## Dmitri Alekseevsky, Andreas Arvanitoyeorgos

*Metrics with homogeneous geodesics on flag manifolds *

Comment.Math.Univ.Carolinae 43,2 (2002) 189-199. **Abstract:**A geodesic of a homogeneous Riemannian manifold $(M=G/K, g)$ is called homogeneous if it is an orbit of an one-parameter subgroup of $G$. In the case when $M=G/H$ is a naturally reductive space, that is the $G$-invariant metric $g$ is defined by some non degenerate biinvariant symmetric bilinear form $B$, all geodesics of $M$ are homogeneous. We consider the case when $M=G/K$ is a flag manifold, i.e. an adjoint orbit of a compact semisimple Lie group $G$, and we give a simple necessary condition that $M$ admits a non-naturally reductive invariant metric with homogeneous geodesics. Using this, we enumerate flag manifolds of a classical Lie group $G$ which may admit a non-naturally reductive $G$-invariant metric with homogeneous geodesics.

**Keywords:** homogeneous Riemannian spaces, homogeneous geodesics, flag manifolds

**AMS Subject Classification:** Primary 53C22, 53C30; Secondary 14M15

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