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 (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 ${\displaystyle \mathrm {ker} (df)^{\perp }}$ 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 the isomorphism ${\displaystyle df:\mathrm {ker} (df)^{\perp }\rightarrow TN}$ is an isometry.^[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 .

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 ${\displaystyle M}$.

Generalizations and variations[]

• Fiber bundle
• Submetry

See also[]

• Fibered manifold
• Geometric topology
• Manifold

Notes[]

References[]

• Gilkey, Peter B.; Leahy, John V.; Park, Jeonghyeong (1998), Spinors, Spectral Geometry, and Riemannian Submersions, Global Analysis Research Center, Seoul National University.
• Barrett O'Neill. The fundamental equations of a submersion. Michigan Math. J. 13 (1966), 459–469. doi:10.1307/mmj/1028999604