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Concept
Fréchet surface
Summary
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.
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
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.
Official source
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