Riemannian submersion explained

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

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

Then, f is called a Riemannian submersion if and only if, for all

x\in M , the vector space isomorphism (df) _{x} : K _{x} → T _{f(x)} N is isometric, i.e., length-preserving.

Examples

G acts isometrically, freely and properly on a Riemannian manifold (M,g) . The projection \pi: M → 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 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:

K _{N(X,Y)=K} _{M(\tilde} X, \tilde Y) + \tfrac 34 |[\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

See also

• Fibered manifold
• Geometric topology
• Manifold

References

• .
• Barrett O'Neill. The fundamental equations of a submersion. Michigan Math. J. 13 (1966), 459–469.