A Riemannian submersion is a smooth submersion 
between Riemannian
manifolds such that 
restricts to an isometry from the orthogonal complement
of 
onto 
at every point 
.
The kernel is the vertical subspace and its orthogonal complement is the horizontal
subspace.
The projection from a Riemannian product 
onto 
is a basic example. More generally, quotients by isometric
group actions often carry a metric making the quotient map a Riemannian submersion.
