I am involved with one question that a friend of mine asked.
If a vector bundle $(E,\pi,M)$ has two sub-bundles, $(F,\xi,M)$, $(L,\psi,M)$, and $\pi^{-1}(x) \cong \xi^{-1}(x) \oplus \psi^{-1}(x),$ then is true that $E$ is isomorphic to the Whitney sum of $F$ and $L$?
It is quite obvious that is true, but I have no expertise on a construction on morphism between bundles, could some one give a hint about how this isomorphism can be written?
Thanks!