Homogeneous spaces

A sphere (from Ancient Greek , ) is a surface analogous to the circle, a curve. In solid geometry, a sphere is the set of points that are all at the same distance from a given point in three-dimensional space. That given point is the center of the sphere, and the distance is the sphere's radius. The earliest known mentions of spheres appear in the work of the ancient Greek mathematicians.

generalization of Euclidean geometry to higher-dimensional vector spaces

research program on the symmetries of geometry

topological space that is the space of cosets of a topological group

homogeneous space that has a constant negative curvature (not any hyperbolic manifold)

pseudo-Riemannian manifold whose symmetry group contains an inversion symmetry about every point

the manifold of all orthonormal k-frames in n-dimensional Euclidean space