# Homogeneous space

A homogeneous space (rare Klein space or Klein's geometry according to Felix Klein ) is in mathematics a space with a transitive group effect . The corresponding group is called the movement group .

This homogeneity clearly means that the room “looks the same in every point”. For example, continuous differentiable manifolds homogeneous, because every two points there is a diffeomorphism , the on maps. Riemann's homogeneous spaces are an important class of homogeneous spaces . ${\ displaystyle x, y}$${\ displaystyle x}$${\ displaystyle y}$

## definition

Let be a set on which the group operates transitively . That is, there is an illustration ${\ displaystyle M}$ ${\ displaystyle G}$

${\ displaystyle G \ times M \ to M \ qquad (g, x) \ mapsto gx}$

with the properties

• for all and all true${\ displaystyle g, h \ in G}$${\ displaystyle x \ in M}$
${\ displaystyle (gh) x = g (hx)}$,
• for all true${\ displaystyle x \ in M}$
${\ displaystyle ex = x}$,
where is the neutral element and${\ displaystyle e \ in G}$
• for every pair there is one with${\ displaystyle x, y \ in M}$${\ displaystyle g \ in G}$
${\ displaystyle y = gx}$.

The tuple is then called homogeneous space and is called the motion group of homogeneous space. ${\ displaystyle (M, G)}$${\ displaystyle G}$

## Examples

Often times, the underlying amount of homogeneous space has an additional structure. In the following three examples, homogeneous spaces in the mathematical sub-areas of group theory , topology and Riemannian differential geometry are considered.

### Sub classroom

An example of a homogeneous space is the minor class of a group with a subgroup . The group operates through ${\ displaystyle G / H}$${\ displaystyle G}$ ${\ displaystyle H}$${\ displaystyle G}$

${\ displaystyle g (aH) = (ga) H}$

to thereby becomes a homogeneous space. ${\ displaystyle G / H}$${\ displaystyle (G / H, G)}$

### Riemann homogeneous space

Often Riemannian homogeneous spaces are meant when homogeneous spaces are mentioned. Here there are two points a isometrics that on maps. Riemannian homogeneous spaces are an important class of examples in Riemannian geometry . Their curvature can often be calculated using algebraic methods. ${\ displaystyle x, y}$${\ displaystyle x}$${\ displaystyle y}$

## properties

If the transitive acting group is finite, the formula applies to the cardinality of the set${\ displaystyle G}$${\ displaystyle X}$

${\ displaystyle \ vert X \ vert = {\ frac {\ vert G \ vert} {\ vert G_ {x} \ vert}}}$,

where denotes the stabilizer of any element . ${\ displaystyle G_ {x}}$${\ displaystyle x \ in X}$

## literature

• Kai Köhler: Differentialgeometrie und homogeneous spaces , p. 151 ff., Wiesbaden: Springer Spectrum, 2014, ISBN 978-3-8348-1569-9
• Jeff Cheeger, David G. Ebin: Comparison theorems in Riemannian geometry. North-Holland Mathematical Library, Vol. 9. North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1975.

## Individual evidence

1. a b Homogeneous space . In: Guido Walz (Ed.): Lexicon of Mathematics . 1st edition. Spectrum Academic Publishing House, Mannheim / Heidelberg 2000, ISBN 3-8274-0439-8 .
2. Kai Köhler: Differentialgeometrie und homogeneous spaces , p. 151, Wiesbaden: Springer Spectrum, 2014, read from Google books , accessed on June 8, 2018