Geodesic symmetries of homogeneous K�hler Manifolds

Abstract

D'Atri and Nickerson [6], [7] have given necessary conditions for the geodesic symmetries of a Riemannian manifold to preserve the volume element. We use their results to show that ifG is a compact simple Lie group,T is a maximal torus ofG, andG/T is not symmetric, then anyG-invariant Kahler metric onG/T does not have volume-preserving geodesic symmetries. From the Kahler/de Rham decomposition of a compact homogeneous Kahler manifold [8], our result extends to the invariant Kahler metrics on a quotient of a compact connected Lie group by a maximal torus. In proving these results we compute directly the Ricci tensor of anyG-invariant Kahler metric onG/T forG compact connected andT a maximal torus ofG. The result is an explicit formula giving the value of the Ricci tensor elements in terms of the root structure of the Lie algebra ofG.