Let $\pi:E\to M$ be a rank $k$ vector bundle over the (compact) manifold $M$ and let $i:M\hookrightarrow E$ denote the zero section. I'm interested in a splitting of $i^*(TE)$, the restriction of the tangent bundle $TE$ to the zero section.
Intuitively I would guess that one could show the following:
$$i^*(TE)\cong TM\oplus E$$
Is this true? If so, how does the proof work?
Any details and references are appreciated!