Homogeneous Geodesics Research Articles

Penrose limits of homogeneous spaces

We prove that the Penrose limit of a spacetime along a homogeneous geodesic is a homogeneous plane wave spacetime and that the Penrose limit of a reductive homogeneous spacetime along a homogeneous geodesic is a Cahen–Wallach space. We then consider several homogenous examples to show that these results are indeed sharp and conclude with a remark about the existence of null homogeneous geodesics.

Homogeneity and plane-wave limits

We explore the plane-wave limit of homogeneous spacetimes. For plane-wave limits along homogeneous geodesics the limit is known to be homogeneous and we exhibit the limiting metric in terms of Lie algebraic data. This simplifies many calculations and we illustrate this with several examples. We also investigate the behaviour of (reductive) homogeneous structures under the plane-wave limit.

Homogeneous geodesics in solvable Lie groups

After the second author and J. Szenthe [10] proved that every homogeneous Riemannian manifold admits a homogeneous geodesic, several authors studied the set of all homogeneous geodesics in various homogeneous spaces. In this paper, we consider special examples of homogeneous spaces of solvable type of arbitrary odd dimension given in [1] and [7] and we show that their sets of homogeneous geodesics have an interesting structure, closely connected to the notion of Hadamard matrices.

On the Existence of Homogeneous Geodesics in Homogeneous Riemannian Manifolds

It is proved that every homogeneous Riemannian manifold admits a geodesic which is an orbit of a one-parameter group of isometries.