# 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 . ${\ displaystyle G_ {x}}$ ${\ displaystyle x \ in X}$ 