In differential geometry, a branch of mathematics, a Riemannian submersion is a submersion from one

Riemannian manifold In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent spac ...

to another that respects the metrics, meaning that it is an

orthogonal projection In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself (an endomorphism) such that P\circ P=P. That is, whenever P is applied twice to any vector, it gives the same result as if i ...

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

\mathrm(df)^

is a sub-bundle of the

tangent bundle In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and ...

TM

which depends both on the projection

f

and on the metric

g

. Then, ''f'' is called a Riemannian submersion if and only if the isomorphism

df : \mathrm(df)^ \rightarrow TN

is an

isometry In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' mea ...

Examples

An example of a Riemannian submersion arises when a

Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the addit ...

G

acts isometrically, freely and properly on a Riemannian manifold

(M,g)

. The projection

\pi: M \rightarrow N

to the

quotient space Quotient space may refer to a quotient set when the sets under consideration are considered as spaces. In particular: *Quotient space (topology), in case of topological spaces * Quotient space (linear algebra), in case of vector spaces *Quotient ...

N = M /G

equipped with the quotient metric is a Riemannian submersion. For example, component-wise multiplication on

S^3 \subset  \mathbb^2

by the group of unit complex numbers yields the

Hopf fibration In the mathematical field of differential topology, the Hopf fibration (also known as the Hopf bundle or Hopf map) describes a 3-sphere (a hypersphere in four-dimensional space) in terms of circles and an ordinary sphere. Discovered by Heinz Ho ...

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)+\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

,* /math> is the

Lie bracket of vector fields In the mathematical field of differential topology, the Lie bracket of vector fields, also known as the Jacobi–Lie bracket or the commutator of vector fields, is an operator that assigns to any two vector fields ''X'' and ''Y'' on a smooth ...

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 In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a ...

* Submetry * co-Lipschitz map

Notes

References