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?
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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 1connected, that the fibers must be connected. These sufficient conditions may be too specific, though. 

