## A metric proof that $\delta $-homogeneous manifolds are geodesic orbit manifolds

### Volume 165 / 2021

#### Abstract

A Riemannian manifold $(M,g)$ is called $\delta $-homogeneous if for any pair of points $p,q\in M$ there is an isometry $f$ such that $f(p)=q$, and such that the points $p,q$ have maximal displacement among all pairs $x,f(x)$ with respect to the Riemannian distance. A result of V. N. Berestovskiĭ and Yu. G. Nikonorov states that any $\delta $-homogeneous manifold $(M,g)$ is a geodesic orbit manifold, i.e. all geodesics in $(M,g)$ are orbits of one-parameter subgroups of isometries. In this paper we give a simple proof of this result, based on a recent metric characterization of geodesics that are orbits.