Homogeneous three-dimensional Riemannian spaces

Discrete Groups and Non-riemannian Homogeneous Spaces

A basic question in geometry is to understand compact locally homogeneous manifolds, i.e., those compact manifolds that can be locally modelled on a homogeneous space J\H of a finite-dimensional Lie group H. This means that there is an atlas on a manifold M consisting of local diffeomorphisms with open sets in J\H where the transition functions between these open sets are given by translations ...

On Riemannian Curvature of Homogeneous Spaces

The Kinematic Formula in Riemannian Homogeneous Spaces

Let G be a Lie group and K a compact subgroup of G. Then the homogeneous space G/K has an invariant Riemannian metric and an invariant volume form ΩG. Let M and N be compact submanifolds of G/K, and I(M ∩ gN) an “integral invariant” of the intersection M ∩ gN . Then the integral

Higher Order Parallel Surfaces in Three-dimensional Homogeneous Spaces

We give a full classification of higher order parallel surfaces in three-dimensional homogeneous spaces with four-dimensional isometry group, i.e. in the so-called Bianchi-CartanVranceanu family. This gives a positive answer to a conjecture formulated in [2]. As a partial result, we prove that totally umbilical surfaces only exist if the space is a Riemannian product of a surface of constant Ga...

Homogeneous pseudo-Riemannian structures of class T2 on low-dimensional generalized symmetric spaces

We give a contribution about the study of homogeneous pseudoRiemannian structures on a type of homogeneous spaces, namely, the pseudo-Riemannian generalized symmetric spaces of low dimensions. It is well known that generalized symmetric Riemannian spaces admit homogeneous structures of the class T2 ⊕ T3. The same property holds in the pseudo-Riemannian context. The aim of the present paper is t...