# Pullback bundle of the tangent bundle through a diffeomorphism and an application to parallel global sections

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)