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 and be two Riemannian manifolds and a 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 the isomorphism is an isometry.

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:
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