Fréchet manifold

In mathematics, in particular in nonlinear analysis, a Fréchet manifold is a topological space modeled on a Fréchet space in much the same way as a manifold is modeled on a Euclidean space.

More precisely, a Fréchet manifold consists of a Hausdorff space X with an atlas of coordinate charts over Fréchet spaces whose transitions are smooth mappings. Thus X has an open cover {Uα}α ε I, and a collection of homeomorphisms φα : UαFα onto their images, where Fα are Fréchet spaces, such that

is smooth for all pairs of indices α, β.

Classification up to homeomorphismEdit

It is by no means true that a finite-dimensional manifold of dimension n is globally homeomorphic to Rn, or even an open subset of Rn. However, in an infinite-dimensional setting, it is possible to classify “well-behaved” Fréchet manifolds up to homeomorphism quite nicely. A 1969 theorem of David Henderson states that every infinite-dimensional, separable, metric Fréchet manifold X can be embedded as an open subset of the infinite-dimensional, separable Hilbert space, H (up to linear isomorphism, there is only one such space).

The embedding homeomorphism can be used as a global chart for X. Thus, in the infinite-dimensional, separable, metric case, up to homeomorphism, the “only” Fréchet manifolds are the open subsets of Hilbert space.

See alsoEdit

ReferencesEdit

  • Hamilton, Richard S. (1982). "The inverse function theorem of Nash and Moser". Bull. Amer. Math. Soc. (N.S.). 7 (1): 65–222. doi:10.1090/S0273-0979-1982-15004-2. ISSN 0273-0979. MR656198
  • Henderson, David W. (1969). "Infinite-dimensional manifolds are open subsets of Hilbert space". Bull. Amer. Math. Soc. 75 (4): 759–762. doi:10.1090/S0002-9904-1969-12276-7. MR0247634