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Suppose we have a manifold $M$ of dimension $m$ and let $E$ be the rank $n$ trivial vector bundle on $M$. Let $\Gamma$ be a section of the trivial bundle $E$. My question is, are there some conditions on the section $\Gamma$ which ensure that it extends to a basis of sections of the vector bundle $E$.

I know that in general this is not possible (for example in case of the trivial bundle on sphere) but I am unable to see what exactly is the obstruction.

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The trivial bundle defines a line subbundle $L \subset E$; you're asking precisely that $E/L$ also be trivial. But as you point out in the question, the tangent bundle $TS^2$ plus a trivial line bundle is the trivial 3-plane bundle, so there is a line subbundle of the trivial 3-plane bundle that gives $TS^2$ as its quotient (and hence every other section of the trivial rank 3 bundle must, at some point, be a scalar multiple of our "special" one.)

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  • $\begingroup$ I don't quite understand your answer If I am not mistaken you seem to be answering the question when the quotient is $TS^2$ but not when is it trivial. $\endgroup$ – happymath Jun 8 '16 at 7:26
  • $\begingroup$ @happymath I'm giving an example where the quotient is not trivial. Being able to extend a section to a basis of sections is the same thing as the quotient being trivial. $\endgroup$ – user98602 Jun 8 '16 at 7:27
  • $\begingroup$ I am sorry for not being clear but I already know of this example that is why I wanted some conditions on section which ensure the quotient is trivial $\endgroup$ – happymath Jun 8 '16 at 7:33
  • $\begingroup$ There is nothing simpler than what you and I have said. $\endgroup$ – user98602 Jun 8 '16 at 7:36
  • $\begingroup$ I am looking for a condition for a vector bundle over any manifold not just the sphere $\endgroup$ – happymath Jun 8 '16 at 7:37

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