TITLE: When does a submersion have connected fibers?
QUESTION [7 upvotes]: Can we characterize when a submersion $F:M \to N$ between two smooth manifolds has connected fibers? If this is too hard, what are some sufficient conditions?

REPLY [8 votes]: If $M$ and $N$ are both compact, then the submersion $F$ can be thought of as a fiber bundle map with fiber $F^{-1}(p)$ for any $p\in N$.  Then one can apply the long exact sequence of homotopy groups of a fiber bundle to learn that if, for example, $M$ is connected and $N$ is 1-connected, that the fibers must be connected.
These sufficient conditions may be too specific, though.