Hilbert manifold

In mathematics, a Hilbert manifold is a manifold modeled on Hilbert spaces. Thus it is a separable Hausdorff space in which each point has a neighbourhood homeomorphic to an infinite dimensional Hilbert space. The concept of a Hilbert manifold provides a possibility of extending the theory of manifolds to infinite-dimensional setting. Analogous to the finite-dimensional situation, one can define a differentiable Hilbert manifold by considering a maximal atlas in which the transition maps are differentiable.

Properties

Many basic constructions of manifold theory, such as the tangent space of a manifold and a tubular neighbourhood of a submanifold (of finite codimension) carry over from the finite dimensional situation to the Hilbert setting with little change. However, in statements involving maps between manifolds, one often has to restrict consideration to Fredholm maps, that is, maps whose differential at every point is Fredholm. The reason for this is that Sard's lemma holds for Fredholm maps, but not in general. Notwithstanding this difference, Hilbert manifolds have several very nice properties.

Kuiper's theorem: If $X$ is a compact topological space or has the homotopy type of a CW complex then every (real or complex) Hilbert space bundle over $X$ is trivial. In particular, every Hilbert manifold is parallelizable.
Every smooth Hilbert manifold can be smoothly embedded onto an open subset of the model Hilbert space.
Every homotopy equivalence between two Hilbert manifolds is homotopic to a diffeomorphism. In particular every two homotopy equivalent Hilbert manifolds are already diffeomorphic. This stands in contrast to lens spaces and exotic spheres, which demonstrate that in the finite-dimensional situation, homotopy equivalence, homeomorphism, and diffeomorphism of manifolds are distinct properties.
Although Sard's Theorem does not hold in general, every continuous map $f:X\to \mathbb {R} ^{n}$ from a Hilbert manifold can be arbitrary closely approximated by a smooth map $g:X\to \mathbb {R} ^{n}$ which has no critical points.

Examples

Any Hilbert space $H$ is a Hilbert manifold with a single global chart given by the identity function on $H.$ Moreover, since $H$ is a vector space, the tangent space $\operatorname {T} _{p}H$ to $H$ at any point $p\in H$ is canonically isomorphic to $H$ itself, and so has a natural inner product, the "same" as the one on $H.$ Thus $H$ can be given the structure of a Riemannian manifold with metric $g(v,w)(p):=\langle v,w\rangle _{H}{\text{ for }}v,w\in \mathrm {T} _{p}H,$ where $\langle \,\cdot ,\cdot \,\rangle _{H}$ denotes the inner product in $H.$
Similarly, any open subset of a Hilbert space is a Hilbert manifold and a Riemannian manifold under the same construction as for the whole space.
There are several mapping spaces between manifolds which can be viewed as Hilbert spaces by only considering maps of suitable Sobolev class. For example we can consider the space $\operatorname {L} M$ of all $H^{1}$ maps from the unit circle $\mathbf {S} ^{1}$ into a manifold $M.$ This can be topologized via the compact open topology as a subspace of the space of all continuous mappings from the circle to $M,$ that is, the free loop space of $M.$ The Sobolev kind mapping space $\operatorname {L} M$ described above is homotopy equivalent to the free loop space. This makes it suited to the study of algebraic topology of the free loop space, especially in the field of string topology. We can do an analogous Sobolev construction for the loop space, making it a codimension $d$ Hilbert submanifold of $\operatorname {L} M,$ where $d$ is the dimension of $M.$

References

Klingenberg, Wilhelm (1982), Riemannian Geometry, Berlin: W. de Gruyter, ISBN 978-3-11-008673-7. Contains a general introduction to Hilbert manifolds and many details about the free loop space.
Lang, Serge (1995), Differential and Riemannian Manifolds, New York: Springer, ISBN 978-0387943381. Another introduction with more differential topology.
N. Kuiper, The homotopy type of the unitary group of Hilbert spaces", Topology 3, 19-30
J. Eells, K. D. Elworthy, "On the differential topology of Hilbert manifolds", Global analysis. Proceedings of Symposia in Pure Mathematics, Volume XV 1970, 41-44.
J. Eells, K. D. Elworthy, "Open embeddings of certain Banach manifolds", Annals of Mathematics 91 (1970), 465-485
D. Chataur, "A Bordism Approach to String Topology", preprint https://arxiv.org/abs/math.at/0306080

External links

Hilbert manifold at the Manifold Atlas

v t e Riemannian geometry (Glossary)
Basic concepts	Curvature tensor Scalar Ricci Sectional Exponential map Geodesic Inner product Metric tensor Levi-Civita connection Covariant derivative Signature Raising and lowering indices/Musical isomorphism Parallel transport Riemannian manifold Pseudo-Riemannian manifold Riemannian volume form
Types of manifolds	Hermitian Hyperbolic Kähler Kenmotsu
Main results	Fundamental theorem of Riemannian geometry Gauss's lemma Gauss–Bonnet theorem Hopf–Rinow theorem Nash embedding theorem Ricci flow Schur's lemma
Generalizations	Finsler Hilbert Sub-Riemannian
Applications	General relativity Geometrization conjecture Poincaré conjecture Uniformization theorem

v t e Analysis in topological vector spaces
Basic concepts	Abstract Wiener space Classical Wiener space Bochner space Convex series Cylinder set measure Infinite-dimensional vector function Matrix calculus Vector calculus
Derivatives	Differentiable vector–valued functions from Euclidean space Differentiation in Fréchet spaces Fréchet derivative Total Functional derivative Gateaux derivative Directional Generalizations of the derivative Hadamard derivative Holomorphic Quasi-derivative
Measurability	Besov measure Cylinder set measure Canonical Gaussian Classical Wiener measure Measure like set functions infinite-dimensional Gaussian measure Projection-valued Vector Bochner / Weakly / Strongly measurable function Radonifying function
Integrals	Bochner Direct integral Dunford Gelfand–Pettis/Weak Regulated Paley–Wiener
Results	Cameron–Martin theorem Inverse function theorem Nash–Moser theorem Feldman–Hájek theorem No infinite-dimensional Lebesgue measure Sazonov's theorem Structure theorem for Gaussian measures
Related	Crinkled arc Covariance operator
Functional calculus	Borel functional calculus Continuous functional calculus Holomorphic functional calculus
Applications	Banach manifold (bundle) Convenient vector space Choquet theory Fréchet manifold Hilbert manifold

v t e Hilbert spaces
Basic concepts	Adjoint Inner product and L-semi-inner product Hilbert space and Prehilbert space Orthogonal complement Orthonormal basis
Main results	Bessel's inequality Cauchy–Schwarz inequality Riesz representation
Other results	Hilbert projection theorem Parseval's identity Polarization identity (Parallelogram law)
Maps	Compact operator on Hilbert space Densely defined Hermitian form Hilbert–Schmidt Normal Self-adjoint Sesquilinear form Trace class Unitary
Examples	Cⁿ(K) with K compact & n<∞ Segal–Bargmann F

Properties

Examples

See also

References

External links