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Homogeneous space
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== Geometry == From the point of view of the [[Erlangen program]], one may understand that "all points are the same", in the [[geometry]] of ''X''. This was true of essentially all geometries proposed before [[Riemannian geometry]], in the middle of the nineteenth century. Thus, for example, [[Euclidean space]], [[affine space]] and [[projective space]] are all in natural ways homogeneous spaces for their respective [[symmetry group]]s. The same is true of the models found of [[non-Euclidean geometry]] of constant [[curvature]], such as [[hyperbolic space]]. A further classical example is the space of lines in projective space of three dimensions (equivalently, the space of two-dimensional subspaces of a four-dimensional [[vector space]]). It is simple [[linear algebra]] to show that GL<sub>4</sub> acts transitively on those. We can parameterize them by ''line co-ordinates'': these are the 2×2 [[minor (linear algebra)|minors]] of the 4×2 matrix with columns two basis vectors for the subspace. The geometry of the resulting homogeneous space is the [[line geometry]] of [[Julius Plücker]].
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