Fréchet surface
In mathematics, a Fréchet surface is an equivalence class of parametrized surfaces in a metric space. In other words, a Fréchet surface is a way of thinking about surfaces independently of how they are "written down" (parametrized). The concept is named after the French mathematician Maurice Fréchet.

Definitions

Let  be a compact 2-dimensional manifold, either closed or with boundary, and let  be a metric space. A parametrized surface in  is a map

that is continuous with respect to the topology on  and the metric topology on  Let

where the infimum is taken over all homeomorphisms  of  to itself. Call two parametrized surfaces  and  in  equivalent if and only if

An equivalence class  of parametrized surfaces under this notion of equivalence is called a Fréchet surface; each of the parametrized surfaces in this equivalence class is called a parametrization of the Fréchet surface

Properties

Many properties of parametrized surfaces are actually properties of the Fréchet surface, that is, of the whole equivalence class, and not of any particular parametrization.

For example, given two Fréchet surfaces, the value of  is independent of the choice of the parametrizations  and  and is called the Fréchet distance between the Fréchet surfaces.

References

 
 

Metric geometry