TITLE: Any line bundle on sphere is trivial
QUESTION [6 upvotes]: I want to prove that any (real) vector bundle of dimension one  on $S^2$ is trivial one. I don't know how to start because I don't have a much technics (this problem was given after 3th lecture on analysis on manifolds). Any suggestions would be appreciated.

REPLY [11 votes]: Here is a more advanced proof which you might appreciate at some later stage.
A real line bundle $L$ on a topological space $X$ is trivial if and only if its first Stiefel-Whitney class $w_1(L)\in H^1(X, \mathbb Z/2\mathbb Z)$ is zero.
Since $H^1(S^2,\mathbb Z/2\mathbb Z)=0$ any line bundle on the 2-sphere must be trivial.
(More generally: for the same reason every real line bundle on a simply connected  space is trivial)