# Fréchet surface

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

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 |

**How to Cite This Entry:**

Fréchet surface.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Fr%C3%A9chet_surface&oldid=23286