Klein geometry

English

Etymology

Named after German mathematician Christian Felix Klein (1849—1925). The concept arose from Klein's Erlangen program (published 1872).

Noun

Klein geometry (plural Klein geometries)

(differential geometry) A type of geometry (mathematical object representing a space and its spatial relationships); a homogeneous space X together with a symmetry group which represents the group action on X of some Lie group;
(more formally) an ordered pair (G, H), where G is a Lie group and H a closed Lie subgroup of G such that the left coset space G / H is connected.
Given a Klein geometry $(G, H)$ , the group $G$ is called the principal group and $G / H$ is called the space of the geometry.

The space of a Klein geometry is a smooth manifold of dimension $\dim G - \dim H$ .
- 1934, American Journal of Mathematics‎^[1], volume 56, Johns Hopkins University Press, page 153:
  The present paper develops the general theory of non-holonomic geometries as generalizations of Klein geometries starting from a set of fundamental assumptions presented in the form of postulates.
- 2006, Luciano Boi, “The Aleph of Space”, in Giandomenico Sica, editor, What is Geometry?, Polimetrica, page 91:
  The kernel of a Klein geometry $(G, H)$ is the largest subgroup $K$ of $H$ that is normal in $G$ . A Klein geometry $(G, H)$ is effective if $K = 1$ and locally effective if $K$ is discrete. A Klein geometry is geometrically oriented if $G$ is connected.
- 2009, Andreas Čap, Jan Slovák, Parabolic Geometries I, American Mathematical Society, page 49:
  A careful geometric study of Klein geometries is available in [Sh97, Chapter 4].
  Given a Klein geometry $(G, H)$ we may first ask whether all of $G$ is “visible” on $G / H$ , i.e. whether the action $𝓁$ of $G$ on $G / H$ is effective. In this case, we call the Klein geometry effective.
(loosely) The coset space G / H.

Related terms

Cayley-Klein geometry

Translations

type of geometry

German: Kleinsche Geometrie f