I read many quiestion about $TS^{2}\ncong S^{2}\times\mathbb{R}^{2}$ where the hint is use Hairy ball theorem and directly is done.
My question is: how do I proof that $TS^{2}\ncong S^{2}\times\mathbb{R}^{2}$ only with theory of vector bundles?
If there is $F:S^{2}\times\mathbb{R}^{2}\longrightarrow TS^{2}$ diffeomorphism then exists a one-one correspondences between $\Gamma(S^{2}\times\mathbb{R}^{2})$ the set of sections on $S^{2}\times\mathbb{R}$ and $\mathfrak{X}(S^{2})$ the vector fields on $S^{2}$ where the Hairy ball theorem appears.
There is a discussion in Hatcher's Vector Bundles, where the tangent bundle is constructed explicitly, it has Euler class=2, which means that the planes are twisted twice as you travel once around the equator.