Isomorphic complex line bundles

Question

I was wondering if there is a result which says that if I have two differentiable manifold that are diffeomorphic then their complex line bundle are isomorphic. For example there is an explicit diffeomorphism between the 2-sphere and the 1-dimensional projective complex space. The tautological line bundle over the 1-dimensional projective line bundle can be used also for The 2-dimensional sphere?

Answer 1

The pullback of bundles along the diffeomorphism gives you a bijection between bundles over the two manifolds. Doesn't really matter what sort of bundles you're dealing with (line bundles, sphere bundles, etc.).

In fact, it suffices that the manifolds are homotopy equivalent, since pullback of bundles only depends on the homotopy class of the map.

However, note that in general the bijection depends on the homotopy class of the map, so different homotopy classes give you different bijections. There is no canonical bijection.

Answer 2

In your precise case, using clutching functions you can easily describe any line/vector bundle on $S^2$, see e.g here at the section "Clutching Functions" where this case is explicitly computed. This is also the way how you build vector bundle on $P^1$ after all, simply by gluing vector bundle on affine space.