# Fréchet surface

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]).