Banach manifold

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 ${\displaystyle X}$ be a set. An atlas of class ${\displaystyle C^r,}$ ${\displaystyle r\ge 0,}$ on ${\displaystyle X}$ is a collection of pairs (called charts) ${\displaystyle \left(U_i,{\phi }_i\right),}$ ${\displaystyle i\in I,}$ such that

1. each ${\displaystyle U_i}$ is a subset of ${\displaystyle X}$ and the union of the ${\displaystyle U_i}$ is the whole of ${\displaystyle X}$;
2. each ${\displaystyle {\phi }_i}$ is a bijection from ${\displaystyle U_i}$ onto an open subset ${\displaystyle {\phi }_i\left(U_i\right)}$ of some Banach space ${\displaystyle E_i,}$ and for any indices ${\displaystyle i\text{ and }j,}$ ${\displaystyle {\phi }_i\left(U_i\cap U_j\right)}$ is open in ${\displaystyle E_i;}$
3. the crossover map $${\displaystyle {\phi }_j\circ {\phi }_i^{-1}:{\phi }_i\left(U_i\cap U_j\right)\to {\phi }_j\left(U_i\cap U_j\right)}$$ is an ${\displaystyle r}$-times continuously differentiable function for every ${\displaystyle i,j\in I;}$ that is, the ${\displaystyle r}$th Fréchet derivative $${\displaystyle {\mathrm{d}}^r\left({\phi }_j\circ {\phi }_i^{-1}\right):{\phi }_i\left(U_i\cap U_j\right)\to \mathrm{L}\mathrm{i}\mathrm{n}\left(E_i^r;E_j\right)}$$ exists and is a continuous function with respect to the ${\displaystyle E_i}$-norm topology on subsets of ${\displaystyle E_i}$ and the operator norm topology on ${\displaystyle \mathrm{Lin}\left(E_i^r;E_j\right).}$

One can then show that there is a unique topology on ${\displaystyle X}$ such that each ${\displaystyle U_i}$ is open and each ${\displaystyle {\phi }_i}$ 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 ${\displaystyle E_i}$ are equal to the same space ${\displaystyle E,}$ the atlas is called an ${\displaystyle E}$-atlas. However, it is not a priori necessary that the Banach spaces ${\displaystyle E_i}$ be the same space, or even isomorphic as topological vector spaces. However, if two charts ${\displaystyle \left(U_i,{\phi }_i\right)}$ and ${\displaystyle \left(U_j,{\phi }_j\right)}$ are such that ${\displaystyle U_i}$ and ${\displaystyle U_j}$ have a non-empty intersection, a quick examination of the derivative of the crossover map $${\displaystyle {\phi }_j\circ {\phi }_i^{-1}:{\phi }_i\left(U_i\cap U_j\right)\to {\phi }_j\left(U_i\cap U_j\right)}$$ shows that ${\displaystyle E_i}$ and ${\displaystyle E_j}$ must indeed be isomorphic as topological vector spaces. Furthermore, the set of points ${\displaystyle x\in X}$ for which there is a chart ${\displaystyle \left(U_i,{\phi }_i\right)}$ with ${\displaystyle x}$ in ${\displaystyle U_i}$ and ${\displaystyle E_i}$ isomorphic to a given Banach space ${\displaystyle E}$ is both open and closed. Hence, one can without loss of generality assume that, on each connected component of ${\displaystyle X,}$ the atlas is an ${\displaystyle E}$-atlas for some fixed ${\displaystyle E.}$

A new chart ${\displaystyle (U,\phi )}$ is called compatible with a given atlas ${\displaystyle \left\{\left(U_i,{\phi }_i\right):i\in I\right\}}$ if the crossover map $${\displaystyle {\phi }_i\circ {\phi }^{-1}:\phi \left(U\cap U_i\right)\to {\phi }_i\left(U\cap U_i\right)}$$ is an ${\displaystyle r}$-times continuously differentiable function for every ${\displaystyle i\in I.}$ 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 ${\displaystyle X.}$

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