differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...

, a branch of

mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 manifold, real, smooth manifold ''M'' equipped with a positive-definite Inner product space, inner product ...

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 it wer ...

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 In differential geometry, in the category of differentiable manifolds, a fibered manifold is a surjective submersion \pi : E \to B\, that is, a surjective differentiable mapping such that at each point y \in U the tangent mapping T_y \pi : T ...

. 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 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 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 additio ...

G

acts isometrically, freely and properly on a Riemannian manifold

(M,g)

. The projection

\pi: M \rightarrow 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  \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 Barrett O'Neill (1924– 16 June 2011) was an American mathematician. He is known for contributions to differential geometry, including two widely-used textbooks on its foundational theory. He was the author of eighteen research articles, the last ...

: :

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 m ...

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 p ...

* Submetry * co-Lipschitz map

Notes

References