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Fréchet manifold

From Wikipedia, the free encyclopedia

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 with an atlas of coordinate charts over Fréchet spaces whose transitions are smooth mappings. Thus has an open cover and a collection of homeomorphisms onto their images, where are Fréchet spaces, such that

is smooth for all pairs of indices

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Transcription

Classification up to homeomorphism

It is by no means true that a finite-dimensional manifold of dimension is globally homeomorphic to or even an open subset of 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 can be embedded as an open subset of the infinite-dimensional, separable Hilbert space, (up to linear isomorphism, there is only one such space).

The embedding homeomorphism can be used as a global chart for 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

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
This page was last edited on 10 July 2021, at 01:48
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