On geodesic graphs of Riemannian g.o. spaces

Abstract

We study homogeneous Riemannian manifolds (GaHY g) on which every geodesic is an orbit of a one-parameter subgroup of G. We analyze the algebraic structure of certain minimal sets of vectors of the corresponding Lie algebra g (called ªgeodesic graphsº) which generate all geodesics through a fixed point. We are particularly interested in the case when the geodesic graphs are of non-linear character. Some structural theorems, many examples and also open problems are presented.

Homogeneity and plane-wave limits

“…This vacuum of minimal five-dimensional supergravity was discovered in [32]. To determine the limit we employ the formulae (28) and (30). We find that This vector is not geodetic, however, unless we add −Y 0 , as in the five-dimensional Gödel universe.…”

“…In Section 4.4 we will discuss an example of a g.o. space, a six-dimensional lorentzian manifold of the type first considered by Kaplan (see, for example, [18]). In a g.o.…”

“…(3) Finally we determine the explicit form of the plane-wave metric. To do this we choose a frame u, V, Y a for m such that u, V = 1 and Y a , Y b = δ ab and then compute the matrices F and A using formulae (28) and (30), respectively.…”

“…The lorentzian version of Kaplan's g.o. space (see, for example, [18]) is a six-dimensional 2-step nilpotent Lie group M with a left invariant metric. The Lie algebra m is spanned by X i for i = 1, .…”

“…It is now a simple exercise to use equations (28) and (30) to show that the plane-wave limit along u 1 is actually flat.…”