Riemannian geodesic orbit spaces (G/H,g) are natural generalizations of symmetric spaces, defined by the property that their geodesics are orbits of one-parameter subgroups of G. We study the geodesic orbit spaces of the form (G/S,g), where G is a compact, connected, semisimple Lie group and S is abelian. We give a simple geometric characterization of those spaces, namely that they are naturally reductive. In turn, this yields the classification of the invariant geodesic orbit (and also the naturally reductive) metrics on any space of the form G/S. Our approach involves simplifying the intricate parameter space of geodesic orbit metrics on G/S by reducing their study to certain submanifolds and generalized flag manifolds, and by studying properties of root systems of simple Lie algebras associated to these manifolds.

中文翻译：

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一类具有阿贝尔各向同性子群的测地轨道空间

黎曼测地线轨道空间 (G/H,g) 是对称空间的自然推广，由它们的测地线是 G 的一个参数子群的轨道的性质定义。我们研究了形式 (G/S,g) 的测地线轨道空间)，其中 G 是紧致、连通、半单李群，S 是阿贝尔群。我们给出了这些空间的简单几何特征，即它们是自然还原的。反过来，这会产生 G/S 形式的任何空间上的不变测地轨道（以及自然还原）度量的分类。我们的方法涉及通过将研究减少到某些子流形和广义标志流形，并通过研究与这些流形相关的简单李代数的根系统的性质，来简化 G/S 上测地轨道度量的复杂参数空间。