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.Comment: In memory of Stanley Hobert, 33 pages. Minor corrections and some simplification of Section 4.3.

Distributions admitting a local basis of homogeneous polynomials

The paper is a survey of several results by the authors, the main one of them being the following characterization of homogeneous algebraic distributions: Let us consider a vertical distribution D on the vector bundle p:E→M locally spanned by vertical vector fields X1,⋯,Xr. Let χ be the Liouville vector field of the vector bundle. Then there exists an r×r invertible matrix with smooth entries (cij) such that the vector fields Yj=∑ri=1cijXi, 1≤j≤r, are homogeneous algebraic of degree d if and only if there exists an r×r matrix A=(aij) of smooth functions given by [χ,Xj]=∑ri=1aijXi such that A restricted to the zero section of E is(d−1) times the identity matrix. Examples and applications are given

Homogeneous quaternionic Kähler structures on eight-dimensional non-compact quaternion-Kähler symmetric spaces

For each non-compact quaternion-Kähler symmetric space of dimension 8, all of its descriptions as a homogeneous Riemannian space (obtained through the Witte’s refined Langlands decomposition) and the associated homogeneous quaternionic Kähler structures are studied