Klein geometry

{{Short description|Type of 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.

==Formal definition== 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 :<math>g \cdot (aH) = (ga)H.</math> 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 ''x''₀ in ''X'' and letting ''H'' be the stabilizer subgroup of ''x''₀ in ''G''. The group ''H'' is necessarily a closed subgroup of ''G'' and ''X'' is naturally diffeomorphic to ''G''/''H''.

Two Klein geometries {{nowrap|(''G''₁, ''H''₁)}} and {{nowrap|(''G''₂, ''H''₂)}} are '''geometrically isomorphic''' if there is a Lie group isomorphism {{nowrap|''φ'' : ''G''₁ → ''G''₂}} so that {{nowrap|1=''φ''(''H''₁) = ''H''₂}}. 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 ''x''₀).

==Bundle description== 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'': :<math>H\to G\to G/H .</math>

==Types of Klein geometries==

===Effective geometries=== 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 :<math>K = \{k \in G : g^{-1}kg \in H\;\;\forall g \in G\}.</math> 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'')}}.

===Geometrically oriented geometries=== 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|(''G''₀, ''G''₀ ∩ ''H'')}} where ''G''₀ is the identity component of ''G''. Note that {{nowrap|1=''G'' = ''G''₀ ''H''}}.

===Reductive geometries=== A Klein geometry {{nowrap|(''G'', ''H'')}} is said to be '''reductive''' and ''G''/''H'' a '''reductive homogeneous space''' if the Lie algebra <math>\mathfrak h</math> of ''H'' has an ''H''-invariant complement in <math>\mathfrak g</math>.

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

{| class="wikitable" border="1"; text-align:center; margin:.5em 0 .5em 1em;" |- | | '''Underlying space''' | '''Transformation group ''G''''' | '''Subgroup ''H''''' | '''Invariants''' |- ! ''Projective geometry'' | Real projective space <math>\mathbb{R}\mathrm{P}^n</math> || Projective group <math>\mathrm{PGL}(n+1)</math>|| A subgroup <math>P</math> fixing a flag <math>\{0\}\subset V_1\subset V_n</math> || Projective lines, cross-ratio |- ! ''Conformal geometry on the sphere'' | Sphere <math>S^n</math> || Lorentz group of an <math>(n+2)</math>-dimensional space <math>\mathrm{O}(n+1,1)</math> || A subgroup <math>P</math> fixing a line in the null cone of the Minkowski metric || Generalized circles, angles |- ! ''Hyperbolic geometry'' | Hyperbolic space <math>H(n)</math>, modelled e.g. as time-like lines through the origin in the Minkowski space <math>\R^{1,n}</math> || Orthochronous Lorentz group <math>\mathrm{O}(1,n)/\mathrm{O}(1)</math> || <math>\mathrm{O}(1)\times \mathrm{O}(n)</math> || Lines, circles, distances, angles |- ! ''Elliptic geometry'' | Elliptic space, modelled e.g. as the lines through the origin in Euclidean space <math>\mathbb{R}^{n+1}</math> || <math>\mathrm{O}(n+1)/\mathrm{O}(1)</math> || <math>\mathrm{O}(n)/\mathrm{O}(1)</math> || Lines, circles, distances, angles |- ! ''Spherical geometry'' | Sphere <math>S^n</math> || Orthogonal group <math>\mathrm{O}(n+1)</math> || Orthogonal group <math>\mathrm{O}(n)</math> || Lines (great circles), circles, distances of points, angles |- ! ''Affine geometry'' | Affine space <math>A(n)\simeq\R^n</math> || Affine group <math>\mathrm{Aff}(n)\simeq \R^n \rtimes \mathrm{GL}(n)</math> || General linear group <math>\mathrm{GL}(n)</math> || Lines, quotient of surface areas of geometric shapes, center of mass of triangles |- ! ''Euclidean geometry'' | Euclidean space <math>E(n)</math> || Euclidean group <math>\mathrm{Euc}(n)\simeq \R^n \rtimes \mathrm{O}(n)</math> || Orthogonal group <math>\mathrm{O}(n)</math> || Distances of points, angles of vectors, areas |- |}

==References== *{{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