Geodesic Orbit Riemannian Structures on $\mathbf{R}^n$

Abstract

A geodesic orbit manifold is a complete Riemannian manifold all of whose geodesics are orbits of one-parameter groups of isometries. We give both a geometric and an algebraic characterization of geodesic orbit manifolds that are diffeomorphic to $\mathbf{R}^n$. Along the way, we establish various structural properties of more general geodesic orbit manifolds.