Riemannian Submersion

In differential geometry, a branch of mathematics, a Riemannian submersion is a submersion from one Riemannian manifold to another that respects the metrics, meaning that it is an orthogonal projection on tangent spaces.

Let (M, g) and (N, h) be two Riemannian manifolds and

a submersion.

Then f is a Riemannian submersion if and only if the isomorphism

is an isometry.

Read more about Riemannian Submersion:  Examples, Properties, Generalizations and Variations