nLab
Klein geometry

Idea

The notion of Klein geometry is essentially that of homogeneous space in the context of differential geometry.

Klein geometries form the local models for Cartan geometries.

For the generalization of Klein geometry to higher category theory see higher Klein geometry.

Definition

A Klein geometry is a pair $(G,H)$ where $G$ is a Lie group and $H$ is a closed Lie subgroup of $G$ such that the (left) coset space $X=G/H$ is connected. $G$ acts transitively on the homogeneous space $X$. We may think of $H$ as the stabilizer of a point in $X$.

Examples

• For $G=E(n)$, the Euclidean group? in $n$-dimensions; $H=O(n)$, the orthogonal group; then, $X$ is $n$-dimensional Cartesian space.

• Analogously, for $G=\mathrm{Iso}(d,1)$ the Poincare group of $(d+1)$-dimensional Minkowski space, and $H=\mathrm{SO}(d,1)$ the special orthogonal group of rotations and Loretz boosts?, then $X={ℝ}^{d+1}$ is Minkowski space itself.

  Passing to the corresponding Cartan geometry – by what physicists call gauging – yields the first order formulation of gravity.

Related concepts

local model	global geometry
Klein geometry	Cartan geometry
Klein 2-geometry	Cartan 2-geometry
higher Klein geometry	higher Cartan geometry

References

The notion of Klein geometry goes back to articles such as

• Felix Klein, A comparative review of recent researches in geometry , trans. M. W. Haskell, Bull. New York Math. Soc. 2, (1892-1893), 215-249. (retyped pdf, scan of original)

in the context of what came to be known as the Erlangen program.

A review is for instance in

• Vladimir Kisil, Erlangen Programme at Large: An Overview (arXiv:1106.1686)