homogeneous space in a sentence

Examples

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