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.
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