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Analytic surface

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in a Euclidean space

An arbitrary two-dimensional analytic submanifold in the space , . However, the term "analytic surface in Rn" is often employed in a wider sense as a manifold which is (locally) analytically parametrizable. This means that the coordinates of the points can be represented by analytic functions of a real parameter which varies in a certain range , . If the rank of the Jacobi matrix , which for an analytic manifold is maximal everywhere in , is equal to , then the dimension of the analytic surface is .

In the complex space the term "analytic surface" is also employed to denote a complex-analytic surface in , i.e. a manifold which allows a holomorphic (complex-analytic) parametrization. This means that the complex coordinates of points can be expressed by holomorphic functions of a parameter which varies within a certain range (it is also usually assumed that ). If and all the functions are linear, one obtains a complex-analytic plane (cf. Analytic plane). If , the term which is sometimes employed is holomorphic curve (complex-analytic curve); if all functions are linear, one speaks of a complex straight line in the parametric representation:

References

[1] B.V. Shabat, "Introduction of complex analysis" , 1–2 , Moscow (1976) (In Russian)
[2] V.S. Vladimirov, "Methods of the theory of functions of several complex variables" , M.I.T. (1966) pp. Chapt. 2 (Translated from Russian)
How to Cite This Entry:
Analytic surface. E.D. SolomentsevE.M. Chirka (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Analytic_surface&oldid=11625
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098