Banach manifold

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

1. each U_i is a subset of X and the union of the U_i is the whole of X;
2. 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;
3. the crossover map

is an r-times continuously differentiable function for every i and j in I, i.e. the rth 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

φ j ∘ φ i − 1 : φ i ( U i ∩ U j ) → φ j ( U i ∩ 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

φ i ∘ φ − 1 : φ ( U ∩ U i ) → φ i ( U ∩ 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 Rⁿ, or even an open subset of Rⁿ. 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.