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

f:M\to N

a submersion.

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

df : \mathrm{ker}(df)^{\perp} \rightarrow TN

is an isometry.

Examples

An example of a Riemannian submersion arises when a Lie group $G$ acts isometrically, freely and properly on a Riemannian manifold $(M,g)$ . The projection $\pi: M \rightarrow N$ to the quotient space $N = M /G$ equipped with the quotient metric is a Riemannian submersion. For example, component-wise multiplication on $S^3 \subset \mathbb{C}^2$ 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:

K_N(X,Y)=K_M(\tilde X, \tilde Y)+\tfrac34|[\tilde X,\tilde Y]^V|^2

where $X,Y$ are orthonormal vector fields on $N$ , $\tilde X, \tilde Y$ their horizontal lifts to $M$ , $[*,*]$ is the Lie bracket of vector fields and $Z^V$ is the projection of the vector field $Z$ to the vertical distribution.

In particular the lower bound for the sectional curvature of $N$ is at least as big as the lower bound for the sectional curvature of $M$ .

Generalizations and variations

Fiber bundle
Submetry
co-Lipschitz map

References

Gilkey, Peter B.; Leahy, John V.; Park, Jeonghyeong (1998), Spinors, Spectral Geometry, and Riemannian Submersions, Global Analysis Research Center, Seoul National University .