 Let

where the infimum is taken over all homeomorphisms "σ" of "M" to itself.Call two parametrized surfaces "f" and "g" in "X" equivalent if and only if

An equivalence class ["f"] 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 ["f"].Many properties of parametrized surfaces are actually properties of the Fréchet surface, i.e.