Local diffeomorphism from $\mathbb R^2$ onto $S^2$ Is there any local diffeomorphism from $\mathbb R^2$ onto $S^2$?

First note that there actually are no covering maps $\mathbb{R}^2 \to S^2$.  This is because $S^2$ is simply connected and hence is its own universal cover; if there were a covering map $\mathbb{R}^2 \to S^2$, then by the universal property of the universal cover there would be a covering map $S^2 \to \mathbb{R}^2$.  But there can't even be a continuous surjection $S^2 \to \mathbb{R}^2$ because $S^2$ is compact and $\mathbb{R}^2$ is not.
Thus any local diffeomorphism $\mathbb{R}^2 \to S^2$ answers your question.  For example, the inverse of the stereographic projection map does the job.
