Hilbert manifold
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Contents |
1 Introduction and Definition
Even if one is only interested in finite-dimensional manifolds, the need for infinite-dimensional manifolds sometimes arises. For example, one approach to study closed geodesics on a manifold is to use Morse theory on its (free) loop space; while for some purposes it is enough to work with finite-dimensional approximations, it is necessary for some finer aspects of the theory to use models of the free loop space which are infinite-dimensional manifolds. While there is up to isomorphism only one vector space of every finite dimension, there are many different kinds of infinite-dimensional topological vector spaces one can choose. Modeling spaces on Fréchet spaces gives the notion of Fréchet manifolds, modelling on Banach spaces gives Banach manifolds, modelling on the Hilbert cube (the countably infinite product of intervals) gives Hilbert cube manifolds. We will stick to Hilbert manifolds (which are not directly related to Hilbert cube manifolds).
Definition 1.1. Let be the (up to isomorphism unique) separable Hilbert space of infinite dimension. Then a Hilbert manifold is a separable metrizable space such that every point has a neighborhood that is homeomorphic to an open subset of .
Some authors have slightly different definitions, leaving out the metrizability or the separability condition. Note that metrizability always implies paracompactness and here also the converse is true. Being metrizable and separable is in this context also equivalent to being second countable and Hausdorff by Uryson's metrization theorem (see also [Eells1966, 4(A)]). A -structure can be defined as an equivalence class of atlases whose chart transition maps are of class . The tangent bundle can be defined as usual and is a Hilbert space bundle with structure group with the norm topology (see [Lang1995, II.1 and III.2]).
2 Properties
2.1 Basic Differential Topology
Many basic theorems of differential topology carry over from the finite dimensional situation to the Hilbert (and even Banach) setting with little change. For example, every closed submanifold of a smooth Hilbert manifold has a tubular neighborhood (unique up to isotopy) (see [Lang1995, IV.5-6]). Also, every Hilbert manifold can be embedded as a closed submanifold into the standard Hilbert space ([Kuiper&Terpstra-Keppler1970]). However, in statements involving maps between manifolds, one often has to restrict consideration to \textit{Fredholm maps}, i.e. maps whose differential at every point has closed image and finite-dimensional kernel and cokernel. The reason for this is that Sard's lemma holds for Fredholm maps, but not in general (see [Smale1965] and [Bonic1966]). The precise statement is:
Theorem 2.1. Let be a smooth Fredholm map between Hilbert manifolds. Then its set of regular values is the intersection of countably many sets with dense interior.
2.2 Homotopy Theory
2.3 Specialties of Infinite Dimension
While proofs are often harder in infinite dimensions, some things are true for Hilbert manifolds that could not be hoped for in finite dimensions.
Although Sard's Theorem does not hold in general, we have also the following theorem:
3 Examples
Example 3.3. Mapping spaces between manifolds can often be viewed as Hilbert manifolds if one considers only maps of suitable Sobolev class. Set to be the Sobolev class of -functions which are -fold weakly differentiable in . Let now be an -dimensional compact smooth manifold, be an arbitrary smooth finite-dimensional or Hilbert manifold and be the space of continuous maps with the compact-open topology. Then the subspace of functions of Sobolev type for can be given the structure of a smooth Hilbert manifold ([Eells1966, 6(D)]). This inclusion is a homotopy equivalence ([Eells1966, 6(E)]). In particular, is diffeomorphic to any other Hilbert manifold homotopy equivalent to and therefore its diffeomorphism type depends only on the homotopy type of and . Hilbert manifold models for mapping spaces (in particular, free loop spaces) have been used, for example, in the study of closed geodesics ([Klingenberg1995], [Klingenberg1978]), string topology ([Chataur2005], [Meier2011]) and fluid dynamics ([Ebin&Marsden1970]).
4 References
- [Azagra&Cepedello Boiso2004] D. Azagra and M. Cepedello Boiso, Uniform approximation of continuous mappings by smooth mappings with no critical points on Hilbert manifolds, Duke Math. J. 124 (2004), no.1, 47–66. MR2072211 (2005h:57037) Zbl 1060.57015
- [Bonic1966] R. Bonic, A note on Sard's theorem in Banach spaces, Proc. Amer. Math. Soc. 17 (1966), 1218. MR0198493 (33 #6648) Zbl 0173.16904
- [Chataur2005] D. Chataur, A bordism approach to string topology, Int. Math. Res. Not. (2005), no.46, 2829–2875. MR2180465 (2007b:55009) Zbl 1086.55004
- [Ebin&Marsden1970] D. G. Ebin and J. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid. , Ann. of Math. (2) 92 (1970), 102–163. MR0271984 (42 #6865) Zbl 0286.76001
- [Eells&Elworthy1970] J. Eells and K. D. Elworthy, Open embeddings of certain Banach manifolds, Ann. of Math. (2) 91 (1970), 465–485. MR0263120 (41 #7725) Zbl 0198.28804
- [Eells1966] J. Eells, A setting for global analysis, Bull. Amer. Math. Soc. 72 (1966), 751–807. MR0203742 (34 #3590) Zbl 0191.44101
- [Henderson1970] D. W. Henderson, Infinite-dimensional manifolds are open subsets of Hilbert space, Topology 9 (1970), 25–33. MR0250342 (40 #3581) Zbl 0167.51904
- [Klingenberg1978] W. Klingenberg, Lectures on closed geodesics, Springer-Verlag, 1978. MR0478069 (57 #17563) Zbl 0517.58004
- [Klingenberg1995] W. P. A. Klingenberg, Riemannian geometry, Walter de Gruyter & Co., 1995. MR1330918 (95m:53003) Zbl 1073.53006
- [Kuiper&Terpstra-Keppler1970] N. H. Kuiper and B. Terpstra-Keppler, Differentiable closed embeddings of Banach manifolds, Essays on Topology and Related Topics (Mémoires dédiés à Georges de Rham), Springer (1970), 118–125. MR0264709 (41 #9300) Zbl 0193.24001
- [Kuiper1965] N. H. Kuiper, The homotopy type of the unitary group of Hilbert space, Topology 3 (1965), 19–30. MR0179792 (31 #4034) Zbl 0129.38901
- [Lang1995] S. Lang, Differential and Riemannian manifolds, Springer-Verlag, 1995. MR1335233 (96d:53001) Zbl 0824.58003
- [Meier2011] L. Meier, Spectral sequences in string topology, Algebr. Geom. Topol. 11 (2011), no.5, 2829–2860. MR2846913 () Zbl 1227.55007
- [Smale1965] S. Smale, An infinite dimensional version of Sard's theorem, Amer. J. Math. 87 (1965), 861–866. MR0185604 (32 #3067) Zbl 0143.35301