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. Formal definition 
Let (M, g) and (N, h) be two Riemannian manifolds and  a (surjective) submersion, i.e., a fibered manifold. The horizontal distribution  is a sub-bundle of the tangent bundle of  which depends both on the projection  and on the metric . Then, f is called a Riemannian submersion if and only if, for all , the vector space isomorphism  is isometric, i.e., length-preserving. Examples
An example of a Riemannian submersion arises when a Lie group  acts isometrically, freely and properly on a Riemannian manifold . The projection  to the quotient space  equipped with the quotient metric is a Riemannian submersion. For example, component-wise multiplication on  by the group of unit complex numbers yields the Hopf fibration. Properties
The sectional curvature of the target space of a Riemannian submersion can be calculated from the curvature of the total space by O'Neill's formula, named for Barrett O'Neill:

where  are orthonormal vector fields on ,  their horizontal lifts to ,  is the Lie bracket of vector fields and  is the projection of the vector field  to the vertical distribution. In particular the lower bound for the sectional curvature of  is at least as big as the lower bound for the sectional curvature of . Generalizations and variations
Fiber bundle
Submetry
co-Lipschitz map

See also
 Fibered manifold
 Geometric topology
 Manifold

Notes

References
. Barrett O'Neill. The fundamental equations of a submersion. Michigan Math. J. 13 (1966), 459–469. Riemannian geometry
Maps of manifolds