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Riemann sphere

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A sphere in the Euclidean space onto which the extended complex plane is conformally and one-to-one transformed under stereographic projection. For example, the unit sphere

can be taken as the Riemann sphere and the plane can be identified with the plane such that the real axis coincides with the axis and the imaginary axis with the axis (see Fig.).

Figure: r082010a

Under stereographic projection to each point there corresponds the point obtained as the point of intersection of the ray drawn from the north pole of the sphere, , to the point with the sphere ; the north pole corresponds to the point at infinity, . Analytically this relation can be expressed by the formulas

(*)

In other words, the correspondence (*) determines a differentiable imbedding of the one-dimensional complex projective space into the space in the form of the sphere . In many questions of the theory of functions, the extended complex plane is identified with the Riemann sphere. The exclusive role of the point at infinity of the plane may be dispensed with if the distance between two points is taken to be the chordal, or spherical, distance between their images :

A higher-dimensional complex projective space , , can be imbedded into the space by a complex -dimensional stereographic projection, generalizing the formulas (*) (see [2]).

References

[1] B.V. Shabat, "Introduction of complex analysis" , 1–2 , Moscow (1976) (In Russian)
[2] B.A. Fuks, "Introduction to the theory of analytic functions of several complex variables" , Amer. Math. Soc. (1965) (Translated from Russian)


Comments

References

[a1] L.V. Ahlfors, "Complex analysis" , McGraw-Hill (1979) pp. Chapt. 8
How to Cite This Entry:
Riemann sphere. E.D. Solomentsev (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Riemann_sphere&oldid=18286
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098