Vector subbundle - Knowlpedia

Let $E\to M$ be a smooth vector bundle. A vector subbundle of rank $k$ is a subset $F\subseteq E$ such that $F_x=F\cap E_x$ is a $k$ -dimensional linear subspace of every fiber and, near each $x\in M$ , there is a vector-bundle trivialization

E|_U\cong U\times\mathbb F^r

that carries $F|_U$ onto $U\times\mathbb F^k$ for a fixed coordinate subspace $\mathbb F^k\subseteq\mathbb F^r$ . These local models give $F\to M$ a unique smooth vector-bundle structure for which the inclusion $F\hookrightarrow E$ is a smooth embedding and is fiberwise linear.

Local criteria

The following conditions are equivalent for a family of $k$ -dimensional subspaces $F_x\subseteq E_x$ :

$F=\bigcup_xF_x$ is a rank- $k$ vector subbundle;
near every point there are smooth sections $s_1,\ldots,s_k$ of $E$ whose values form a basis of $F_x$ ;
locally, $F$ is the image of a smooth field of projections of constant rank $k$ .

The constant-dimension requirement is essential. A family of linear subspaces whose dimension jumps is generally not a vector bundle.

Kernels and images

Let $\Phi:E\to E'$ be a smooth vector-bundle map over the identity of $M$ . If the fiberwise rank of $\Phi$ is locally constant, then

\ker\Phi=\bigcup_{x\in M}\ker\Phi_x \quad\text{and}\quad \operatorname{im}\Phi=\bigcup_{x\in M}\operatorname{im}\Phi_x

are vector subbundles of $E$ and $E'$ , respectively. Without constant rank, these sets need not be subbundles even though each individual fiber is a linear subspace.

Examples

The tangent bundle of an embedded submanifold $N\subseteq M$ is a subbundle $TN\subseteq TM|_N$ .
A smooth distribution of constant rank is a vector subbundle of $TM$ ; integrability is an additional condition and is not part of the definition.
For a smooth bundle metric, the orthogonal complement $F^\perp$ of a vector subbundle $F\subseteq E$ is another vector subbundle, and $E=F\oplus F^\perp$ .
The vertical and horizontal distributions associated with a bundle and a connection are standard subbundles.

References

Dale Husemoller, Fibre Bundles, 3rd ed., Springer, 1994. DOI record. Relevant: vector bundles and subbundles.
John M. Lee, Introduction to Smooth Manifolds, 2nd ed., Springer, 2013. DOI record. Relevant: vector bundles and constant-rank constructions.