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 ${\displaystyle X}$ with an atlas of coordinate charts over Fréchet spaces whose transitions are smooth mappings. Thus ${\displaystyle X}$ has an open cover ${\displaystyle \left\{U_{\alpha }\right\}_{\alpha \in I},}$ and a collection of homeomorphisms ${\displaystyle \phi _{\alpha }:U_{\alpha }\to F_{\alpha }}$ onto their images, where ${\displaystyle F_{\alpha }}$ are Fréchet spaces, such that

$${\displaystyle \phi _{\alpha \beta }:=\phi _{\alpha }\circ \phi _{\beta }^{-1}|_{\phi _{\beta }\left(U_{\beta }\cap U_{\alpha }\right)}}$$

is smooth for all pairs of indices ${\displaystyle \alpha ,\beta .}$

Classification up to homeomorphism

It is by no means true that a finite-dimensional manifold of dimension ${\displaystyle n}$  is globally homeomorphic to ${\displaystyle \mathbb {R} ^{n}}$  or even an open subset of ${\displaystyle \mathbb {R} ^{n}.}$  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 ${\displaystyle X}$  can be embedded as an open subset of the infinite-dimensional, separable Hilbert space, ${\displaystyle H}$  (up to linear isomorphism, there is only one such space).

The embedding homeomorphism can be used as a global chart for ${\displaystyle X.}$  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^{[citation needed]}.

See also

• Banach manifold – Manifold modeled on Banach spaces, of which a Fréchet manifold is a generalization
• Manifolds of mappings – locally convex vector spaces satisfying a very mild completeness conditionPages displaying wikidata descriptions as a fallback
• Differentiation in Fréchet spaces
• Hilbert manifold – Manifold modelled on Hilbert spaces

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