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A note on a family of reductive Riemannian homogeneous spaces whose geodesic symmetries fail to be divergence-preserving. (English) Zbl 1063.53042
Fernández-Núñez, José (ed.) et al., Proceedings of the XI fall workshop on geometry and physics, Oviedo, Spain, September 23–25, 2002. Madrid: Real Sociedad Matemática Española (ISBN 84-933610-1-1/pbk). Publicaciones de la Real Sociedad Matemática Española 6, 35-45 (2004).
In this paper the authors show that in some flag manifolds in the quaternionic plane there exist reductive Riemannian homogeneous spaces, whose geodesic symmetries fail to be divergence preserving, and others which satisfy the D’Atri and Nickerson’s first necessary condition but not the second one. In fact they study the 12-dimensional manifold \(Sp(3)/SU(2)\times SU(2)\times SU(2)\). They give three invariant almost complex structures \(I,J,K\), such that both \(I,J\) are nearly Kähler and \(K\) is Hermitian. They do many computations with help of Mathematica to get an equation that implies the statement.
For the entire collection see [Zbl 1052.53004].

53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
53B21 Methods of local Riemannian geometry
53C30 Differential geometry of homogeneous manifolds