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

→ Main article : 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 . ${\displaystyle G_{x}}$ ${\displaystyle x\in X}$

See also

• Erlangen program

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