Fréchet surface explained
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 ifAn 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
- Fréchet, M.. Maurice Fréchet. Sur quelques points du calcul fonctionnel. Rend. Circolo Mat. Palermo. 22. 1906. 1–72. 10.1007/BF03018603. 10338.dmlcz/100655. free.