In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a "'homogeneous space "'for a " G "-orbit.
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Thus, for example, Euclidean space, affine space and projective space are all in natural ways homogeneous spaces for their respective symmetry groups.
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On the other hand, [ [ n-sphere | ] ] is a homogeneous space for, and one has the following fiber bundle:
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They showed that in homogeneous spaces of finite volume, orbits of a Zariski dense subgroup of a semisimple group equidistribute towards algebraic measures.
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Cartan geometries & mdash; those with zero curvature & mdash; are locally equivalent to homogeneous spaces, hence geometries in the sense of Klein.
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Mahler's compactness theorem in the geometry of numbers characterises relatively compact subsets in certain non-compact homogeneous spaces ( specifically spaces of lattices ).
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Klein's aim was then to study objects living on the homogeneous space which were " congruent " by the action of " G ".
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Hyperbolic space is a homogeneous space that can be characterized by a exponentially with respect to the radius of the ball, rather than polynomially.
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It also draws on the inspiration of Felix Klein's Erlangen programme, in which a " geometry " is defined to be a homogeneous space.
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It follows that the " n "-dimensional hyperbolic space can be exhibited as the homogeneous space and a Riemannian symmetric space of rank 1,
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