Klein geometry

In mathematics, a Klein geometry is a type of geometry motivated by Felix Klein in his influential Erlangen program. More specifically, it is a homogeneous space X together with a transitive action on X by a Lie group G, which acts as the symmetry group of the geometry.

For background and motivation see the article on the Erlangen program.

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 G/H is connected. The group G is called the principal group of the geometry and G/H is called the space of the geometry (or, by an abuse of terminology, simply the Klein geometry). The space X = G/H of a Klein geometry is a smooth manifold of dimension

There is a natural smooth left action of G on X given by

Clearly, this action is transitive (take a = 1), so that one may then regard X as a homogeneous space for the action of G. The stabilizer of the identity coset H ∈ X is precisely the group H.

Given any connected smooth manifold X and a smooth transitive action by a Lie group G on X, we can construct an associated Klein geometry (G, H) by fixing a basepoint x₀ in X and letting H be the stabilizer subgroup of x₀ in G. The group H is necessarily a closed subgroup of G and X is naturally diffeomorphic to G/H.

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