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In the comments on the MathSE question "Is Seifert-Weber space homogeneous for a Lie group?", it is claimed that if $ M $ is a manifold which admits a finite volume hyperbolic metric (...

Let $ U_1 \to E \to B $ be a $ U_1 $ principal bundle. Suppose that $ B $ is homogenous (admits a transitive action by a Lie group) and compact. Then must it be the case that $ E $, the total space of ...

Let $ M $ be a smooth manifold. Recall that a manifold $ M $ is smooth homogeneous if there exists a Lie group acting transitively on $ M $. Recall that a manifold $ M $ is Riemannian homogeneous if ...

Here are some examples of compact homogeneous 3 manifolds for different Thurston geometries: (1) Spherical: $\mathbb{S}^3 \cong \mathrm{SU}_2$ modulo any finite subgroup (2) Euclidean: 3 torus $\...

as a beginner,I am not sure whether this question is too basic to post here./-\。 Many textbook will talk about the prototypical example SL(n,Z)\SL(n,R), which can be identified with the space of ...