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A property of Wallach's flag manifolds

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*
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.