Klein geometry

A Klein geometry is a pair {{nowrap|(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 {{nowrap|1=X = G/H}} of a Klein geometry is a smooth manifold of dimension

:dim X = dim G − dim H.

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

:$g\; \backslash cdot\; (aH)\; =\; (ga)H.$

Clearly, this action is transitive (take {{nowrap|1=a = 1}}), so that one may then regard X as a homogeneous space for the action of G. The stabilizer of the identity coset {{nowrap|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 {{nowrap|(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.

Two Klein geometries {{nowrap|(G₁, H₁)}} and {{nowrap|(G₂, H₂)}} are geometrically isomorphic if there is a Lie group isomorphism {{nowrap|φ : G₁ → G₂}} so that {{nowrap|1=φ(H₁) = H₂}}. In particular, if φ is conjugation by an element {{nowrap|g ∈ G}}, we see that {{nowrap|(G, H)}} and {{nowrap|(G, gHg⁻¹)}} are isomorphic. The Klein geometry associated to a homogeneous space X is then unique up to isomorphism (i.e. it is independent of the chosen basepoint x₀).

Given a Lie group G and closed subgroup H, there is natural right action of H on G given by right multiplication. This action is both free and proper. The orbits are simply the left cosets of H in G. One concludes that G has the structure of a smooth principal H-bundle over the left coset space G/H:

:$H\backslash to\; G\backslash to\; G/H\; .$

The action of G on {{nowrap|1=X = G/H}} need not be effective. The kernel of a Klein geometry is defined to be the kernel of the action of G on X. It is given by

:$K\; =\; \backslash \{k\; \backslash in\; G\; :\; g^\{-1\}kg\; \backslash in\; H\backslash ;\backslash ;\backslash forall\; g\; \backslash in\; G\backslash \}.$

The kernel K may also be described as the core of H in G (i.e. the largest subgroup of H that is normal in G). It is the group generated by all the normal subgroups of G that lie in H.

A Klein geometry is said to be effective if {{nowrap|1=K = 1}} and locally effective if K is discrete. If {{nowrap|(G, H)}} is a Klein geometry with kernel K, then {{nowrap|(G/K, H/K)}} is an effective Klein geometry canonically associated to {{nowrap|(G, H)}}.

A Klein geometry {{nowrap|(G, H)}} is geometrically oriented if G is connected. (This does not imply that G/H is an oriented manifold). If H is connected it follows that G is also connected (this is because G/H is assumed to be connected, and {{nowrap|G → G/H}} is a fibration).

Given any Klein geometry {{nowrap|(G, H)}}, there is a geometrically oriented geometry canonically associated to {{nowrap|(G, H)}} with the same base space G/H. This is the geometry {{nowrap|(G₀, G₀ ∩ H)}} where G₀ is the identity component of G. Note that {{nowrap|1=G = G₀ H}}.

A Klein geometry {{nowrap|(G, H)}} is said to be reductive and G/H a reductive homogeneous space if the Lie algebra $\backslash mathfrak\; h$ of H has an H-invariant complement in $\backslash mathfrak\; g$.

In the following table, there is a description of the classical geometries, modeled as Klein geometries.

• {{cite book | author=R. W. Sharpe | title=Differential Geometry: Cartan's Generalization of Klein's Erlangen Program | publisher=Springer-Verlag | year=1997 | isbn=0-387-94732-9}}

Category:Differential geometry

Category:Lie groups

Category:Homogeneous spaces