Kernel and image bundles of a constant-rank morphism

Definition

Let $\Phi:E\to F$ be a smooth vector bundle morphism over the identity of a manifold $M$ , and suppose the fiber maps $\Phi_x:E_x\to F_x$ have locally constant rank. Its kernel bundle and image bundle are

\ker\Phi=\coprod_{x\in M}\ker\Phi_x\subseteq E, \qquad \operatorname{im}\Phi=\coprod_{x\in M}\operatorname{im}\Phi_x\subseteq F.

The constant-rank bundle-map theorem gives these sets unique smooth vector subbundle structures. If $E$ and $F$ have ranks $e$ and $f$ , and $\Phi$ has rank $r$ on a connected component, then their ranks are $e-r$ and $r$ , respectively.

Why constant rank is sufficient

In local frames, $\Phi$ is represented by a smooth matrix-valued function. A nonvanishing $r\times r$ minor can be used, after smooth changes of local frame, to put that matrix into block form

\begin{pmatrix}I_r&0\\0&0\end{pmatrix}.

The coordinate spans of the zero block and the $I_r$ block then give smooth local frames for the kernel and image. This is the vector-bundle analogue of the constant-rank normal form; see Lee, Chapter 10.

Exact sequences and induced isomorphism

The inclusion and projection associated with $\Phi$ yield exact sequences

0\longrightarrow\ker\Phi\longrightarrow E\longrightarrow\operatorname{im}\Phi\longrightarrow0

and

0\longrightarrow\operatorname{im}\Phi\longrightarrow F\longrightarrow F/\operatorname{im}\Phi\longrightarrow0.

Fiberwise, the first isomorphism theorem induces $E/\ker\Phi\cong\operatorname{im}\Phi$ ; the local block form shows that this is an isomorphism of smooth vector bundles, not merely a collection of vector-space isomorphisms.

Failure when rank jumps

The constant-rank hypothesis cannot be omitted. Multiplication by $x$ defines a morphism of trivial line bundles over $\mathbb R$ ,

(x,v)\longmapsto(x,xv).

Its kernel is zero away from $0$ and one-dimensional over $0$ , while its image has the opposite rank jump. Neither family is a vector bundle over $\mathbb R$ .

References

J. M. Lee, Introduction to Smooth Manifolds, 2nd ed., Springer, 2012. DOI record. Relevant: Chapter 10, bundle homomorphisms and the vector-bundle rank theorem.
D. Husemoller, Fibre Bundles, 3rd ed., Springer, 1994. DOI record. Relevant: Chapter 3, vector-bundle exact sequences and subbundles.