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 {U_α}_{α ε I}, and a collection of homeomorphisms φ_α : U_α → F_α onto their images, where F_α are Fréchet spaces, such that

$\phi_{\alpha\beta} := \phi_\alpha \circ \phi_\beta^{-1}|_{\phi_\beta(U_\beta\cap U_\alpha)}$ is smooth for all pairs of indices α, β.

Classification up to homeomorphism

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 also

• Banach manifold, of which a Fréchet manifold is a generalization
• Manifolds of mappings

References

• 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. MR 656198
• 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. MR 0247634