Split Banach submanifold

Let $k\ge 1$, let $M$ be a $C^k$ Banach manifold modelled on the real Banach space $E$ (Countable base Banach manifold and smooth map), and let $S\subseteq M$ be a subset. Then $S$ is a split $C^k$ submanifold of $M$ when for every $p\in S$ there are

• a chart $(\phi ,U)$ of $M$ with $p\in U$, and
• a decomposition $E=E_0\oplus E_1$ of the model space into two closed subspaces with bounded coordinate projections (A complemented closed subspace of a normed space, Linear subspace of a vector space),

such that

$\phi [U\cap S]=\phi [U]\cap (E_0\oplus \{0\})\mathrm{.}$

In other words, in the chart the submanifold is exactly the slice of the open set $\phi [U]$ cut out by setting the $E_1$-coordinate equal to zero. The subspace topology on $S$ (Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace) carries the resulting componentwise $C^k$ structure: the charts $(\phi {\mathrm{\mid }}_{U\cap S},U\cap S)$ take values in open subsets of the local space $E_0$, and their transition maps are restrictions of the $C^k$ transition maps of $M$. On an overlap, the derivative of such a transition map is a bounded linear isomorphism between the two local model spaces. Hence the isomorphism type of $E_0$ is locally constant on $S$, and every connected component of $S$ is a $C^k$ Banach manifold modelled on one fixed representative of that type. Different components need not have isomorphic model spaces; without an additional uniform-model hypothesis, $S$ as a whole need not be modelled on one Banach space in the global convention of Countable base Banach manifold and smooth map.

For $p\in S$ the tangent space $T_{p}S$ is the tangent space of the component of $S$ containing $p$, and the differential of the inclusion identifies it with a subspace of $T_{p}M$.

• Splitness is a local condition, and the complement is part of the local data. The definition does not assert that an arbitrary closed subspace of a Banach space is a submanifold of it: the complement $E_1$ is produced along with the chart, and the companion page exhibits a closed subspace $c_0$ of ${\mathrm{\ell }}^{\mathrm{\infty }}$ that is not complemented in it. For a closed subspace $E_0\le E$ with $E$ second countable, the split charts with $M=E$ and $\phi =\mathrm{i}\mathrm{d}$ exist exactly when $E_0$ is complemented in $E$; for $E={\mathrm{\ell }}^{\mathrm{\infty }}$ the ambient is not a Banach manifold in the library's sense, so the example separates closedness from complementedness at the level of Banach spaces rather than exhibiting a split-submanifold failure for a Banach manifold.

• The local model can vary between components. For example, $\{0\}\cup (1,2)\subseteq \mathbb{R}$ satisfies the slice condition, with local model $\{0\}$ at the isolated point and $\mathbb{R}$ on the interval. Thus it is split in the componentwise sense above, but it is not modelled on one fixed Banach space.

• The tangent space of a split submanifold is complemented. In a chart at $p$ the tangent space of $S$ corresponds to $E_0$ and that of $M$ to $E$, so the inclusion $T_{p}S\to T_{p}M$ is, in that chart, the inclusion of the complemented subspace $E_0$ into $E$. This is why the regular value theorem below demands a complemented kernel rather than mere surjectivity of the derivative.

• Automatic cases. If $E_0$ is finite dimensional or of finite codimension in $E$, then every closed subspace of that kind is complemented, so the only obstruction to splitness in these cases is the local product structure of $S$ itself (Finite-dimensional subspaces are complemented). In particular finite-dimensional level sets of submersions are automatically split when the derivative is surjective.