Namespaces
Variants
Actions

Difference between revisions of "Analytic surface"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
(TeX)
 
Line 1: Line 1:
 +
{{TEX|done}}
 
''in a Euclidean space''
 
''in a Euclidean space''
  
An arbitrary two-dimensional analytic submanifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012440/a0124401.png" /> in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012440/a0124402.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012440/a0124403.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012440/a0124404.png" /> can be represented by analytic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012440/a0124405.png" /> of a real parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012440/a0124406.png" /> which varies in a certain range <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012440/a0124407.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012440/a0124408.png" />. If the rank of the Jacobi matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012440/a0124409.png" />, which for an analytic manifold is maximal everywhere in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012440/a01244010.png" />, is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012440/a01244011.png" />, then the dimension of the analytic surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012440/a01244012.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012440/a01244013.png" />.
+
An arbitrary two-dimensional analytic submanifold $X$ in the space $\mathbf R^n$, $n>2$. However, the term  "analytic surface in $\mathbf R^n$"  is often employed in a wider sense as a manifold which is (locally) analytically parametrizable. This means that the coordinates of the points $x=(x_1,\dots,x_n)\in X$ can be represented by analytic functions $x_i=x_i(t)$ of a real parameter $t=(t_1,\dots,t_k)$ which varies in a certain range $\Delta\subset\mathbf R^k$, $1\leq k<n$. If the rank of the Jacobi matrix $|\partial x/\partial t|$, which for an analytic manifold is maximal everywhere in $\Delta$, is equal to $k$, then the dimension of the analytic surface $X$ is $k$.
  
In the complex space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012440/a01244014.png" /> the term  "analytic surface"  is also employed to denote a complex-analytic surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012440/a01244015.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012440/a01244016.png" />, i.e. a manifold which allows a holomorphic (complex-analytic) parametrization. This means that the complex coordinates of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012440/a01244017.png" /> can be expressed by holomorphic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012440/a01244018.png" /> of a parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012440/a01244019.png" /> which varies within a certain range <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012440/a01244020.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012440/a01244021.png" /> (it is also usually assumed that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012440/a01244022.png" />). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012440/a01244023.png" /> and all the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012440/a01244024.png" /> are linear, one obtains a complex-analytic plane (cf. [[Analytic plane|Analytic plane]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012440/a01244025.png" />, the term which is sometimes employed is holomorphic curve (complex-analytic curve); if all functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012440/a01244026.png" /> are linear, one speaks of a complex straight line in the parametric representation:
+
In the complex space $\mathbf C^n$ the term  "analytic surface"  is also employed to denote a complex-analytic surface $X$ in $\mathbf C^n$, i.e. a manifold which allows a holomorphic (complex-analytic) parametrization. This means that the complex coordinates of points $z=(z_1,\dots,z_n)\in X$ can be expressed by holomorphic functions $z_i=z_i(\tau)$ of a parameter $\tau=(\tau_1,\dots,\tau_k)$ which varies within a certain range $\Delta\subset\mathbf C^k$, $1\leq k<n$ (it is also usually assumed that $\rank\|\partial z/\partial\tau\|\equiv k$). If $\Delta=\mathbf C^k$ and all the functions $z_i(\tau)$ are linear, one obtains a complex-analytic plane (cf. [[Analytic plane|Analytic plane]]). If $k=1$, the term which is sometimes employed is holomorphic curve (complex-analytic curve); if all functions $z_i(\tau)$ are linear, one speaks of a complex straight line in the parametric representation:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012440/a01244027.png" /></td> </tr></table>
+
$$z_i=a_i\tau+b_i;\quad a_i,b_i\in\mathbf C,\quad i=1,\dots,n,\quad(a_1,\dots,a_n)\neq0.$$
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  B.V. Shabat,  "Introduction of complex analysis" , '''1–2''' , Moscow  (1976)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.S. Vladimirov,  "Methods of the theory of functions of several complex variables" , M.I.T.  (1966)  pp. Chapt. 2  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  B.V. Shabat,  "Introduction of complex analysis" , '''1–2''' , Moscow  (1976)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.S. Vladimirov,  "Methods of the theory of functions of several complex variables" , M.I.T.  (1966)  pp. Chapt. 2  (Translated from Russian)</TD></TR></table>

Latest revision as of 10:18, 30 December 2018

in a Euclidean space

An arbitrary two-dimensional analytic submanifold $X$ in the space $\mathbf R^n$, $n>2$. However, the term "analytic surface in $\mathbf R^n$" is often employed in a wider sense as a manifold which is (locally) analytically parametrizable. This means that the coordinates of the points $x=(x_1,\dots,x_n)\in X$ can be represented by analytic functions $x_i=x_i(t)$ of a real parameter $t=(t_1,\dots,t_k)$ which varies in a certain range $\Delta\subset\mathbf R^k$, $1\leq k<n$. If the rank of the Jacobi matrix $|\partial x/\partial t|$, which for an analytic manifold is maximal everywhere in $\Delta$, is equal to $k$, then the dimension of the analytic surface $X$ is $k$.

In the complex space $\mathbf C^n$ the term "analytic surface" is also employed to denote a complex-analytic surface $X$ in $\mathbf C^n$, i.e. a manifold which allows a holomorphic (complex-analytic) parametrization. This means that the complex coordinates of points $z=(z_1,\dots,z_n)\in X$ can be expressed by holomorphic functions $z_i=z_i(\tau)$ of a parameter $\tau=(\tau_1,\dots,\tau_k)$ which varies within a certain range $\Delta\subset\mathbf C^k$, $1\leq k<n$ (it is also usually assumed that $\rank\|\partial z/\partial\tau\|\equiv k$). If $\Delta=\mathbf C^k$ and all the functions $z_i(\tau)$ are linear, one obtains a complex-analytic plane (cf. Analytic plane). If $k=1$, the term which is sometimes employed is holomorphic curve (complex-analytic curve); if all functions $z_i(\tau)$ are linear, one speaks of a complex straight line in the parametric representation:

$$z_i=a_i\tau+b_i;\quad a_i,b_i\in\mathbf C,\quad i=1,\dots,n,\quad(a_1,\dots,a_n)\neq0.$$

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. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Analytic_surface&oldid=43566
This article was adapted from an original article by E.D. SolomentsevE.M. Chirka (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article