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)


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.