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 [math]\displaystyle{ X }[/math] with an atlas of coordinate charts over Fréchet spaces whose transitions are smooth mappings. Thus [math]\displaystyle{ X }[/math] has an open cover [math]\displaystyle{ \left\{ U_{\alpha} \right\}_{\alpha \in I}, }[/math] and a collection of homeomorphisms [math]\displaystyle{ \phi_{\alpha} : U_{\alpha} \to F_{\alpha} }[/math] onto their images, where [math]\displaystyle{ F_{\alpha} }[/math] are Fréchet spaces, such that [math]\displaystyle{ \phi_{\alpha\beta} := \phi_\alpha \circ \phi_\beta^{-1}|_{\phi_\beta\left(U_\beta\cap U_\alpha\right)} }[/math] is smooth for all pairs of indices [math]\displaystyle{ \alpha, \beta. }[/math]

Classification up to homeomorphism

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

The embedding homeomorphism can be used as a global chart for [math]\displaystyle{ X. }[/math] 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