Definition: A submersion $f:M\to N$ between Riemannian manifolds is called a Riemannian submersion if for each $p\in M$, $d_pf:(\ker d_pf)^\perp\to T_{f(p)}N$ is an isometry.
In contrast to Riemannian immersions I find it difficult to get an intuitive understanding of what Riemannian submersions are. At the moment I think of them as something like orthogonal projections but I am not sure if this intuition is always correct.
So my questions are:
$\bullet$ How can one think of Riemannian submersions?
$\bullet$ What nice properties do Riemannian submersions have in contrast to arbitrary submersions?