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.

DefinitionEdit

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  

ExamplesEdit

  • 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 homeomorphismEdit

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 alsoEdit

  • Finsler manifold – smooth manifold equipped with a Minkowski functional at each tangent space
  • Banach bundle – vector bundle whose fibres form Banach spaces
  • Differentiation in Fréchet spaces
  • Fréchet manifold – topological space modeled on a Fréchet space in much the same way as a manifold is modeled on a Euclidean space
  • Hilbert manifold – manifold modeled on Hilbert spaces; separable Hausdorff space in which each point has a neighborhood homeomorphic to an infinite dimensional Hilbert space

ReferencesEdit

  • 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.