Klein geometry

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

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 ${\displaystyle (G,H)}$, the group ${\displaystyle G}$ is called the principal group and ${\displaystyle G/H}$ is called the space of the geometry.

   The space of a Klein geometry is a smooth manifold of dimension ${\displaystyle \operatorname {dim} G-\operatorname {dim} H}$.

   • 1934, American Journal of Mathematics‎, 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 ${\displaystyle (G,H)}$ is the largest subgroup ${\displaystyle K}$ of ${\displaystyle H}$ that is normal in ${\displaystyle G}$. A Klein geometry ${\displaystyle (G,H)}$ is effective if ${\displaystyle K=1}$ and locally effective if ${\displaystyle K}$ is discrete. A Klein geometry is geometrically oriented if ${\displaystyle G}$ 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 ${\displaystyle (G,H)}$ we may first ask whether all of ${\displaystyle G}$ is “visible” on ${\displaystyle G/H}$, i.e. whether the action ${\displaystyle {\mathcal {l}}}$ of ${\displaystyle G}$ on ${\displaystyle G/H}$ is effective. In this case, we call the Klein geometry effective.

2. (loosely) The coset space G / H.

• Cayley-Klein geometry