I have read in some lecture notes in the internet (without reference) the following result:
Let $E$ be a differentiable sphere bundle whose base $B$ has dimension $n\geq 2$ and whose fibers $F$ have dimension $n-1$. If $B$ is non-compact, then it admits a global smooth cross-section.
The proof is somewhat technical (involves sections with isolate singularities and how to remove them by sending them to infinity) and seems to be correct, but I would like to know a reference in the published literature for this result. In fact, the case $n=2$ would be enough. Thanks!