Show that $S^n \times R$ is parallelizable for all n

Question

A manifold $M^n$ is called parallelizable if and only if there exist n vector fields on $M^n$ which are independent at each point of $M^n$. Could you help me figure out the vector fields for $S^n \times R$?

Answer 1

Consider $f:S^n\times \mathbb{R}\rightarrow \mathbb{R}^{n+1}-\{0\}$ defined by $f(x,t)=e^tx$, it is a diffeomorphism where you identify $S^n $ with the unit sphere of $\mathbb{R}^{n+1}-\{0\}$; $\mathbb{R}^{n+1}-\{0\}$ is parallelizable since it is an open subset of $\mathbb{R}^{n+1}$.