A property of Wallach’s flag manifolds. (English) Zbl 1199.53092

Arias-Marco, Teresa

Arch. Math., Brno 43, No. 5, 307-319 (2007).

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

Keywords:

Riemannian manifold; naturally reductive Riemannian homogeneous space; D’Atri space; flag manifold

Citations:

Zbl 0261.53033; Zbl 0285.53019; Zbl 1063.53042