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Difference between revisions of "Vekua method"

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''in the theory of infinitesimal deformations''
 
''in the theory of infinitesimal deformations''
  
A method which can be applied to the case when certain quantities which characterize the deformation of surfaces with positive Gaussian curvature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096540/v0965401.png" /> are, in a conjugate-isothermal parametrization, generalized analytic functions (cf. [[Generalized analytic function|Generalized analytic function]]). This makes it possible to reduce the study of the deformation of surfaces with variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096540/v0965402.png" /> to a definite problem concerning surfaces with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096540/v0965403.png" />, whose infinitesimal deformations (cf. [[Infinitesimal deformation|Infinitesimal deformation]]) are described by ordinary analytic functions, thus establishing a far-reaching analogy between the properties of deformations of surfaces with variable and constant positive Gaussian curvature.
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A method which can be applied to the case when certain quantities which characterize the deformation of surfaces with positive Gaussian curvature $  K $
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are, in a conjugate-isothermal parametrization, generalized analytic functions (cf. [[Generalized analytic function|Generalized analytic function]]). This makes it possible to reduce the study of the deformation of surfaces with variable $  K > 0 $
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to a definite problem concerning surfaces with $  K = \textrm{ const } > 0 $,  
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whose infinitesimal deformations (cf. [[Infinitesimal deformation|Infinitesimal deformation]]) are described by ordinary analytic functions, thus establishing a far-reaching analogy between the properties of deformations of surfaces with variable and constant positive Gaussian curvature.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I.N. Vekua,  "Generalized analytic functions" , Pergamon  (1962)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I.N. Vekua,  "Generalized analytic functions" , Pergamon  (1962)  (Translated from Russian)</TD></TR></table>

Latest revision as of 08:28, 6 June 2020


in the theory of infinitesimal deformations

A method which can be applied to the case when certain quantities which characterize the deformation of surfaces with positive Gaussian curvature $ K $ are, in a conjugate-isothermal parametrization, generalized analytic functions (cf. Generalized analytic function). This makes it possible to reduce the study of the deformation of surfaces with variable $ K > 0 $ to a definite problem concerning surfaces with $ K = \textrm{ const } > 0 $, whose infinitesimal deformations (cf. Infinitesimal deformation) are described by ordinary analytic functions, thus establishing a far-reaching analogy between the properties of deformations of surfaces with variable and constant positive Gaussian curvature.

References

[1] I.N. Vekua, "Generalized analytic functions" , Pergamon (1962) (Translated from Russian)
How to Cite This Entry:
Vekua method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Vekua_method&oldid=49143
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article