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A generalization of the concept of a surface in a Euclidean space to the case of an arbitrary metric space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041390/f0413901.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041390/f0413902.png" /> be a compact two-dimensional manifold (either closed or with a boundary). The points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041390/f0413903.png" /> play the role of parameter. Continuous mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041390/f0413904.png" /> are called parametrized surfaces. Two parametrized surfaces are regarded as equivalent if
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A generalization of the concept of a surface in a Euclidean space to the case of an arbitrary metric space $A$. Let $M^2$ be a compact two-dimensional manifold (either closed or with a boundary). The points of $M^2$ play the role of parameter. Continuous mappings $f : M^2 \rightarrow A$ are called parametrized surfaces. Two parametrized surfaces are regarded as equivalent if
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$$
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\rho(f_1,f_2) \equiv \inf_\sigma \max_{m \in M^2} d(f_1(x),f_2(\sigma(x))) = 0
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$$
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where $d$ is the distance in $A$ and the $\sigma$ are all possible homeomorphisms of $M^2$ onto itself. A class of equivalent parametrized surfaces is called a Fréchet surface (see [[#References|[1]]]), and each of the parametrized surfaces in this class is called a parametrization of the Fréchet surface. Many properties of parametrized surfaces are properties of the Fréchet surface, and not of its concrete parametrization. For two Fréchet surfaces, the value of $\rho(f_1,f_2)$ is independent of the choice of the parametrizations $f_1$ and $f_2$; it is called the Fréchet distance between the Fréchet surfaces. If one changes the domain $M^2$ of the parameter in the definition of a Fréchet surface to a circle or a closed interval, one obtains the definition of a Fréchet curve (see [[#References|[2]]]).
  
<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/f/f041/f041390/f0413905.png" /></td> </tr></table>
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====References====
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  M. Fréchet,  ''Ann. Soc. Polon. Math.'' , '''3'''  (1924)  pp. 4–19</TD></TR>
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<TR><TD valign="top">[2]</TD> <TD valign="top">  M. Fréchet,  "Sur quelques points du calcul fonctionnel" ''Rend. Circolo Mat. Palermo'' , '''74'''  (1906)  pp. 1–74</TD></TR>
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</table>
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041390/f0413906.png" /> is the distance in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041390/f0413907.png" /> and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041390/f0413908.png" /> are all possible homeomorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041390/f0413909.png" /> onto itself. A class of equivalent parametrized surfaces is called a Fréchet surface (see [[#References|[1]]]), and each of the parametrized surfaces in this class is called a parametrization of the Fréchet surface. Many properties of parametrized surfaces are properties of the Fréchet surface, and not of its concrete parametrization. For two Fréchet surfaces, the value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041390/f04139010.png" /> is independent of the choice of the parametrizations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041390/f04139011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041390/f04139012.png" />; it is called the Fréchet distance between the Fréchet surfaces. If one changes the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041390/f04139013.png" /> of the parameter in the definition of a Fréchet surface to a circle or a closed interval, one obtains the definition of a Fréchet curve (see [[#References|[2]]]).
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====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M. Fréchet,  ''Ann. Soc. Polon. Math.'' , '''3'''  (1924)  pp. 4–19</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M. Fréchet,  "Sur quelques points du calcul fonctionnel"  ''Rend. Circolo Mat. Palermo'' , '''74'''  (1906)  pp. 1–74</TD></TR></table>
 

Latest revision as of 20:15, 20 December 2016

A generalization of the concept of a surface in a Euclidean space to the case of an arbitrary metric space $A$. Let $M^2$ be a compact two-dimensional manifold (either closed or with a boundary). The points of $M^2$ play the role of parameter. Continuous mappings $f : M^2 \rightarrow A$ are called parametrized surfaces. Two parametrized surfaces are regarded as equivalent if $$ \rho(f_1,f_2) \equiv \inf_\sigma \max_{m \in M^2} d(f_1(x),f_2(\sigma(x))) = 0 $$ where $d$ is the distance in $A$ and the $\sigma$ are all possible homeomorphisms of $M^2$ onto itself. A class of equivalent parametrized surfaces is called a Fréchet surface (see [1]), and each of the parametrized surfaces in this class is called a parametrization of the Fréchet surface. Many properties of parametrized surfaces are properties of the Fréchet surface, and not of its concrete parametrization. For two Fréchet surfaces, the value of $\rho(f_1,f_2)$ is independent of the choice of the parametrizations $f_1$ and $f_2$; it is called the Fréchet distance between the Fréchet surfaces. If one changes the domain $M^2$ of the parameter in the definition of a Fréchet surface to a circle or a closed interval, one obtains the definition of a Fréchet curve (see [2]).

References

[1] M. Fréchet, Ann. Soc. Polon. Math. , 3 (1924) pp. 4–19
[2] M. Fréchet, "Sur quelques points du calcul fonctionnel" Rend. Circolo Mat. Palermo , 74 (1906) pp. 1–74
How to Cite This Entry:
Fréchet surface. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fr%C3%A9chet_surface&oldid=22461
This article was adapted from an original article by V.A. Zalgaller (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article