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Riemannian submersion

## Summary

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 ${\displaystyle f:M\to N}$  a (surjective) submersion, i.e., a fibered manifold. The horizontal distribution ${\displaystyle K:=\mathrm {ker} (df)^{\perp }}$  is a sub-bundle of the tangent bundle of ${\displaystyle TM}$  which depends both on the projection ${\displaystyle f}$  and on the metric ${\displaystyle g}$ .

Then, f is called a Riemannian submersion if and only if, for all ${\displaystyle x\in M}$ , the vector space isomorphism ${\displaystyle (df)_{x}:K_{x}\rightarrow T_{f(x)}N}$  is isometric, i.e., length-preserving.[1]

## Examples

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

${\displaystyle K_{N}(X,Y)=K_{M}({\tilde {X}},{\tilde {Y}})+{\tfrac {3}{4}}|[{\tilde {X}},{\tilde {Y}}]^{V}|^{2}}$

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

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