This tag is for questions relating to "homogeneous-spaces", a particular class of manifolds that behave per construction very symmetrically under the action of some groups, and they can be fully reconstructed just by looking at their behaviour under curtain actions.
Homogeneous spaces are, in a sense, the nicest examples of Riemannian manifolds and are good spaces on which to test conjecture. Also those are as important in connection with Lie groups and their applications as sets of cosets are in ordinary group theory. Indeed, in the Kleinian view, a geometry consists of a homogeneous space with the group acting as its symmetry group.
Definition: Given a topological group or algebraic group or Lie group, etc., $G$, a homogeneous $G$-space is a topological space or scheme, or smooth manifold etc. with transitive $G$-action.
E.g., A special case of homogeneous spaces are coset spaces arising from the quotient $G/H$ of a group $G$ by a subgroup. For the case of Lie groups this is also called Klein geometry.
References:
Baker A. (2002) Homogeneous Spaces. In: Matrix Groups. Springer Undergraduate Mathematics Series. Springer, London
