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Metrics with homogeneous geodesics on ag manifolds

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Dmitri Alekseevsky, Andreas Arvanitoyeorgos

Metrics with homogeneous geodesics on ag 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 whenM =G/H is a naturally reductive space, that is theG-invariant metric g is defined by some non degenerate biinvariant symmetric bilinear form B, all geodesics ofM are homogeneous. We consider the case whenM =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 groupGwhich may admit a non-naturally reductiveG-invariant metric with homogeneous geodesics.

Keywords: homogeneous Riemannian spaces, homogeneous geodesics, flag mani- folds

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

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