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 .
definition
Let be a set on which the group operates transitively . That is, there is an illustration
with the properties
- for all and all true
- ,
- for all true
- ,
- where is the neutral element and
- for every pair there is one with
- .
The tuple is then called homogeneous space and is called the motion group of homogeneous space.
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
to thereby becomes a homogeneous space.
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.
properties
If the transitive acting group is finite, the formula applies to the cardinality of the set
- ,
where denotes the stabilizer of any element .
See also
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
- ↑ ^{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 .
- ↑ Kai Köhler: Differentialgeometrie und homogeneous spaces , p. 151, Wiesbaden: Springer Spectrum, 2014, read from Google books , accessed on June 8, 2018