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From the point of view of the Erlangen programme we are saying that, in the geometry of X, all points are the same. That was true, one could say, of all geometries proposed before Riemannian geometryIn mathematics, Riemannian geometry has at least two meanings, one of which is described in this article and another also called elliptic geometry. In differential geometry, Riemannian geometry is the study of smooth manifolds with Riemannian metrics; i.. Therefore Euclidean spaceEuclidean space is the usual n dimensional mathematical space, a generalization of the 2- and 3-dimensional spaces studied by Euclid. Formally, for any non-negative integer n n dimensional Euclidean space is the set R n (where R is the set of real numbers, affine spaceIn mathematics, an affine space may be defined somewhat abstractly as a set on which a vector space acts transitively. Albeit somewhat jocular, the following characterization may be easier to understand: an affine space is what is left of a vector space a and projective spaceProjective geometry In mathematics, a projective space is a fundamental construction from any vector space. It generalises the projective plane that may be constructed from a three-dimensional vector space, over any field. While the theory of projective p are all in natural ways homogeneous spaces for respective symmetry groupThe symmetry group of a geometric figure is the group of congruencies under which it is invariant, with composition as the operation. The article on group theory also contains an explanation of the concept. In Euclidean geometry, discrete symmetry groupss. The same is true of the models found of non-Euclidean geometryThe term non-Euclidean geometry (also spelled: non-Euclidian geometry describes both hyperbolic and elliptic geometry, which are contrasted with Euclidean geometry. The essential difference between Euclidean and non-Euclidean geometry is the nature of par, of constant curvatureCurvature is the amount by which an geometric object deviates from being flat''. The word flat might have very different meaning depending on the object considered (for curves it is a straight line and for surfaces it is a Euclidean plane). In this articl.

A further classical example is the space of lines in projective space of three dimensions (equivalently, the space of two-dimensionsional subspaces of a four-dimensional vector spaceThe fundamental concept in linear algebra is that of a vector space or linear space . This is a generalization of the set of all geometrical vectors and is used throughout modern mathematics. Formal definition A set V is a vector space over a field F (for). It is simple linear algebra to show that GL₄ acts transitively on those. We can parametrize them by line co-ordinates: these are the 2×2 minors of the 2×4 matrix with columns two basis vectors for the subspace. The geometry of the resulting homogeneous space is the line geometry of Julius Plücker.

In general, if X is a homogeneous space, and H is the stabilizer of some fixed x in X, the points of X correspond to the cosets G/H. We can assume that H is a closed subgroup of G, for a continuous action: when it is the identity subgroup {e}, we have a principal homogeneous space. For example in the line geometry example we can identify H as a 12-dimensional subgroup of the 16-dimensional group GL₄, defined by conditions on the matrix entries h₁₃ = h₁₄ = h₂₃ = h₂₄ = 0, by looking for the stabilizer of the subspace spanned by the first two standard basis vectors. That shows that X has dimension 4. Since the homogeneous coordinates given by the minors are 6 in number, this means that the latter are not indepedent of each other. In fact a single quadratic relation holds between the six minors, as was known to the geometrys of the nineteenth century.

This example is a first example of a Grassmannian, other than a projective space. There are many further homogeneous spaces of the classical linear groups in common use in mathematics.

The idea of a prehomogeneous vector space was introduced by Mikio Sato. It is a finite-dimensional vector space V with a group action of an algebraic group G, such that there is an orbit of G that is open for the Zariski topology (and so, dense). An example is GL₁ acting on a one-dimensional space. The definition is more restrictive than it initially appears: such spaces have remarkable properties, and there is a classification.