A property of Wallach's flag manifolds

Teresa Arias-Marco

Address. Department of Geometry and Topology, University of Valencia
Vicente Andres, Estelles 1, 46100 Burjasot, Valencia, Spain

E-mail: Teresa.Arias@uv.es

Abstract. In this note we study the Ledger conditions on the families of flag manifold $(M^{6}=SU(3)/SU(1)\times SU(1) \times SU(1), g_{(c_1,c_2,c_3)})$, $\big(M^{12}=Sp(3)/SU(2) \times SU(2) \times SU(2), g_{(c_1,c_2,c_3)}\big)$, constructed by N.\,R. Wallach in \cite{W}. In both cases, we conclude that every member of the both families of Riemannian flag manifolds is a D'Atri space if and only if it is naturally reductive. Therefore, we finish the study of $M^6$ made by D'Atri and Nickerson in \cite{D'A-N2}. Moreover, we correct and improve the result given by the author and A. M. Naveira in \cite{AM-N1} about $M^{12}$.

AMSclassification. 53C21, 53B21, 53C25, 53C30.

Keywords. Riemannian manifold, naturally reductive Riemannian homogeneous space, D'Atri space, flag manifold.