I have read in some lecture notes in the internet (without reference) the following result:



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> 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.
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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!



Triviality of a differentiable sphere bundle
I am guessing that you misunderstand the claim.


Here are some facts that seem to be in the right ballpark: a bundle whose fibers are spheres of dimension $n$ over a base whose dimension is $n$ or less will certainly have a section. And if the base $B$ has dimension $n+1$ then it will have a section if and only if the Euler class is zero; this is a cohomology class in $H^{n+1}(B)$, or a twisted version of this if the bundle is nonorientable. If the $(n+1)$-manifold $B$ is connected and noncompact then this cohomology group is zero, so a section exists. 


So if you assume $n$ is meant to be dimension of fiber rather than total space then the claim makes sense but with $n+1$ rather than $n-1$ for the base. If the sphere bundle is the unit sphere bundle of a vector bundle and $n$ is the fiber dimension of the vector bundle, then you can change the $n+1$ to $n$, but that's still off by $1$.


