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) 



In Diff, are the surjective submersions precisely the local-section-admitting maps?
There are two possible meanings for the sentence "*f* : *M* → *N* admits local sections",
so let's first disambiguate.


**Meaning 1:** For every point of *N*, there exists a neighborhood of that points and a section from that neighborhood back to *M*.


That's what people typically check in order to verify that, say, a map is a $G$-principal bundle.


**Meaning 2:** For every point *m* ∈ *M*, there exists a neighborhood of $f(m)$, and a section *s* from that neighborhood back to *M*, subject to the extra condition that $s(f(m))=m$.


Clearly, you care about the ***second*** meaning of that sentence.




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It is correct that a map is a submersion (not necessarily surjective!) iff it admits local sections.


If a map has local sections, then the maps on tangent spaces are sujective: that's just obvious.


Conversely, if a map is surjective at the level of tangent spaces, you first pick a local section of the maps of tangent spaces. Then, to finish the argument, you use the fact that
any subspace of the tangent space $T\_mM$ is the tangent space of a submanifold of *M*, and apply the implicit function theorem.


**Note:** if you care about infinite dimensional Banach manifolds, then the existence of a section for the map to tangent spaces needs to be assumed a separate condition. Indeed, it's not enough to assume that the map of tangent spaces is surjective, since it's not true that any surjective map of Banach spaces has a section.


**Note:** For complex varieties, you don't have the implicit function theorem, so it doens't work. Counterexample: the map $z\mapsto z^2$ from ℂ\* to itself. The fix is to pass the the "étale topology"... but that's another story.


