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 ${\left\{U_{\alpha }\right\}}_{\alpha \in I},$ and a collection of homeomorphisms ${\varphi }_{\alpha }:U_{\alpha }\to F_{\alpha }$ onto their images, where $F_{\alpha }$ are Fréchet spaces, such that $${\varphi }_{\alpha \beta }:={\varphi }_{\alpha }\circ {\varphi }_{\beta }^{-1}{|}_{{\varphi }_{\beta }(U_{\beta }\cap U_{\alpha })}$$ is smooth for all pairs of indices $\alpha ,\beta .$

Classification up to homeomorphism

It is by no means true that a finite-dimensional manifold of dimension $n$ is globally homeomorphic to ${\mathbb{ℝ}}^n$ or even an open subset of ${\mathbb{ℝ}}^n.$ 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 $X$ 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 – Manifold modeled on Banach spaces, of which a Fréchet manifold is a generalization
• 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