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 : Uα → Fα onto their images, where Fα are Fréchet spaces, such that
- 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” topological Fréchet manifolds are the open subsets of the separable infinite-dimensional Hilbert space. But in the case of differentiable or smooth Fréchet manifolds (up to the appropriate notion of diffeomorphism) this fails.
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. 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