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)

1. (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 $\operatorname {dim} G-\operatorname {dim} H$.

   • 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 ${\mathcal {l}}$ of $G$ on $G/H$ is effective. In this case, we call the Klein geometry effective.

2. (loosely) The coset space G / H.

Related terms

• Cayley-Klein geometry

Translations

type of geometry

• German: Kleinsche Geometrie f