TITLE: In Diff, are the surjective submersions precisely the local-section-admitting maps?
QUESTION [12 upvotes]: Question as in title (Diff = category of smooth manifolds and smooth maps)
I thought I'd convinced myself this is true, so this is just a sanity check.
Also, what about for settings other than smooth manifolds? (like analytic manifolds, complex manifolds, or less differentiable - say, $C^2$ manifolds)

REPLY [3 votes]: In case any one in interested, the following result is given in Lee's Introduction to smooth manifolds. It is theorem $4.26$ named Local section theorem. 

Suppose $M$ and $N$ are smooth manifolds and $\pi:M\rightarrow N$ is a smooth map. Then $\pi:M\rightarrow N$ is a smooth submersion if and only if every point in $M$ is in the image of a smooth local section of $\pi$.