Is every real vector bundle over the circle necessarily trivial? If yes - could you please point to a reference. If no - what are sufficient conditions?


I am particularly concerned with the case of a smooth map $\gamma:S^1\rightarrow Q$ and the vector bundle $\gamma^\* TQ$.



Is every real vector bundle over the circle necessarily trivial?
In the spirit of Theo's comment, I'll say something about sufficient conditions.


A real vector bundle over the circle is trivial if and only if it is orientable. I discussed this a bit  [here.](https://mathoverflow.net/questions/21649/how-to-prove-that-w-1ew-1dete/21660#21660) 


The main point is that up to isomorphism, every real vector bundle over the circle is either trivial, or the Whitney sum of a trivial bundle with the Mobius bundle. The latter is not orientable.


The other answers at the question I linked to above may also be helpful to you.


