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Riemannian flag manifolds with homogeneous geodesics. (English) Zbl 1148.53038
In this paper a complete classification of Riemannian flag manifolds with homogeneous geodesics is obtained. A Riemannian flag manifold is an adjoint orbit \(G/K\) of a compact semisimple Lie group \(G\) carrying an invariant Riemannian metric. A geodesic is called homogeneous if it is an orbit of an one-parameter subgroup of the Lie group \(G.\) The authors introduce a necessary condition for the manifold to admit a non-standard \(G\)-invariant metric with homogeneous geodesics. It is shown that among all flag manifolds \(G/K\) of a simple Lie group \(G\) only the manifold \(\text{Com}(\mathbb{R}^{2l+2}= \text{SO} (2l+1)/ U(l)\) of complex structures in \(\mathbb{R}^{2l+2}\) and the complex projective space \(\mathbb{C}P^{2l-1}= \text{Sp}(l)/U(1) \cdot \text{Sp}(l-1)\) admit a non-naturally reductive invariant metric with homogeneous geodesics.

MSC:
53C30 Differential geometry of homogeneous manifolds
53C22 Geodesics in global differential geometry
14M15 Grassmannians, Schubert varieties, flag manifolds
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