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| 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 [[group action|transitive action]] on ''X'' by a [[Lie group]] ''G'', which acts as the [[symmetry group]] of the geometry.
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| For background and motivation see the article on the [[Erlangen program]].
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| ==Formal definition==
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| A '''Klein geometry''' is a pair (''G'', ''H'') where ''G'' is a [[Lie group]] and ''H'' is a [[closed set|closed]] [[Lie subgroup]] of ''G'' such that the (left) [[coset space]] ''G''/''H'' is [[connected space|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
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| :dim ''X'' = dim ''G'' − dim ''H''.
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| There is a natural smooth [[group action|left action]] of ''G'' on ''X'' given by
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| :<math>g\cdot(aH) = (ga)H.</math>
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| 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 (group theory)|stabilizer]] of the identity coset ''H'' ∈ ''X'' is precisely the group ''H''.
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| 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 ''x''<sub>0</sub> in ''X'' and letting ''H'' be the stabilizer subgroup of ''x''<sub>0</sub> in ''G''. The group ''H'' is necessarily a closed subgroup of ''G'' and ''X'' is naturally [[diffeomorphic]] to ''G''/''H''.
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| Two Klein geometries (''G''<sub>1</sub>, ''H''<sub>1</sub>) and (''G''<sub>2</sub>, ''H''<sub>2</sub>) are '''geometrically isomorphic''' if there is a [[Lie group isomorphism]] φ : ''G''<sub>1</sub> → ''G''<sub>2</sub> so that φ(''H''<sub>1</sub>) = ''H''<sub>2</sub>. In particular, if φ is [[conjugacy class|conjugation]] by an element ''g'' ∈ ''G'', we see that (''G'', ''H'') and (''G'', ''gHg''<sup>−1</sup>) 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''<sub>0</sub>).
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| ==Bundle description==
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| Given a Lie group ''G'' and closed subgroup ''H'', there is natural [[group action|right action]] of ''H'' on ''G'' given by right multiplication. This action is both free and [[proper action|proper]]. The [[orbit (group theory)|orbits]] are simply the left [[coset]]s of ''H'' in ''G''. One concludes that ''G'' has the structure of a smooth [[principal bundle|principal ''H''-bundle]] over the left coset space ''G''/''H'':
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| :<math>H\to G\to G/H.\,</math>
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| ==Types of Klein geometries==
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| ===Effective geometries===
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| The action of ''G'' on ''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
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| :<math>K = \{k \in G : g^{-1}kg \in H\;\;\forall g \in G\}.</math>
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| The kernel ''K'' may also be described as the [[core (group)|core]] of ''H'' in ''G'' (i.e. the largest subgroup of ''H'' that is [[normal subgroup|normal]] in ''G''). It is the group generated by all the normal subgroups of ''G'' that lie in ''H''.
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| A Klein geometry is said to be '''effective''' if ''K'' = 1 and '''locally effective''' if ''K'' is [[discrete group|discrete]]. If (''G'', ''H'') is a Klein geometry with kernel ''K'', then (''G''/''K'', ''H''/''K'') is an effective Klein geometry canonically associated to (''G'', ''H'').
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| ===Geometrically oriented geometries===
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| A Klein geometry (''G'', ''H'') is '''geometrically oriented''' if ''G'' is [[connected space|connected]]. (This does ''not'' imply that ''G''/''H'' is an [[orientability|oriented manifold]]). If ''H'' is connected it follows that ''G'' is also connected (this is because ''G''/''H'' is assumed to be connected, and ''G'' → ''G''/''H'' is a [[fibration]]).
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| Given any Klein geometry (''G'', ''H''), there is a geometrically oriented geometry canonically associated to (''G'', ''H'') with the same base space ''G''/''H''. This is the geometry (''G''<sub>0</sub>, ''G''<sub>0</sub> ∩ ''H'') where ''G''<sub>0</sub> is the [[identity component]] of ''G''. Note that ''G'' = ''G''<sub>0</sub> ''H''.
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| ===Reductive geometries===
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| A Klein geometry (''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>.
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| == Examples ==
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| In the following table, there is a description of the classical geometries, modeled as Klein geometries.
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| {| class="wikitable" border="1"; text-align:center; margin:.5em 0 .5em 1em;"
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| | '''Underlying space'''
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| | '''Transformation group ''G'''''
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| | '''Subgroup ''H'''''
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| | '''Invariants'''
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| ! ''[[Euclidean geometry]]''
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| | [[Euclidean space]] <math>E(n)</math> || [[Euclidean group]] <math>\mathrm{Euc}(n)\simeq \mathrm{O}(n)\rtimes \R^n</math> || [[Orthogonal group]] <math>\mathrm{O}(n)</math> || Distances of [[Euclidean group|points]], [[angle]]s of [[Euclidean vector|vectors]]
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| ! ''[[Spherical geometry]]''
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| | [[Sphere]] <math>S^n</math> || Orthogonal group <math>\mathrm{O}(n+1)</math> || Orthogonal group <math>\mathrm{O}(n)</math> || Distances of points, angles of vectors
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| ! ''[[Conformal geometry]] on the sphere''
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| | [[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 (geometry)|line]] in the [[null cone]] of the Minkowski metric || Angles of vectors
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| ! ''[[Projective geometry]]''
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| | [[Real projective space]] <math>\mathbb{RP}^n</math> || [[Projective group]] <math>\mathrm{PGL}(n+1)</math>|| A subgroup <math>P</math> fixing a [[Flag (linear algebra)|flag]] <math>\{0\}\subset V_1\subset V_n</math> || [[Projective line]]s, [[Cross-ratio]]
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| ! ''[[Affine geometry]]''
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| | [[Affine space]] <math>A(n)\simeq\R^n</math> || [[Affine group]] <math>\mathrm{Aff}(n)\simeq \mathrm{GL}(n)\rtimes \R^n</math> || [[General linear group]] <math>\mathrm{GL}(n)</math> || Lines, Quotient of surface areas of geometric shapes, [[Center of mass]] of [[triangles]].
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| ! ''[[Hyperbolic geometry]]''
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| | [[Hyperbolic space]] <math>H(n)</math>, modeled e.g. as time-like lines in the [[Minkowski space]] <math>\R^{1,n}</math> || Lorentz group <math>\mathrm{O}(1,n)</math> || <math>\mathrm{O}(1)\times \mathrm{O}(n)</math> || Hyperbolic lines, hyperbolic circles, angles.
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| |}
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| ==References==
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| *{{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}}
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| [[Category:Differential geometry]]
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| [[Category:Lie groups]]
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| [[Category:Homogeneous spaces]]
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The writer's name is Christy. He works as a bookkeeper. As a lady what she really likes is fashion and she's been performing it for fairly a whilst. Alaska is where he's always been residing.
My weblog: psychic phone