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Fréchet surface

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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 $$ \rho(f_1,f_2) \equiv \inf_\sigma \max_{m \in M^2} d(f_1(x),f_2(\sigma(x))) = 0 $$ 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://www.encyclopediaofmath.org/index.php?title=Fr%C3%A9chet_surface&oldid=40064
This article was adapted from an original article by V.A. Zalgaller (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article