Geodesic
Metrics with homogeneous geodesics on flag manifolds
Commentationes Mathematicae Universitatis Carolinae, Tome 43 (2002) no. 2, pp. 189-199 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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.
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.
Classification : 03E25, 14M15, 53C22, 53C30
Keywords: homogeneous Riemannian spaces; homogeneous geodesics; flag manifolds 
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     author = {Alekseevsky, Dmitri and Arvanitoyeorgos, Andreas},
     title = {Metrics with homogeneous geodesics on flag manifolds},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     pages = {189--199},
     year = {2002},
     volume = {43},
     number = {2},
     mrnumber = {1922121},
     zbl = {1090.53044},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMUC_2002_43_2_a0/}
}
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Alekseevsky, Dmitri; Arvanitoyeorgos, Andreas. Metrics with homogeneous geodesics on flag manifolds. Commentationes Mathematicae Universitatis Carolinae, Tome 43 (2002) no. 2, pp. 189-199. http://geodesic.mathdoc.fr/item/CMUC_2002_43_2_a0/