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.

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.

Fundamentals of differential geometry(which is expanded from his earlier DG book) was partially an attempt to expand the Fascicule des Résultats out to a full book. $\endgroup$ – Harry Gindi Sep 28 '10 at 14:43