We construct homogeneous flat pseudo-Riemannian manifolds with non-abelian fundamental group. In the compact case, all homogeneous flat pseudo-Riemannian manifolds are complete and have abelian linear holonomy group. To the contrary, we show that there do exist non-compact and noncomplete examples, where the linear holonomy is non-abelian, starting in dimensions ≥ 8, which is the lowest possible dimension. We also construct a complete flat pseudo-Riemannian homogeneous manifold of dimension 14 with non-abelian linear holonomy. Furthermore, we derive a criterion for the properness of the action of an affine transformation group with transitive centralizer.

Let M be a compact connected pseudo-Riemannian manifold on which a solvable connected Lie group G of isometries acts transitively. We show that G acts almost freely on M and that the metric on M is induced by a bi-invariant pseudo-Riemannian metric on G. Furthermore, we show that the identity component of the isometry group of M coincides with G.

Abstract. We show that every n-dimensional locally homogeneous pp-wave is a plane wave, provided it is indecomposable and its curvature operator, when acting on 2-forms, has rank greater than one. As a consequence we obtain that indecomposable, Ricci-flat locally homogeneous pp-waves are plane waves. This generalises a classical result by Jordan, Ehlers and Kundt in dimension 4. Several examples show that our assumptions on indecomposability and the rank of the curvature are essential. Background and main resultsA semi-Riemannian manifold (M, g) is homogeneous if it admits a transitive action by a group of isometries. This means that for each pair of points p and q in M there is an isometry of (M, g) that maps p to q. In the spirit of Felix Klein's Erlanger Programm to characterise geometries by their symmetry group, homogeneous manifolds are fundamental building blocks in geometry. Homogeneity is strongly tied to the geometry and the curvature of a manifold. For example, homogeneous Riemannian manifolds are geodescially complete, and, as an example for the link to curvature, we recall the celebrated result that any Ricci-flat homogeneous Riemannian manifold is flat [3]. A weaker version of homogeneity which still guarantees that the manifold looks the same everywhere is local homogeneity: a semi-Riemannian manifold is locally homogeneous if for each pair of points p and q in M there is an isometry defined on a neighbourhood of p that maps p to q.Here we will study local homogeneity for a certain class of Lorentzian manifolds, the so-called pp-waves and the plane waves. Locally, an (n + 2)-dimensional pp-wave admits coordinates (x − , x 1 , . . . , x n , x + ) such that (1.1) g := 2dxwhere H = H(x 1 , . . . , x n , x + ) is a function not depending on x − . For a plane wave, this function is required to be quadratic in the x i 's with x + -dependent coefficients. In general, they are not homogeneous, but they admit a parallel null (i.e. non-zero and light-like) vector field. An invariant definition of pp-waves and plane waves is given as follows: A Lorentzian manifold (M, g) is a pp-wave if it admits a parallel null vector field V ∈ Γ(T M), i.e., V = 0, g(V, V ) = 0 and ∇V = 0, and if its curvature endomorphism R : Λ 2 T M → Λ 2 T M is non-zero and satisfies where V ⊥ := {X ∈ T M | g(X, V ) = 0}. A plane wave is a pp-wave with the additional conditionFour-dimensional pp-waves were discovered in a mathematical context by Brinkmann [7] as one class of Einstein spaces that can be mapped conformally onto each other. In physics, plane waves and pp-waves appeared in general relativity [11], where they continue to play an important role (see for example [6,17] for more references) as metrics for which the Einstein equations become linear and, when they solve these equations, describe the propagation of gravitational waves with flat surfaces as wave fronts. Later Penrose discovered that when "zooming in on null geodesics" every space-times has a plane wave as limit [22]. More recently, the conditions under which the hom...

Abstract. Let M be pseudo-Riemannian homogeneous Einstein manifold of finite volume, and suppose a connected Lie group G acts transitively and isometrically on M . In this situation, the metric on M induces a bilinear form ⟨⋅, ⋅⟩ on the Lie algebra g of G which is nil-invariant, a property closely related to invariance. We study such spaces M in three important cases. First, we assume ⟨⋅, ⋅⟩ is invariant, in which case the Einstein property requires that G is either solvable or semisimple. Next, we investigate the case where G is solvable. Here, M is compact and M = G Γ for a lattice Γ in G. We show that in dimensions less or equal to 7, compact quotients M = G Γ exist only for nilpotent groups G. We conjecture that this is true for any dimension. In fact, this holds if Schanuel's Conjecture on transcendental numbers is true. Finally, we consider semisimple Lie groups G, and find that M splits as a pseudo-Riemannian product of Einstein quotients for the compact and the non-compact factors of G.

The family N of n-variate normal distributions is parameterized by the cone of positive definite symmetric n × n-matrices and the n-dimensional real vector space. Equipped with the Fisher information metric, N becomes a Riemannian manifold. As such, it is diffeomorphic, but not isometric, to the Riemannian symmetric space Pos 1 (n+1, R) of unimodular positive definite symmetric (n +1)×(n +1)-matrices. As the computation of distances in the Fisher metric for n > 1 presents some difficulties, Lovrič et al. (J Multivar Anal 74:36-48, 2000) proposed to use the Killing metric on Pos 1 (n+1, R) as an alternative metric in which distances are easier to compute. In this work, we survey the geometric properties of the space N and provide a quantitative analysis of the defect of certain geodesics for the Killing metric to be geodesics for the Fisher metric. We find that for these geodesics the use of the Killing metric as an approximation for the Fisher metric is indeed justified for long distances. Keywords Gaussian distributions • Fisher metric • Cone of positive definite matrices • Symmetric spaces • Geodesics.