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It is probably a stupid question and it is probably poorly written. Hopefully, it is not already answered somewhere else.

(1) Let $f:M \to N$ be a diffeomorphism. Is it true that the pullback bundle of the tangent bundle $f^*(TN)\simeq TM$?

My attempt is:

since the differential $df:TM \to TN$ is a bundle isomorphism and also $f^*(TN)\simeq TN$ is a bundle isomorphism we have that $TM\simeq df(TM)=TN\simeq f^*(TN) $

(2) In particular, given a manifold ${M}$ which is diffeomorphic to the standard sphere $S^n$, is it true that the bundle $E=T{M}\oplus\epsilon^1$ is trivial? ($\epsilon^1$ is the trivial line bundle).

My attempt of the proof of this second fact, assuming (1) is:

$\epsilon^{m+1}=f^*(\epsilon^{m+1})=f^*(TS^m\oplus\epsilon^1)=f^*(TS^m)\oplus f^*\epsilon^1=T{M}\oplus\epsilon^1$

(3) Suppose now we have a flat connection $\nabla$ on E as above (so hopefully trivial). Is it true that there exist m+1 linearly independent parallel global sections and that they span the bundle at every point?

By page 110, section 4-1, of Chern's Lectures on Differential Geometry and by the fact that E is trivial it should hold that exist m+1 linearly independent parallel global sections. Being parallel implies that these sections are globally linearly independent? (I believe it should be the case by a parallel transport argument)

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