Compact homogeneous hypersurfaces

Homogeneous Hypersurfaces in Complex Hyperbolic Spaces

We study the geometry of homogeneous hypersurfaces and their focal sets in complex hyperbolic spaces. In particular, we provide a characterization of the focal set in terms of its second fundamental form and determine the principal curvatures of the homogeneous hypersurfaces together with their multiplicities.

Topology of Polar Weighted Homogeneous Hypersurfaces

Polar weighted homogeneous polynomials is the class of special polynomials of real variables xi, yi, i = 1, . . . , n with zi = xi + √ −1 yi which enjoys a “ polar action”. In many aspects, their behavior looks like that of complex weighted homogeneous polynomials. We study basic properties of hypersurfaces which are defined by polar weighted homogeneous polynomials.

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...

Compact null hypersurfaces and collapsing Riemannian manifolds

Restrictions are obtained on the topology of a compact divergence-free null hypersurface in a four-dimensional Lorentzian manifold whose Ricci tensor is zero or satisfies some weaker conditions. This is done by showing that each null hypersurface of this type can be used to construct a family of three-dimensional Riemannian metrics which collapses with bounded curvature and applying known resul...