Constructing a vector bundle built out of kernels of the Jacobian?

Question

A smooth map $f: M \to N$ between smooth manifolds is a submersion if each Jacobian$$Df_x: DM_x \to DN_{f(x)}$$is surjective. How do I construct a vector bundle $\kappa_f$ built out of the kernels of the $Df_x$?

Answer 1

Clearly the subset of $TM$ defined by this has a vector space above each fiber and a projection map to $M$; what you want to prove is local triviality of the projection. But this follows from the implicit function theorem: in a chart $U \subset M$, the map $f$ is of the form of a standard projection $\Bbb R^n \to \Bbb R^k$. Then the kernel of the Jacobian here is $\Bbb R^n \times \Bbb R^{n-k} \subset \Bbb R^n \times \Bbb R^n = T\Bbb R^n$, which is trivial - hence the vector bundle is indeed locally trivial, as desired.

(Actually, if you put a Riemannian metric on $M$, you can construct an isomorphism $\kappa_f \oplus f^*TN \cong TM$.)