HOMOGENEOUS COMPACT GEOMETRIES

Homogeneous Geometries

For the purposes of this paper, a geometry will consist of a set together with a closure operation on that set, satisfying the exchange condition, and under which singletons are closed (for more precise definitions, see §2.1). The geometry is homogeneous if in the automorphism group of the geometry, the pointwise stabilizer of any finitedimensional closed subset of the geometry is transitive on...

Geometries of homogeneous spaces

θ. This definition is deficient as it depends on a choice of basis. A definition in R is that a rotation is a linear map g with an axis, a line L fixed by g, and on the orthogonal complement L⊥ of L the restriction of g is a two-dimensional rotation. For this to make sense, one must have understood that the two-dimensional definition is independent of basis, and that g does stabilize the orthog...

Large Homogeneous Compact Spaces

Compact Homogeneous Universes

A thorough classification of the topologies of compact homogeneous universes is given except for the hyperbolic spaces, and their global degrees of freedom are completely worked out. To obtain compact universes, spatial points are identified by discrete subgroups of the isometry group of the generalized Thurston geometries, which are related to the Bianchi and the Kantowski-Sachs-Nariai univers...

The homogeneous geometries of real hyperbolic space

We describe the holonomy algebras of all canonical connections of homogeneous structures on real hyperbolic spaces in all dimensions. The structural results obtained then lead to a determination of the types, in the sense of Tricerri and Vanhecke, of the corresponding homogeneous tensors. We use our analysis to show that the moduli space of homogeneous structures on real hyperbolic space has tw...