Banach manifold

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 modelled on Hilbert spaces.

Definition

Let "X" be a set. An atlas of class "C"^"r", "r" ≥ 0, on "X" is a collection of pairs (called charts) ("U"_"i", "φ"_"i"), "i" ∈ "I", such that
# each "U"_"i" is a subset of "X" and the union of the "U"_"i" is the whole of "X";
# each "φ"_"i" is a bijection from "U"_"i" onto an open subset "φ"_"i"("U"_"i") of some Banach space "E"_"i", and for any "i" and "j", "φ"_"i"("U"_"i" ∩ "U"_"j") is open in "E"_"i";
# the crossover map

::$varphi\_\{j\}\; circ\; varphi\_\{i\}^\{-1\}\; :\; varphi\_\{i\}\; (U\_\{i\}\; cap\; U\_\{j\})\; o\; varphi\_\{j\}\; (U\_\{i\}\; cap\; U\_\{j\})$

: is an "r"-times continuously differentiable function for every "i" and "j" in "I", i.e. the "r"^th Fréchet derivative

::

: exists and is a continuous function with respect to the "E"_"i"-norm topology on subsets of "E"_"i" and the operator norm topology on Lin("E"_"i"^"r"; "E"_"j".)

One can then show that there is a unique topology on "X" such that each "U"_"i" is open and each "φ"_"i" 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 "E"_"i" are equal to the same space "E", the atlas is called an "E"-atlas. However, it is not "a priori" necessary that the Banach spaces "E"_"i" be the same space, or even isomorphic as topological vector spaces. However, if two charts ("U"_"i", "φ"_"i") and ("U"_"j", "φ"_"j") are such that "U"_"i" and "U"_"j" have a non-empty intersection, a quick examination of the derivative of the crossover map

:$varphi\_\{j\}\; circ\; varphi\_\{i\}^\{-1\}\; :\; varphi\_\{i\}\; (U\_\{i\}\; cap\; U\_\{j\})\; o\; varphi\_\{j\}\; (U\_\{i\}\; cap\; U\_\{j\})$

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

A new chart ("U", "φ") is called compatible with a given atlas { ("U"_"i", "φ"_"i") | "i" ∈ "I" } if the crossover map

:$varphi\_\{i\}\; circ\; varphi^\{-1\}\; :\; varphi\; (U\; cap\; U\_\{i\})\; o\; varphi\_\{i\}\; (U\; cap\; U\_\{i\})$

is an "r"-times continuously differentiable function for every "i" ∈ "I". 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 "X".

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

Examples

* If ("X", || ||) is a Banach space, then "X" is a Banach manifold with an atlas containing a single, globally-defined chart (the identity map).
* Similarly, if "U" is an open subset of some Banach space, then "U" 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 "n" is "globally" homeomorphic to R^"n", or even an open subset of R^"n". 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 "X" can be embedded as an open subset of the infinite-dimensional, separable Hilbert space, "H" (up to linear isomorphism, there is only one such space). 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 "X". Thus, in the infinite-dimensional, separable, metric case, the “only” Banach manifolds are the open subsets of Hilbert space.

References

* cite journal
last = Henderson
first = David W.
title = Infinite-dimensional manifolds are open subsets of Hilbert space
journal = Bull. Amer. Math. Soc.
volume = 75
year = 1969
pages = 759–762
doi = 10.1090/S0002-9904-1969-12276-7 MathSciNet|id=0247634
* cite book
last = Lang
first = Serge
authorlink = Serge Lang
title = Differential manifolds
publisher = Addison-Wesley Publishing Co., Inc.
location = Reading, Mass.–London–Don Mills, Ont.
year = 1972