Riemannian submersion

From Wikipedia, the free encyclopedia
Jump to: navigation, search

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.

Examples[edit]

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[edit]

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[edit]

See also[edit]

References[edit]