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A property of Wallach’s flag manifolds. (English) Zbl 1199.53092
Summary: 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 [Ann. Math. (2) 96, 277–295 (1972; Zbl 0261.53033)]. 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 J. E. D’Atri and H. K. Nickerson in [J. Differ. Geom. 9, 251–262 (1974; Zbl 0285.53019)]. Moreover, we correct and improve the result given by the author and A. M. Naveira in [Publicaciones de la Real Sociedad Matemática Española 6, 35–45 (2004; Zbl 1063.53042)] about $$M^{12}$$.
##### MSC:
 53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions 53B21 Methods of local Riemannian geometry 53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.) 53C30 Differential geometry of homogeneous manifolds
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