If $M$ is Riemannian, then $\kappa_f \oplus f^*TN \cong TM$, where $\kappa_f$ is built out of kernels of the $Df_x$?

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. I know how to construct a vector bundle $\kappa_f$ built out of the kernels of the $Df_x$. My question is, if $M$ is Riemannian, how do I see that$$\kappa_f \oplus f^*TN \cong TM?$$

Answer 1

You have $TM = \kappa_f \oplus \kappa_f^{\perp}$ so it is enough to show that $\kappa_f^{\perp}$ is isomorphic to $f^{*}(TN)$. Then note that the map $\Phi \colon \kappa_f^{\perp} \rightarrow f^{*}(TN)$ given by

$$ \Phi(p, v) = (p, df_p(v)) $$

is a bundle map which is an isomorphism on each fiber (as $df_p|_{(\kappa_f)_p^{\perp}} \colon (\kappa_f)_p^{\perp} \rightarrow T_{f(p)}N$ is an injective linear map between two vector spaces of the same dimension).