# Trivial tangent bundle and parallelizability of a $n$-sphere

So, I want to show that a $n$-sphere $S$ is parallelizable iff it has trivial tangent bundle. For "$\Leftarrow$" I would like to take a trivialization $\varphi:S\times \mathbb{R}^n\rightarrow TS$ and consider the vector fields $v_i(x):=\varphi(x,e_i)$.

It seems to me like these vector fields are independent in every $x\in S$, but how can I deduce this exactly? The proof should somehow use that $\varphi$ is a homeomorphism, right? I don't see how to get from "homeomorphism" to linear independence.

1) The equivalence you mention has nothing to do with spheres: it is valid for any differential manifold. 2) The crucial point which solves your problem is that $\phi$ restricts at every point of the manifold to a vector space isomorphism between the fiber of the trivial bundle and that of the tangent bundle . – Georges Elencwajg Jan 15 '12 at 12:45