Banach manifold

Summary

In mathematics, a Banach manifold is a manifold modeled on Banach spaces. Thus it is a topological space in which each point has a neighbourhood homeomorphic to an open set in a Banach space (a more involved and formal definition is given below). Banach manifolds are one possibility of extending manifolds to infinite dimensions.

A further generalisation is to Fréchet manifolds, replacing Banach spaces by Fréchet spaces. On the other hand, a Hilbert manifold is a special case of a Banach manifold in which the manifold is locally modeled on Hilbert spaces.

Definition

Let be a set. An atlas of class on is a collection of pairs (called charts) such that

  1. each is a subset of and the union of the is the whole of ;
  2. each is a bijection from onto an open subset of some Banach space and for any indices is open in
  3. the crossover map
    is an -times continuously differentiable function for every that is, the th Fréchet derivative
    exists and is a continuous function with respect to the -norm topology on subsets of and the operator norm topology on

One can then show that there is a unique topology on such that each is open and each is a homeomorphism. Very often, this topological space is assumed to be a Hausdorff space, but this is not necessary from the point of view of the formal definition.

If all the Banach spaces are equal to the same space the atlas is called an -atlas. However, it is not a priori necessary that the Banach spaces be the same space, or even isomorphic as topological vector spaces. However, if two charts and are such that and have a non-empty intersection, a quick examination of the derivative of the crossover map

shows that and must indeed be isomorphic as topological vector spaces. Furthermore, the set of points for which there is a chart with in and isomorphic to a given Banach space is both open and closed. Hence, one can without loss of generality assume that, on each connected component of the atlas is an -atlas for some fixed

A new chart is called compatible with a given atlas if the crossover map

is an -times continuously differentiable function for every Two atlases are called compatible if every chart in one is compatible with the other atlas. Compatibility defines an equivalence relation on the class of all possible atlases on

A -manifold structure on is then defined to be a choice of equivalence class of atlases on of class If all the Banach spaces are isomorphic as topological vector spaces (which is guaranteed to be the case if is connected), then an equivalent atlas can be found for which they are all equal to some Banach space is then called an -manifold, or one says that is modeled on

Examples

  • If is a Banach space, then is a Banach manifold with an atlas containing a single, globally-defined chart (the identity map).
  • Similarly, if is an open subset of some Banach space then is a Banach manifold. (See the classification theorem below.)

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" Banach manifolds up to homeomorphism quite nicely. A 1969 theorem of David Henderson states that every infinite-dimensional, separable, metric Banach 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, usually identified with ). In fact, Henderson's result is stronger: the same conclusion holds for any metric manifold modeled on a separable infinite-dimensional Fréchet space.

The embedding homeomorphism can be used as a global chart for Thus, in the infinite-dimensional, separable, metric case, the "only" Banach manifolds are the open subsets of Hilbert space.

See also

References

  • 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. MR 0247634.
  • Lang, Serge (1972). Differential manifolds. Reading, Mass.–London–Don Mills, Ont.: Addison-Wesley Publishing Co., Inc.
  • Zeidler, Eberhard (1997). Nonlinear functional analysis and its Applications. Vol.4. Springer-Verlag New York Inc.
  • Abraham, Ralph; Marsden, J. E.; Ratiu, Tudor (1988). Manifolds, Tensor Analysis, and Applications. New York: Springer. ISBN 0-387-96790-7.