TITLE: Splitting of the tangent bundle of a vector bundle
QUESTION [9 upvotes]: 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!

REPLY [8 votes]: [Note to moderators:  This is mostly a comment on the previous answer.]
A reference for the short exact sequence 
$$ 0 \to \pi^*E\to TE \to \pi^* TM \to 0 $$ 
is [1, Vol. I, Chapter 3, Exercise 29 (page 103)]. Its restriction
$ 0 \to E \to TE_{|M} \to TM \to 0 $
splits canonically, irrespective of any compactness assumption on the base, since $\pi\colon E\to M$ is split by the zero section.  So we always have a canonical isomorphism of vector bundles:
$$TE_{|M} \cong TM \oplus E$$
As $\pi$ is a homotopy equivalence, it follows that the unrestricted sequence also splits, so that we have a (non-canonical) isomorphism
$$TE \cong \pi^*TM \oplus \pi^*E$$
A reference for this argument is [2, Kapitel IX, Satz 6.9 (page 365)]. 
[1] Michael Spivak, A Comprehensive Introduction to Differential Geometry
[2] Tammo tom Dieck, Topologie