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A generalization of the concept of a surface in a Euclidean space to the case of an arbitrary metric space . Let
be a compact two-dimensional manifold (either closed or with a boundary). The points of
play the role of parameter. Continuous mappings
are called parametrized surfaces. Two parametrized surfaces are regarded as equivalent if
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where is the distance in
and the
are all possible homeomorphisms of
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
is independent of the choice of the parametrizations
and
; it is called the Fréchet distance between the Fréchet surfaces. If one changes the domain
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 |
Fréchet surface. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fr%C3%A9chet_surface&oldid=11702