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 ${\displaystyle g}$. The expression ${\displaystyle \mathrm {ker} (df)^{\perp }}$ denotes the subbundle of ${\displaystyle TM}$ that is the orthogonal complement of ${\displaystyle \mathrm {ker} (df_{x})\subset T_{x}M}$ at each point x of M.

Then, f is called a Riemannian submersion if and only if, for all ${\displaystyle x\in M}$, the vector space isomorphism ${\displaystyle (df)_{x}:\mathrm {ker} (df_{x})^{\perp }\rightarrow T_{f(x)}N}$ is an isometry, or in other words it carries each vector to one of the same length.^[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 ${\displaystyle M}$.

Generalizations and variations

• Fiber bundle
• Submetry
• co-Lipschitz map

See also

• Fibered manifold
• Geometric topology
• Manifold