere minus the point
3036:onto a point α of the Riemann sphere.
2947:. Then by the notion of equivalence,
1308:with complex coefficients, such that
1230:is the ratio of polynomial functions
328:, making it a fundamental example in
74:. This extended plane represents the
7:
5895:from the original on October 8, 2021
4690:, is also 3-dimensional, but unlike
3099:in the three-dimensional real space
2426:"; in this sense, the origin of the
106:. With the Riemann model, the point
6136:"Theorie der Abel'schen Functionen"
5142:on the complex projective line. In
3092:{\displaystyle x^{2}+y^{2}+z^{2}=1}
2226:{\displaystyle f(z)={\frac {1}{z}}}
2032:be a complex number in one copy of
1835:, since the denominator is zero at
1613:, which may be finite or infinite.
6990:Riemannian connection on a surface
6895:Measurable Riemann mapping theorem
6808:Vector bundles on algebraic curves
6742:Weber's theorem (Algebraic curves)
6339:Hasse's theorem on elliptic curves
6329:Counting points on elliptic curves
6060:Complex Variables and Applications
5959:it lacks sufficient corresponding
4932:. These are functions of the form
4746:denote the sphere (as an abstract
3303:{\displaystyle (\theta ,\varphi )}
2513:
2453:
1941:
1871:
1822:
1706:form an algebraic field, known as
1662:
1600:
1538:
1512:
1116:
1108:
1054:
1026:
1012:
961:
946:
907:
864:
836:
810:
804:
784:
778:
772:
729:
723:
667:
661:
511:
416:
385:
178:
113:
89:
25:
5381:of Möbius transformations is the
4794:is conformally equivalent to the
1032:{\displaystyle \infty /0=\infty }
673:{\displaystyle z+\infty =\infty }
638:{\displaystyle z\in \mathbf {C} }
38:– details are given below).
7063:
7062:
6195:Moebius Transformations Revealed
6098:Principles of Algebraic Geometry
6006:
5944:
5564:{\displaystyle \mathbf {R} ^{3}}
5551:
5456:
5408:
4886:{\displaystyle \mathbf {P} ^{3}}
4873:
4204:{\displaystyle \mathbf {R} ^{3}}
4191:
4104:.) Conversely, any metric on an
3121:{\displaystyle \mathbf {R} ^{3}}
3108:
2796:
2655:{\displaystyle \mathbf {C} ^{2}}
2642:
2572:
2248:
2174:
2138:with the nonzero complex number
2124:
2082:
2040:
1998:
1828:{\displaystyle f(\pm 5)=\infty }
1713:For example, given the function
1683:
1624:
985:for all nonzero complex numbers
897:
745:for all nonzero complex numbers
631:
569:
554:{\displaystyle \mathbf {C} ^{*}}
541:
506:
485:
468:
440:
406:
364:
313:{\displaystyle \mathbf {C} ^{2}}
300:
266:
252:
233:, the sphere is an example of a
202:on the Riemann sphere, with the
6975:Riemann's differential equation
6885:Hirzebruch–Riemann–Roch theorem
6430:Hurwitz's automorphisms theorem
5847:Projectively extended real line
5499:{\displaystyle {\mbox{PSU}}(2)}
5138:The Möbius transformations are
3940:-chart, the unit circle in the
1699:{\displaystyle \mathbf {C} (z)}
1640:{\displaystyle \mathbf {C} (z)}
1122:{\displaystyle \infty /\infty }
955:
949:
842:{\displaystyle 0\times \infty }
816:{\displaystyle \infty -\infty }
503:
497:
481:
356:consist of the complex numbers
146:is near to very small numbers.
7000:Riemann–Hilbert correspondence
6870:Generalized Riemann hypothesis
6655:Gonality of an algebraic curve
6566:Differential of the first kind
6077:Goldman, William Mark (1999).
5686:
5674:
5534:{\displaystyle {\mbox{SO}}(3)}
5528:
5522:
5493:
5487:
5460:
5446:
5412:
5398:
5354:
5342:
5336:
5330:
5256:
5223:
5175:
5160:
5085:are complex numbers such that
4951:
4945:
4853:{\displaystyle {\mbox{SO}}(3)}
4847:
4841:
4716:{\displaystyle {\mbox{SO}}(3)}
4710:
4704:
4676:
4670:
4647:{\displaystyle {\mbox{SO}}(3)}
4641:
4635:
4347:
4335:
4310:
4301:
4116:in any given conformal class.
4038:
4030:
3976:
3968:
3759:
3738:
3718:
3700:
3498:
3477:
3297:
3285:
3262:
3244:
3157:
3139:
2966:
2954:
2868:
2856:
2832:
2820:
2782:for some non-zero coefficient
2769:
2751:
2745:
2733:
2713:
2701:
2681:
2669:
2617:As the complex projective line
2207:
2201:
1938:
1912:
1909:
1903:
1874:
1868:
1816:
1807:
1732:
1726:
1693:
1687:
1675:everywhere. The functions of
1634:
1628:
1597:
1574:
1568:
1541:
1535:
1492:
1479:
1456:
1443:
1420:
1407:
1353:
1347:
1324:
1318:
1275:
1269:
1246:
1240:
1217:
1211:
1194:{\displaystyle f(z)=g(z)/h(z)}
1188:
1182:
1171:
1165:
1156:
1150:
270:
262:
1:
7035:Riemann–Siegel theta function
6798:Birkhoff–Grothendieck theorem
6508:Nagata's conjecture on curves
6379:Schoof–Elkies–Atkin algorithm
6253:Five points determine a conic
5607:of two holomorphic functions
4682:{\displaystyle {\mbox{O}}(3)}
4175:{\displaystyle 1/{\sqrt {K}}}
3989:{\displaystyle |\zeta |<1}
3906:{\displaystyle \xi =1/\zeta }
3872:{\displaystyle \zeta =1/\xi }
2552:, the resulting space is the
2345:{\displaystyle \zeta =1/\xi }
237:and can be thought of as the
225:, and is one of the simplest
7050:Riemann–von Mangoldt formula
6369:Supersingular elliptic curve
4419:
4385:
3805:The transition maps between
3128:. To this end, consider the
2633:in the complex vector space
2579:{\displaystyle \mathbf {C} }
2255:{\displaystyle \mathbf {C} }
2181:{\displaystyle \mathbf {C} }
2131:{\displaystyle \mathbf {C} }
2089:{\displaystyle \mathbf {C} }
2047:{\displaystyle \mathbf {C} }
2005:{\displaystyle \mathbf {C} }
1954:. Using these definitions,
1947:{\displaystyle z\to \infty }
1886:{\displaystyle f(\infty )=3}
1606:{\displaystyle z\to \infty }
595:, it can lead to ambiguity.
489:
447:{\displaystyle \mathbf {C} }
371:{\displaystyle \mathbf {C} }
6576:Riemann's existence theorem
6503:Hilbert's sixteenth problem
6395:Elliptic curve cryptography
6308:Fundamental pair of periods
6183:Encyclopedia of Mathematics
6079:Complex Hyperbolic Geometry
5116:{\displaystyle ad-bc\neq 0}
4723:is not a connected space.)
4616:Up to scaling, this is the
4468:{\displaystyle \zeta =u+iv}
4051:{\displaystyle |\xi |<1}
3227:{\displaystyle \zeta =x+iy}
1066:{\displaystyle 0/\infty =0}
184:{\displaystyle 1/0=\infty }
7122:
7045:Riemann–Stieltjes integral
7040:Riemann–Silberstein vector
7015:Riemann–Liouville integral
6706:Moduli of algebraic curves
6134:Riemann, Bernhard (1857).
4916:
3571:{\displaystyle \xi =x-iy,}
2554:one-point compactification
2240:between the two copies of
1547:{\displaystyle f(\infty )}
1433:is zero but the numerator
7058:
6980:Riemann's minimal surface
6860:
6155:. New York: McGraw–Hill.
6153:Real and Complex Analysis
6100:. John Wiley & Sons.
6062:. New York: McGraw-Hill.
3020:, later in this article.
2506:-chart plays the role of
2446:-chart plays the role of
1921:{\displaystyle f(z)\to 3}
7005:Riemann–Hilbert problems
6910:Riemann curvature tensor
6875:Grand Riemann hypothesis
6865:Cauchy–Riemann equations
6473:Cayley–Bacharach theorem
6400:Elliptic curve primality
4911:stereographic projection
4150:to the sphere of radius
3765:{\displaystyle (0,0,-1)}
3504:{\displaystyle (0,0,-1)}
3347:{\displaystyle \varphi }
3130:stereographic projection
3014:complex projective plane
1498:{\displaystyle f(z_{0})}
1462:{\displaystyle g(z_{0})}
1426:{\displaystyle h(z_{0})}
354:extended complex numbers
348:Extended complex numbers
235:complex projective space
76:extended complex numbers
36:stereographic projection
6930:Riemann mapping theorem
6732:Riemann–Hurwitz formula
6696:Gromov–Witten invariant
6556:Compact Riemann surface
6344:Mazur's torsion theorem
6115:Penrose, Roger (2007).
6015:This article cites its
5974:more precise citations.
5383:projective linear group
3917:to the Riemann sphere.
3724:{\displaystyle (0,0,1)}
3323:{\displaystyle \theta }
3268:{\displaystyle (x,y,z)}
3163:{\displaystyle (0,0,1)}
2940:{\displaystyle w\neq 0}
2662:: two non-null vectors
2623:complex projective line
2589:On the other hand, the
2519:{\displaystyle \infty }
2459:{\displaystyle \infty }
1668:{\displaystyle \infty }
1518:{\displaystyle \infty }
870:{\displaystyle \infty }
683:for any complex number
391:{\displaystyle \infty }
214:is the Riemann sphere.
153:because they allow for
119:{\displaystyle \infty }
95:{\displaystyle \infty }
7030:Riemann–Siegel formula
7010:Riemann–Lebesgue lemma
6945:Riemann series theorem
6349:Modular elliptic curve
6151:Rudin, Walter (1987).
5768:
5719:
5693:
5661:
5641:
5621:
5601:
5565:
5535:
5500:
5467:
5419:
5364:
5144:projective coordinates
5117:
5079:
5059:
5039:
5019:
4996:
4930:Möbius transformations
4914:
4887:
4854:
4820:
4788:
4768:
4740:
4717:
4683:
4648:
4600:
4469:
4431:
4251:
4225:
4224:{\displaystyle \zeta }
4205:
4176:
4140:
4078:
4077:{\displaystyle z>0}
4052:
4022:, while the unit disk
4016:
4015:{\displaystyle z<0}
3990:
3954:
3934:
3933:{\displaystyle \zeta }
3907:
3873:
3839:
3819:
3818:{\displaystyle \zeta }
3792:
3766:
3725:
3684:
3572:
3534:
3505:
3461:
3348:
3324:
3304:
3269:
3228:
3193:
3164:
3122:
3093:
3037:
3003:
3002:{\displaystyle =\left}
2941:
2921:must be non-zero, say
2915:
2895:
2875:
2847:projective coordinates
2839:
2804:
2776:
2720:
2688:
2656:
2591:uniformization theorem
2580:
2540:
2520:
2500:
2499:{\displaystyle \zeta }
2480:
2479:{\displaystyle \zeta }
2460:
2440:
2420:
2392:
2391:{\displaystyle \zeta }
2372:
2371:{\displaystyle \xi =0}
2346:
2312:
2292:
2291:{\displaystyle \zeta }
2256:
2227:
2182:
2160:
2159:{\displaystyle 1/\xi }
2132:
2110:
2109:{\displaystyle \zeta }
2090:
2068:
2048:
2026:
2025:{\displaystyle \zeta }
2006:
1968:
1948:
1922:
1887:
1852:
1829:
1788:
1700:
1669:
1641:
1607:
1581:
1554:can be defined as the
1548:
1519:
1499:
1463:
1427:
1391:
1360:
1331:
1302:
1282:
1253:
1224:
1195:
1123:
1095:
1067:
1033:
999:
976:
917:
883:multiplicative inverse
871:
843:
817:
791:
759:
736:
697:
674:
639:
604:extended complex plane
589:
555:
523:
448:
426:
392:
372:
314:
277:
185:
140:
120:
96:
60:extended complex plane
39:
6970:Riemann zeta function
6263:Rational normal curve
5769:
5720:
5694:
5692:{\displaystyle (f,g)}
5662:
5642:
5622:
5602:
5566:
5536:
5501:
5468:
5420:
5365:
5146:, the transformation
5118:
5080:
5060:
5040:
5020:
4997:
4919:Möbius transformation
4907:Möbius transformation
4904:
4888:
4855:
4821:
4789:
4769:
4741:
4718:
4684:
4649:
4601:
4470:
4432:
4252:
4226:
4206:
4177:
4141:
4098:conformal equivalence
4079:
4053:
4017:
3991:
3955:
3935:
3908:
3874:
3840:
3820:
3793:
3767:
3726:
3685:
3573:
3535:
3506:
3462:
3358:), the projection is
3349:
3325:
3305:
3277:spherical coordinates
3270:
3236:Cartesian coordinates
3229:
3194:
3165:
3123:
3094:
3031:
3004:
2942:
2916:
2896:
2876:
2840:
2805:
2777:
2721:
2719:{\displaystyle (u,v)}
2689:
2687:{\displaystyle (w,z)}
2657:
2581:
2541:
2521:
2501:
2481:
2461:
2441:
2421:
2393:
2373:
2347:
2313:
2293:
2257:
2228:
2183:
2161:
2133:
2111:
2091:
2069:
2049:
2027:
2007:
1982:As a one-dimensional
1978:As a complex manifold
1969:
1949:
1923:
1888:
1853:
1851:{\displaystyle \pm 5}
1830:
1789:
1701:
1670:
1649:holomorphic functions
1642:
1608:
1582:
1549:
1520:
1500:
1464:
1428:
1392:
1390:{\displaystyle z_{0}}
1361:
1332:
1303:
1283:
1254:
1225:
1196:
1124:
1096:
1068:
1034:
1000:
977:
918:
872:
844:
818:
792:
760:
737:
698:
675:
640:
610:Arithmetic operations
590:
556:
524:
449:
427:
393:
373:
315:
278:
186:
141:
121:
97:
33:
7020:Riemann–Roch theorem
6803:Stable vector bundle
6675:Weil reciprocity law
6665:Riemann–Roch theorem
6645:Brill–Noether theory
6581:Riemann–Roch theorem
6498:Genus–degree formula
6359:Mordell–Weil theorem
6334:Division polynomials
5750:
5703:
5671:
5651:
5631:
5611:
5583:
5546:
5512:
5477:
5436:
5388:
5157:
5089:
5069:
5049:
5029:
5009:
4939:
4868:
4831:
4810:
4778:
4758:
4752:topological manifold
4730:
4694:
4660:
4625:
4482:
4444:
4440:In real coordinates
4264:
4235:
4215:
4186:
4154:
4130:
4121:Gauss–Bonnet theorem
4062:
4026:
4000:
3964:
3953:{\displaystyle \xi }
3944:
3924:
3883:
3849:
3838:{\displaystyle \xi }
3829:
3809:
3776:
3735:
3697:
3585:
3544:
3533:{\displaystyle z=0,}
3515:
3474:
3365:
3338:
3314:
3310:on the sphere (with
3282:
3241:
3203:
3192:{\displaystyle z=0,}
3174:
3136:
3103:
3044:
2951:
2925:
2905:
2885:
2853:
2817:
2786:
2730:
2698:
2666:
2637:
2568:
2539:{\displaystyle \xi }
2530:
2510:
2490:
2470:
2450:
2439:{\displaystyle \xi }
2430:
2402:
2382:
2356:
2322:
2311:{\displaystyle \xi }
2302:
2282:
2244:
2195:
2170:
2142:
2120:
2100:
2078:
2067:{\displaystyle \xi }
2058:
2036:
2016:
1994:
1958:
1932:
1897:
1862:
1839:
1801:
1720:
1679:
1659:
1620:
1591:
1580:{\displaystyle f(z)}
1562:
1529:
1509:
1473:
1437:
1401:
1374:
1359:{\displaystyle h(z)}
1341:
1330:{\displaystyle g(z)}
1312:
1292:
1281:{\displaystyle h(z)}
1263:
1252:{\displaystyle g(z)}
1234:
1223:{\displaystyle f(z)}
1205:
1144:
1129:are left undefined.
1105:
1077:
1043:
1009:
989:
930:
893:
861:
827:
801:
769:
749:
714:
687:
652:
621:
565:
536:
461:
436:
402:
382:
360:
295:
247:
208:meromorphic function
200:holomorphic function
194:. For example, any
161:
130:
110:
86:
64:closed complex plane
7096:Projective geometry
6995:Riemannian geometry
6905:Riemann Xi function
6890:Local zeta function
6626:Structure of curves
6518:Quartic plane curve
6440:Hyperelliptic curve
6420:De Franchis theorem
6364:Nagell–Lutz theorem
6117:The Road to Reality
5928:, pp. 428–430.
5767:{\displaystyle 1/2}
5718:{\displaystyle g=0}
5600:{\displaystyle f/g}
5430:Fubini–Study metric
4804:Riemannian manifold
4611:Fubini–Study metric
4250:{\displaystyle K=1}
3791:{\displaystyle z=0}
2726:are equivalent iff
2627:equivalence classes
2419:{\displaystyle 1/0}
1647:—form all possible
1368:continuous function
1094:{\displaystyle 0/0}
342:branches of physics
231:projective geometry
6915:Riemann hypothesis
6633:Divisors on curves
6425:Faltings's theorem
6374:Schoof's algorithm
6354:Modularity theorem
5812:Conformal geometry
5764:
5715:
5689:
5657:
5637:
5617:
5597:
5561:
5531:
5520:
5496:
5485:
5463:
5444:
5415:
5396:
5360:
5305:
5208:
5113:
5075:
5055:
5035:
5015:
4992:
4915:
4883:
4850:
4839:
4816:
4800:conformal manifold
4784:
4764:
4736:
4713:
4702:
4679:
4668:
4644:
4633:
4596:
4465:
4427:
4247:
4221:
4201:
4172:
4136:
4114:constant curvature
4074:
4048:
4012:
3986:
3950:
3930:
3903:
3869:
3835:
3815:
3788:
3762:
3721:
3680:
3648:
3568:
3530:
3501:
3457:
3428:
3344:
3320:
3300:
3265:
3224:
3189:
3160:
3118:
3089:
3038:
2999:
2937:
2911:
2891:
2871:
2849:. Given any point
2835:
2800:
2772:
2716:
2684:
2652:
2576:
2536:
2516:
2496:
2476:
2456:
2436:
2416:
2388:
2368:
2352:. The point where
2342:
2308:
2288:
2252:
2223:
2178:
2156:
2128:
2106:
2086:
2064:
2044:
2022:
2002:
1964:
1944:
1918:
1883:
1848:
1825:
1784:
1696:
1665:
1637:
1603:
1577:
1544:
1515:
1505:can be defined as
1495:
1459:
1423:
1387:
1356:
1327:
1298:
1278:
1249:
1220:
1191:
1133:Rational functions
1119:
1091:
1063:
1029:
995:
972:
913:
867:
839:
813:
787:
755:
732:
707:may be defined by
693:
670:
635:
585:
551:
519:
444:
422:
388:
368:
330:algebraic geometry
310:
273:
181:
136:
116:
92:
40:
7078:
7077:
6985:Riemannian circle
6925:Riemann invariant
6820:
6819:
6816:
6815:
6727:Hasse–Witt matrix
6670:Weierstrass point
6617:Smooth completion
6586:TeichmĂĽller space
6488:Cubic plane curve
6408:
6407:
6322:Arithmetic theory
6303:Elliptic integral
6298:Elliptic function
6199:Douglas N. Arnold
6126:978-0-679-77631-4
6054:
6053:
6046:
6021:does not provide
6000:
5999:
5992:
5827:Directed infinity
5660:{\displaystyle g}
5640:{\displaystyle g}
5620:{\displaystyle f}
5519:
5484:
5443:
5395:
5350:
5329:
5323:
5312:
5304:
5267:
5261:
5243:
5222:
5216:
5171:
5078:{\displaystyle d}
5058:{\displaystyle c}
5038:{\displaystyle b}
5018:{\displaystyle a}
4987:
4838:
4819:{\displaystyle 1}
4787:{\displaystyle S}
4767:{\displaystyle S}
4739:{\displaystyle S}
4701:
4667:
4632:
4552:
4475:, the formula is
4422:
4401:
4388:
4321:
4170:
4139:{\displaystyle K}
4094:Riemannian metric
3825:-coordinates and
3647:
3621:
3427:
3401:
2914:{\displaystyle z}
2894:{\displaystyle w}
2562:biholomorphically
2558:complex structure
2378:should then have
2221:
1967:{\displaystyle f}
1782:
1469:is nonzero, then
1301:{\displaystyle z}
1201:(in other words,
1139:rational function
998:{\displaystyle z}
964:
953:
941:
877:does not have an
758:{\displaystyle z}
696:{\displaystyle z}
501:
492:
475:
338:quantum mechanics
227:complex manifolds
196:rational function
139:{\displaystyle 0}
72:point at infinity
62:(also called the
16:(Redirected from
7113:
7106:Bernhard Riemann
7091:Riemann surfaces
7066:
7065:
6920:Riemann integral
6900:Riemann (crater)
6854:Bernhard Riemann
6847:
6840:
6833:
6824:
6660:Jacobian variety
6630:
6533:Riemann surfaces
6523:Real plane curve
6483:Cramer's paradox
6463:BĂ©zout's theorem
6288:
6237:algebraic curves
6229:
6222:
6215:
6206:
6191:
6178:"Riemann sphere"
6166:
6147:
6130:
6111:
6092:
6073:
6049:
6042:
6038:
6035:
6029:
6010:
6009:
6002:
5995:
5988:
5984:
5981:
5975:
5970:this article by
5961:inline citations
5948:
5947:
5940:
5929:
5923:
5917:
5911:
5905:
5904:
5902:
5900:
5885:
5879:
5873:
5788:celestial sphere
5773:
5771:
5770:
5765:
5760:
5724:
5722:
5721:
5716:
5698:
5696:
5695:
5690:
5666:
5664:
5663:
5658:
5646:
5644:
5643:
5638:
5626:
5624:
5623:
5618:
5606:
5604:
5603:
5598:
5593:
5570:
5568:
5567:
5562:
5560:
5559:
5554:
5540:
5538:
5537:
5532:
5521:
5517:
5505:
5503:
5502:
5497:
5486:
5482:
5472:
5470:
5469:
5464:
5459:
5445:
5441:
5424:
5422:
5421:
5416:
5411:
5397:
5393:
5369:
5367:
5366:
5361:
5348:
5327:
5321:
5320:
5316:
5310:
5306:
5303:
5289:
5275:
5265:
5259:
5241:
5220:
5214:
5213:
5212:
5169:
5122:
5120:
5119:
5114:
5084:
5082:
5081:
5076:
5064:
5062:
5061:
5056:
5044:
5042:
5041:
5036:
5024:
5022:
5021:
5016:
5001:
4999:
4998:
4993:
4988:
4986:
4972:
4958:
4892:
4890:
4889:
4884:
4882:
4881:
4876:
4863:projective space
4859:
4857:
4856:
4851:
4840:
4836:
4825:
4823:
4822:
4817:
4793:
4791:
4790:
4785:
4773:
4771:
4770:
4765:
4745:
4743:
4742:
4737:
4726:Conversely, let
4722:
4720:
4719:
4714:
4703:
4699:
4688:
4686:
4685:
4680:
4669:
4665:
4653:
4651:
4650:
4645:
4634:
4630:
4605:
4603:
4602:
4597:
4592:
4588:
4587:
4586:
4571:
4570:
4553:
4551:
4550:
4545:
4541:
4540:
4539:
4527:
4526:
4502:
4497:
4496:
4474:
4472:
4471:
4466:
4436:
4434:
4433:
4428:
4423:
4415:
4402:
4400:
4399:
4394:
4390:
4389:
4381:
4361:
4356:
4355:
4350:
4338:
4332:
4331:
4326:
4322:
4320:
4319:
4318:
4313:
4304:
4289:
4279:
4278:
4256:
4254:
4253:
4248:
4230:
4228:
4227:
4222:
4210:
4208:
4207:
4202:
4200:
4199:
4194:
4181:
4179:
4178:
4173:
4171:
4166:
4164:
4145:
4143:
4142:
4137:
4106:oriented surface
4083:
4081:
4080:
4075:
4057:
4055:
4054:
4049:
4041:
4033:
4021:
4019:
4018:
4013:
3995:
3993:
3992:
3987:
3979:
3971:
3959:
3957:
3956:
3951:
3939:
3937:
3936:
3931:
3912:
3910:
3909:
3904:
3899:
3878:
3876:
3875:
3870:
3865:
3844:
3842:
3841:
3836:
3824:
3822:
3821:
3816:
3797:
3795:
3794:
3789:
3771:
3769:
3768:
3763:
3730:
3728:
3727:
3722:
3689:
3687:
3686:
3681:
3676:
3675:
3659:
3658:
3649:
3640:
3637:
3636:
3630:
3622:
3620:
3609:
3595:
3577:
3575:
3574:
3569:
3539:
3537:
3536:
3531:
3510:
3508:
3507:
3502:
3466:
3464:
3463:
3458:
3453:
3452:
3439:
3438:
3429:
3420:
3417:
3416:
3410:
3402:
3400:
3389:
3375:
3353:
3351:
3350:
3345:
3329:
3327:
3326:
3321:
3309:
3307:
3306:
3301:
3274:
3272:
3271:
3266:
3233:
3231:
3230:
3225:
3198:
3196:
3195:
3190:
3169:
3167:
3166:
3161:
3127:
3125:
3124:
3119:
3117:
3116:
3111:
3098:
3096:
3095:
3090:
3082:
3081:
3069:
3068:
3056:
3055:
3008:
3006:
3005:
3000:
2998:
2994:
2990:
2946:
2944:
2943:
2938:
2920:
2918:
2917:
2912:
2900:
2898:
2897:
2892:
2880:
2878:
2877:
2874:{\displaystyle }
2872:
2844:
2842:
2841:
2838:{\displaystyle }
2836:
2809:
2807:
2806:
2801:
2799:
2781:
2779:
2778:
2773:
2725:
2723:
2722:
2717:
2693:
2691:
2690:
2685:
2661:
2659:
2658:
2653:
2651:
2650:
2645:
2631:non-null vectors
2609:surface without
2599:hyperbolic plane
2595:simply-connected
2585:
2583:
2582:
2577:
2575:
2564:identified with
2545:
2543:
2542:
2537:
2525:
2523:
2522:
2517:
2505:
2503:
2502:
2497:
2485:
2483:
2482:
2477:
2465:
2463:
2462:
2457:
2445:
2443:
2442:
2437:
2425:
2423:
2422:
2417:
2412:
2397:
2395:
2394:
2389:
2377:
2375:
2374:
2369:
2351:
2349:
2348:
2343:
2338:
2317:
2315:
2314:
2309:
2297:
2295:
2294:
2289:
2261:
2259:
2258:
2253:
2251:
2232:
2230:
2229:
2224:
2222:
2214:
2188:. Then the map
2187:
2185:
2184:
2179:
2177:
2165:
2163:
2162:
2157:
2152:
2137:
2135:
2134:
2129:
2127:
2115:
2113:
2112:
2107:
2095:
2093:
2092:
2087:
2085:
2073:
2071:
2070:
2065:
2053:
2051:
2050:
2045:
2043:
2031:
2029:
2028:
2023:
2011:
2009:
2008:
2003:
2001:
1984:complex manifold
1973:
1971:
1970:
1965:
1953:
1951:
1950:
1945:
1927:
1925:
1924:
1919:
1892:
1890:
1889:
1884:
1857:
1855:
1854:
1849:
1834:
1832:
1831:
1826:
1793:
1791:
1790:
1785:
1783:
1781:
1774:
1773:
1760:
1753:
1752:
1739:
1705:
1703:
1702:
1697:
1686:
1674:
1672:
1671:
1666:
1646:
1644:
1643:
1638:
1627:
1612:
1610:
1609:
1604:
1586:
1584:
1583:
1578:
1553:
1551:
1550:
1545:
1524:
1522:
1521:
1516:
1504:
1502:
1501:
1496:
1491:
1490:
1468:
1466:
1465:
1460:
1455:
1454:
1432:
1430:
1429:
1424:
1419:
1418:
1396:
1394:
1393:
1388:
1386:
1385:
1365:
1363:
1362:
1357:
1336:
1334:
1333:
1328:
1307:
1305:
1304:
1299:
1287:
1285:
1284:
1279:
1258:
1256:
1255:
1250:
1229:
1227:
1226:
1221:
1200:
1198:
1197:
1192:
1178:
1128:
1126:
1125:
1120:
1115:
1100:
1098:
1097:
1092:
1087:
1073:. The quotients
1072:
1070:
1069:
1064:
1053:
1038:
1036:
1035:
1030:
1019:
1004:
1002:
1001:
996:
981:
979:
978:
973:
965:
957:
954:
951:
942:
934:
922:
920:
919:
914:
900:
876:
874:
873:
868:
848:
846:
845:
840:
822:
820:
819:
814:
796:
794:
793:
788:
764:
762:
761:
756:
741:
739:
738:
733:
702:
700:
699:
694:
679:
677:
676:
671:
644:
642:
641:
636:
634:
594:
592:
591:
586:
572:
560:
558:
557:
552:
550:
549:
544:
528:
526:
525:
520:
515:
514:
509:
502:
499:
493:
488:
483:
477:
476:
471:
466:
453:
451:
450:
445:
443:
431:
429:
428:
423:
409:
397:
395:
394:
389:
377:
375:
374:
369:
367:
319:
317:
316:
311:
309:
308:
303:
285:projective space
282:
280:
279:
274:
269:
261:
260:
255:
190:
188:
187:
182:
171:
155:division by zero
151:complex analysis
145:
143:
142:
137:
125:
123:
122:
117:
101:
99:
98:
93:
52:Bernhard Riemann
21:
7121:
7120:
7116:
7115:
7114:
7112:
7111:
7110:
7081:
7080:
7079:
7074:
7054:
6965:Riemann surface
6940:Riemann problem
6856:
6851:
6821:
6812:
6784:
6775:Delta invariant
6746:
6715:
6679:
6640:Abel–Jacobi map
6621:
6595:
6591:Torelli theorem
6561:Dessin d'enfant
6541:Belyi's theorem
6527:
6513:PlĂĽcker formula
6444:
6435:Hurwitz surface
6404:
6383:
6317:
6291:Analytic theory
6283:Elliptic curves
6277:
6258:Projective line
6245:Rational curves
6239:
6233:
6176:
6173:
6163:
6150:
6133:
6127:
6114:
6108:
6095:
6089:
6076:
6070:
6057:
6050:
6039:
6033:
6030:
6027:
6023:page references
6011:
6007:
5996:
5985:
5979:
5976:
5966:Please help to
5965:
5949:
5945:
5938:
5933:
5932:
5924:
5920:
5912:
5908:
5898:
5896:
5887:
5886:
5882:
5874:
5870:
5865:
5822:Dessin d'enfant
5808:
5748:
5747:
5701:
5700:
5669:
5668:
5649:
5648:
5629:
5628:
5609:
5608:
5581:
5580:
5577:
5549:
5544:
5543:
5510:
5509:
5508:rotation group
5475:
5474:
5434:
5433:
5386:
5385:
5290:
5276:
5272:
5268:
5207:
5206:
5201:
5195:
5194:
5189:
5179:
5155:
5154:
5150:can be written
5087:
5086:
5067:
5066:
5047:
5046:
5027:
5026:
5007:
5006:
4973:
4959:
4937:
4936:
4921:
4899:
4871:
4866:
4865:
4829:
4828:
4808:
4807:
4776:
4775:
4756:
4755:
4728:
4727:
4692:
4691:
4658:
4657:
4623:
4622:
4578:
4562:
4558:
4554:
4531:
4518:
4511:
4507:
4506:
4488:
4480:
4479:
4442:
4441:
4370:
4366:
4365:
4345:
4308:
4293:
4284:
4283:
4270:
4262:
4261:
4233:
4232:
4213:
4212:
4189:
4184:
4183:
4152:
4151:
4128:
4127:
4102:smooth function
4090:
4060:
4059:
4024:
4023:
3998:
3997:
3962:
3961:
3942:
3941:
3922:
3921:
3881:
3880:
3847:
3846:
3827:
3826:
3807:
3806:
3774:
3773:
3733:
3732:
3695:
3694:
3661:
3610:
3596:
3583:
3582:
3542:
3541:
3513:
3512:
3511:onto the plane
3472:
3471:
3441:
3390:
3376:
3363:
3362:
3336:
3335:
3312:
3311:
3280:
3279:
3239:
3238:
3201:
3200:
3172:
3171:
3170:onto the plane
3134:
3133:
3106:
3101:
3100:
3073:
3060:
3047:
3042:
3041:
3026:
2976:
2972:
2949:
2948:
2923:
2922:
2903:
2902:
2883:
2882:
2851:
2850:
2815:
2814:
2784:
2783:
2728:
2727:
2696:
2695:
2664:
2663:
2640:
2635:
2634:
2619:
2566:
2565:
2528:
2527:
2508:
2507:
2488:
2487:
2468:
2467:
2448:
2447:
2428:
2427:
2400:
2399:
2380:
2379:
2354:
2353:
2320:
2319:
2300:
2299:
2280:
2279:
2272:Riemann surface
2242:
2241:
2193:
2192:
2168:
2167:
2140:
2139:
2118:
2117:
2098:
2097:
2076:
2075:
2056:
2055:
2034:
2033:
2014:
2013:
1992:
1991:
1980:
1956:
1955:
1930:
1929:
1895:
1894:
1860:
1859:
1837:
1836:
1799:
1798:
1765:
1761:
1744:
1740:
1718:
1717:
1677:
1676:
1657:
1656:
1653:Riemann surface
1618:
1617:
1589:
1588:
1560:
1559:
1527:
1526:
1507:
1506:
1482:
1471:
1470:
1446:
1435:
1434:
1410:
1399:
1398:
1377:
1372:
1371:
1339:
1338:
1310:
1309:
1290:
1289:
1261:
1260:
1232:
1231:
1203:
1202:
1142:
1141:
1135:
1103:
1102:
1075:
1074:
1041:
1040:
1007:
1006:
987:
986:
928:
927:
891:
890:
859:
858:
825:
824:
799:
798:
767:
766:
747:
746:
712:
711:
685:
684:
650:
649:
619:
618:
612:
563:
562:
539:
534:
533:
504:
459:
458:
434:
433:
400:
399:
380:
379:
358:
357:
350:
326:algebraic curve
320:. As with any
298:
293:
292:
250:
245:
244:
241:projective line
223:Riemann surface
159:
158:
128:
127:
108:
107:
84:
83:
80:complex numbers
78:, that is, the
28:
23:
22:
15:
12:
11:
5:
7119:
7117:
7109:
7108:
7103:
7098:
7093:
7083:
7082:
7076:
7075:
7073:
7072:
7059:
7056:
7055:
7053:
7052:
7047:
7042:
7037:
7032:
7027:
7022:
7017:
7012:
7007:
7002:
6997:
6992:
6987:
6982:
6977:
6972:
6967:
6962:
6957:
6955:Riemann sphere
6952:
6950:Riemann solver
6947:
6942:
6937:
6932:
6927:
6922:
6917:
6912:
6907:
6902:
6897:
6892:
6887:
6882:
6877:
6872:
6867:
6861:
6858:
6857:
6852:
6850:
6849:
6842:
6835:
6827:
6818:
6817:
6814:
6813:
6811:
6810:
6805:
6800:
6794:
6792:
6790:Vector bundles
6786:
6785:
6783:
6782:
6777:
6772:
6767:
6762:
6756:
6754:
6748:
6747:
6745:
6744:
6739:
6734:
6729:
6723:
6721:
6717:
6716:
6714:
6713:
6708:
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6268:Riemann sphere
6265:
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6171:External links
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6034:September 2010
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5800:twistor theory
5786:model for the
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4917:Main article:
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2268:Riemann sphere
2250:
2238:transition map
2236:is called the
2234:
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2209:
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2166:of the second
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1797:we may define
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705:multiplication
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600:Riemann sphere
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50:, named after
48:Riemann sphere
26:
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18:Complex sphere
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6752:Singularities
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6600:Constructions
6598:
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6584:
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6577:
6574:
6572:
6571:Klein quartic
6569:
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6551:Bolza surface
6549:
6547:
6546:Bring's curve
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6478:Conic section
6476:
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6471:
6469:
6466:
6464:
6461:
6459:
6458:AF+BG theorem
6456:
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6273:Twisted cubic
6271:
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6162:0-07-100276-6
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6142:(in German).
6141:
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6122:
6118:
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6107:0-471-32792-1
6103:
6099:
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6088:0-19-853793-X
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6069:0-07-010905-2
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5809:
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5797:
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5792:string theory
5789:
5785:
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5777:
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4908:
4903:
4897:Automorphisms
4896:
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4009:
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4003:
3983:
3980:
3972:
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3927:
3918:
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3915:diffeomorphic
3900:
3896:
3892:
3889:
3886:
3866:
3862:
3858:
3855:
3852:
3832:
3812:
3803:
3801:
3798:, because an
3785:
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3715:
3712:
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3606:
3603:
3600:
3597:
3591:
3588:
3581:
3580:
3579:
3565:
3562:
3559:
3556:
3553:
3550:
3547:
3527:
3524:
3521:
3518:
3495:
3492:
3489:
3486:
3483:
3480:
3454:
3449:
3446:
3442:
3430:
3424:
3421:
3407:
3403:
3397:
3394:
3391:
3386:
3383:
3380:
3377:
3371:
3368:
3361:
3360:
3359:
3357:
3341:
3333:
3317:
3294:
3291:
3288:
3278:
3259:
3256:
3253:
3250:
3247:
3237:
3221:
3218:
3215:
3212:
3209:
3206:
3186:
3183:
3180:
3177:
3154:
3151:
3148:
3145:
3142:
3131:
3113:
3086:
3083:
3078:
3074:
3070:
3065:
3061:
3057:
3052:
3048:
3035:
3030:
3023:
3021:
3019:
3018:automorphisms
3015:
3010:
2995:
2991:
2987:
2983:
2980:
2977:
2973:
2969:
2963:
2960:
2957:
2934:
2931:
2928:
2908:
2888:
2865:
2862:
2859:
2848:
2829:
2826:
2823:
2811:
2792:
2789:
2766:
2763:
2760:
2757:
2754:
2748:
2742:
2739:
2736:
2710:
2707:
2704:
2678:
2675:
2672:
2647:
2632:
2628:
2624:
2616:
2614:
2612:
2608:
2604:
2600:
2596:
2592:
2587:
2563:
2559:
2555:
2551:
2550:Topologically
2547:
2533:
2493:
2473:
2433:
2413:
2409:
2405:
2385:
2365:
2362:
2359:
2339:
2335:
2331:
2328:
2325:
2305:
2285:
2275:
2273:
2269:
2265:
2239:
2218:
2215:
2210:
2204:
2198:
2191:
2190:
2189:
2153:
2149:
2145:
2116:of the first
2103:
2061:
2019:
1989:
1985:
1977:
1975:
1961:
1935:
1915:
1906:
1900:
1880:
1877:
1865:
1845:
1842:
1819:
1813:
1810:
1804:
1778:
1775:
1770:
1766:
1762:
1757:
1754:
1749:
1745:
1741:
1735:
1729:
1723:
1716:
1715:
1714:
1711:
1709:
1690:
1654:
1650:
1631:
1614:
1594:
1571:
1565:
1557:
1532:
1525:. Moreover,
1487:
1483:
1476:
1451:
1447:
1440:
1415:
1411:
1404:
1382:
1378:
1369:
1350:
1344:
1321:
1315:
1295:
1272:
1266:
1243:
1237:
1214:
1208:
1185:
1179:
1175:
1168:
1162:
1159:
1153:
1147:
1140:
1132:
1130:
1112:
1088:
1084:
1080:
1060:
1057:
1050:
1046:
1023:
1020:
1016:
992:
969:
966:
958:
943:
938:
935:
926:
925:
924:
901:
888:
884:
880:
856:
852:
833:
830:
807:
781:
775:
752:
726:
720:
717:
710:
709:
708:
706:
690:
664:
658:
655:
648:
647:
646:
627:
624:
616:
609:
607:
605:
601:
596:
579:
546:
532:The notation
516:
494:
478:
472:
457:
456:
455:
410:
355:
347:
345:
343:
340:and in other
339:
335:
331:
327:
323:
305:
290:
289:complex lines
286:
257:
243:
242:
236:
232:
228:
224:
220:
215:
213:
209:
205:
201:
197:
193:
175:
172:
168:
164:
156:
152:
147:
133:
105:
82:plus a value
81:
77:
73:
69:
68:complex plane
65:
61:
57:
53:
49:
45:
37:
32:
19:
7067:
6954:
6935:Riemann form
6737:Prym variety
6711:Stable curve
6701:Hodge bundle
6691:ELSV formula
6493:Fermat curve
6450:Plane curves
6413:Higher genus
6388:Applications
6313:Modular form
6267:
6181:
6152:
6143:
6139:
6116:
6097:
6078:
6059:
6040:
6031:
6020:
5986:
5977:
5958:
5926:Penrose 2007
5921:
5916:, p. 1.
5914:Goldman 1999
5909:
5899:December 12,
5897:. Retrieved
5883:
5876:Riemann 1857
5871:
5857:Wheel theory
5837:Möbius plane
5784:relativistic
5780:Bloch sphere
5733:polarization
5727:
5578:
5575:Applications
5427:
5372:
5147:
5140:homographies
5137:
5133:translations
5004:
4922:
4796:round metric
4725:
4617:
4615:
4608:
4439:
4257:is given by
4118:
4111:
4091:
3919:
3804:
3692:
3469:
3039:
3033:
3011:
2812:
2630:
2622:
2620:
2588:
2548:
2298:value and a
2276:
2271:
2267:
2235:
1981:
1796:
1712:
1707:
1615:
1136:
984:
797:. Note that
744:
682:
613:
603:
599:
597:
531:
353:
351:
334:Bloch sphere
238:
216:
192:well-behaved
148:
75:
63:
59:
47:
41:
6960:Riemann sum
6612:Polar curve
5980:August 2010
5972:introducing
5852:Smith chart
5832:Hopf bundle
5817:Cross-ratio
5796:worldsheets
5776:Quantum bit
5375:determinant
3800:orientation
3578:is written
3024:As a sphere
2264:holomorphic
44:mathematics
7085:Categories
6607:Dual curve
6235:Topics in
6146:: 115–155.
5955:references
5936:References
5739:states of
2054:, and let
454:, such as
6720:Morphisms
6468:Bitangent
6188:EMS Press
5744:particles
5473:, namely
5340:ζ
5295:ζ
5281:ζ
5248:ζ
5230:ζ
5164:ζ
5129:rotations
5125:dilations
5108:≠
5099:−
4978:ζ
4964:ζ
4949:ζ
4448:ζ
4420:¯
4417:ζ
4408:ζ
4386:¯
4383:ζ
4378:ζ
4343:ζ
4306:ζ
4219:ζ
4148:isometric
4125:curvature
4035:ξ
3973:ζ
3948:ξ
3928:ζ
3901:ζ
3887:ξ
3867:ξ
3853:ζ
3833:ξ
3813:ζ
3754:−
3673:φ
3667:−
3651:θ
3601:−
3589:ξ
3557:−
3548:ξ
3493:−
3450:φ
3431:θ
3395:−
3369:ζ
3342:φ
3318:θ
3295:φ
3289:θ
3207:ζ
2932:≠
2793:∈
2790:λ
2764:λ
2755:λ
2534:ξ
2514:∞
2494:ζ
2474:ζ
2454:∞
2434:ξ
2386:ζ
2360:ξ
2340:ξ
2326:ζ
2306:ξ
2286:ζ
2154:ξ
2104:ζ
2062:ξ
2020:ζ
1942:∞
1939:→
1913:→
1872:∞
1843:±
1823:∞
1811:±
1776:−
1663:∞
1601:∞
1598:→
1539:∞
1513:∞
1117:∞
1109:∞
1055:∞
1027:∞
1013:∞
962:∞
947:∞
908:∞
902:∪
865:∞
851:undefined
849:are left
837:∞
834:×
811:∞
808:−
805:∞
785:∞
779:∞
776:×
773:∞
730:∞
724:∞
721:×
668:∞
662:∞
628:∈
574:∖
547:∗
512:∞
490:¯
473:^
417:∞
411:∪
386:∞
179:∞
114:∞
90:∞
70:plus one
7069:Category
5893:Archived
5806:See also
5746:of spin
5735:states,
4802:, not a
2611:boundary
2546:-chart.
2398:-value "
887:division
879:additive
857:, since
615:Addition
239:complex
219:geometry
212:codomain
104:infinity
54:, is a
7101:Spheres
6780:Tacnode
6765:Crunode
6190:, 2001
6017:sources
5968:improve
5741:massive
3356:azimuth
2607:compact
2526:in the
2466:in the
765:, with
322:compact
287:of all
66:): the
58:of the
6760:Acnode
6684:Moduli
6159:
6123:
6104:
6085:
6066:
5957:, but
5794:, the
5730:photon
5349:
5328:
5322:
5311:
5266:
5260:
5242:
5221:
5215:
5170:
5065:, and
5005:where
4748:smooth
4088:Metric
3332:zenith
2845:using
2012:. Let
1988:charts
1893:since
1858:, and
703:, and
283:, the
46:, the
6197:, by
5889:"C^*"
5863:Notes
5790:. In
5379:group
4925:group
3234:. In
1556:limit
1005:with
855:field
229:. In
204:poles
56:model
6770:Cusp
6157:ISBN
6121:ISBN
6102:ISBN
6083:ISBN
6064:ISBN
6019:but
5901:2021
5778:and
5737:spin
5627:and
4618:only
4069:>
4043:<
4007:<
3981:<
3879:and
3354:the
3334:and
3330:the
3275:and
2901:and
2694:and
1337:and
1259:and
1137:Any
1101:and
1039:and
881:nor
823:and
602:(or
352:The
102:for
5483:PSU
5442:PGL
5394:PGL
4893:.)
4750:or
4182:in
3628:tan
3408:cot
2629:of
2605:(a
1928:as
1587:as
1558:of
1288:of
952:and
923:by
889:on
606:).
336:of
291:in
217:In
42:In
7087::
6186:,
6180:,
6144:54
5891:.
5802:.
5518:SO
5425:.
5131:,
5127:,
5045:,
5025:,
4905:A
4837:SO
4700:SO
4631:SO
4084:.
2810:.
2586:.
2274:.
1779:50
1710:.
645:,
500:or
344:.
6846:e
6839:t
6832:v
6228:e
6221:t
6214:v
6165:.
6129:.
6110:.
6091:.
6072:.
6047:)
6041:(
6036:)
6032:(
6026:.
5993:)
5987:(
5982:)
5978:(
5964:.
5903:.
5878:.
5762:2
5758:/
5754:1
5713:0
5710:=
5707:g
5687:)
5684:g
5681:,
5678:f
5675:(
5655:g
5635:g
5615:f
5595:g
5591:/
5587:f
5557:3
5552:R
5529:)
5526:3
5523:(
5494:)
5491:2
5488:(
5461:)
5457:C
5453:,
5450:2
5447:(
5413:)
5409:C
5405:,
5402:2
5399:(
5358:.
5355:]
5352:1
5346:,
5343:)
5337:(
5334:f
5331:[
5325:=
5318:]
5314:1
5308:,
5301:d
5298:+
5292:c
5287:b
5284:+
5278:a
5270:[
5263:=
5257:]
5254:d
5251:+
5245:c
5239:,
5236:b
5233:+
5227:a
5224:[
5218:=
5210:)
5204:d
5199:b
5192:c
5187:a
5181:(
5176:]
5173:1
5167:,
5161:[
5148:f
5111:0
5105:c
5102:b
5096:d
5093:a
5073:d
5053:c
5033:b
5013:a
4990:,
4984:d
4981:+
4975:c
4970:b
4967:+
4961:a
4955:=
4952:)
4946:(
4943:f
4913:.
4879:3
4874:P
4848:)
4845:3
4842:(
4814:1
4782:S
4762:S
4734:S
4711:)
4708:3
4705:(
4677:)
4674:3
4671:(
4666:O
4642:)
4639:3
4636:(
4594:.
4590:)
4584:2
4580:v
4576:d
4573:+
4568:2
4564:u
4560:d
4556:(
4548:2
4543:)
4537:2
4533:v
4529:+
4524:2
4520:u
4516:+
4513:1
4509:(
4504:4
4499:=
4494:2
4490:s
4486:d
4463:v
4460:i
4457:+
4454:u
4451:=
4425:.
4412:d
4405:d
4397:2
4392:)
4375:+
4372:1
4368:(
4363:4
4358:=
4353:2
4348:|
4340:d
4336:|
4329:2
4324:)
4316:2
4311:|
4302:|
4298:+
4295:1
4291:2
4286:(
4281:=
4276:2
4272:s
4268:d
4245:1
4242:=
4239:K
4197:3
4192:R
4168:K
4162:/
4158:1
4134:K
4072:0
4066:z
4046:1
4039:|
4031:|
4010:0
4004:z
3984:1
3977:|
3969:|
3897:/
3893:1
3890:=
3863:/
3859:1
3856:=
3786:0
3783:=
3780:z
3760:)
3757:1
3751:,
3748:0
3745:,
3742:0
3739:(
3719:)
3716:1
3713:,
3710:0
3707:,
3704:0
3701:(
3678:.
3670:i
3663:e
3656:)
3645:2
3642:1
3634:(
3624:=
3618:z
3615:+
3612:1
3607:y
3604:i
3598:x
3592:=
3566:,
3563:y
3560:i
3554:x
3551:=
3528:,
3525:0
3522:=
3519:z
3499:)
3496:1
3490:,
3487:0
3484:,
3481:0
3478:(
3455:.
3447:i
3443:e
3436:)
3425:2
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3414:(
3404:=
3398:z
3392:1
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3384:i
3381:+
3378:x
3372:=
3298:)
3292:,
3286:(
3263:)
3260:z
3257:,
3254:y
3251:,
3248:x
3245:(
3222:y
3219:i
3216:+
3213:x
3210:=
3187:,
3184:0
3181:=
3178:z
3158:)
3155:1
3152:,
3149:0
3146:,
3143:0
3140:(
3114:3
3109:R
3087:1
3084:=
3079:2
3075:z
3071:+
3066:2
3062:y
3058:+
3053:2
3049:x
3034:A
2996:]
2992:w
2988:/
2984:z
2981:,
2978:1
2974:[
2970:=
2967:]
2964:z
2961:,
2958:w
2955:[
2935:0
2929:w
2909:z
2889:w
2869:]
2866:z
2863:,
2860:w
2857:[
2833:]
2830:z
2827:,
2824:w
2821:[
2797:C
2770:)
2767:v
2761:,
2758:u
2752:(
2749:=
2746:)
2743:z
2740:,
2737:w
2734:(
2714:)
2711:v
2708:,
2705:u
2702:(
2682:)
2679:z
2676:,
2673:w
2670:(
2648:2
2643:C
2573:C
2414:0
2410:/
2406:1
2366:0
2363:=
2336:/
2332:1
2329:=
2249:C
2219:z
2216:1
2211:=
2208:)
2205:z
2202:(
2199:f
2175:C
2150:/
2146:1
2125:C
2083:C
2041:C
1999:C
1962:f
1936:z
1916:3
1910:)
1907:z
1904:(
1901:f
1881:3
1878:=
1875:)
1869:(
1866:f
1846:5
1820:=
1817:)
1814:5
1808:(
1805:f
1771:2
1767:z
1763:2
1758:1
1755:+
1750:2
1746:z
1742:6
1736:=
1733:)
1730:z
1727:(
1724:f
1694:)
1691:z
1688:(
1684:C
1635:)
1632:z
1629:(
1625:C
1595:z
1575:)
1572:z
1569:(
1566:f
1542:)
1536:(
1533:f
1493:)
1488:0
1484:z
1480:(
1477:f
1457:)
1452:0
1448:z
1444:(
1441:g
1421:)
1416:0
1412:z
1408:(
1405:h
1383:0
1379:z
1354:)
1351:z
1348:(
1345:h
1325:)
1322:z
1319:(
1316:g
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1273:z
1270:(
1267:h
1247:)
1244:z
1241:(
1238:g
1218:)
1215:z
1212:(
1209:f
1189:)
1186:z
1183:(
1180:h
1176:/
1172:)
1169:z
1166:(
1163:g
1160:=
1157:)
1154:z
1151:(
1148:f
1113:/
1089:0
1085:/
1081:0
1061:0
1058:=
1051:/
1047:0
1024:=
1021:0
1017:/
993:z
970:0
967:=
959:z
944:=
939:0
936:z
911:}
905:{
898:C
831:0
782:=
753:z
727:=
718:z
691:z
665:=
659:+
656:z
632:C
625:z
583:}
580:0
577:{
570:C
542:C
517:.
507:C
495:,
486:C
479:,
469:C
441:C
420:}
414:{
407:C
365:C
306:2
301:C
271:)
267:C
263:(
258:1
253:P
176:=
173:0
169:/
165:1
134:0
20:)
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