Dictionary

Klein geometry en · NOUN

Etymology

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

Meanings

1. (qualifier: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;
   • 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#47;H, i.e. whether the action #92;mathcal#123;l#125; of G on G#47;H is effective. In this case, we call the Klein geometry effective. 2009, Andreas Čap, Jan Slovák, Parabolic Geometries I, American Mathematical Society, page 49:
   • The space of a Klein geometry is a smooth manifold of dimension #92;operatorname#123;dim#125;G-#92;operatorname#123;dim#125;H.
   • 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. 1934, American Journal of Mathematics, volume 56, Johns Hopkins University Press, page 153:
   • 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#61;1 and locally effective if K is discrete. A Klein geometry is geometrically oriented if G is connected. 2006, Luciano Boi, “The Aleph of Space”, in Giandomenico Sica, editor, What is Geometry?, Polimetrica, page 91:
   • Given a Klein geometry (G,H), the group G is called the principal group and G#47;H is called the space of the geometry.
2. (qualifier:differential geometry; more formally) 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. ▸ 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.
3. (broadly) The coset space G / H.

Forms

Relateds

Cayley-Klein geometry

Translations (1)

de Kleinsche Geometrie (type of geometry)

wikipedia: Felix Klein · wikipedia: Erlangen program