Hilbert Manifold - Examples

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 T_pH to H at any point p ∈ 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

where 〈·, ·〉_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 LM of all H1 maps from the unit circle S1 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, i.e. the free loop space of M. The Sobolev kind mapping space LM 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 LM, where d is the dimension of M.