Let $\pi: E \to M$ a vector bundle. Is $E$ a direct summnad of $M\times\mathbb{R}^{d}$, for some $d$?

Question

Let $\pi: E \to M$ a vector bundle over a smooth manifold $M$.

$E$ is direct summand of $M\times\mathbb{R}^{d}$, if there exist a vector bundle morphisms $f:E\to M\times\mathbb{R}^{d}$ and $h:M\times\mathbb{R}^{d}\to E$ such that $\pi_{1}\circ f =Id_{M}$,$\pi\circ h= Id_{M}$ and $h\circ f=Id_{E}$.

Answer 1

Yes, Proposition 1.4 in Hatcher's book here.