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 (Wallach, N. R., Compact homogeneous Riemannian manifols with strictly positive curvature, Ann. of Math. 96 (1972), 276–293.). 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 (D’Atri, J. E., Nickerson, H. K., Geodesic symmetries in spaces with special curvature tensors, J. Differenatial Geom. 9 (1974), 251–262.). Moreover, we correct and improve the result given by the author and A. M. Naveira in (Arias-Marco, T., Naveira, A. M., A note on a family of reductive Riemannian homogeneous spaces whose geodesic symmetries fail to be divergence-preserving, Proceedings of the XI Fall Workshop on Geometry and Physics. Publicaciones de la RSME 6 (2004), 35–45.) about $M^{12}$.
@article{108076,
author = {Teresa Arias-Marco},
title = {A property of Wallach's flag manifolds},
journal = {Archivum Mathematicum},
volume = {043},
year = {2007},
pages = {307-319},
zbl = {1199.53092},
mrnumber = {2381780},
language = {en},
url = {http://dml.mathdoc.fr/item/108076}
}
Arias-Marco, Teresa. A property of Wallach's flag manifolds. Archivum Mathematicum, Tome 043 (2007) pp. 307-319. http://gdmltest.u-ga.fr/item/108076/
The classification of 4-dimensional homogeneous D’Atri spaces revisited, Differential Geometry and its Applications 25 (2007), 29–34. | MR 2293639 | Zbl 1121.53026
The classification of 4-dimensional homogeneous D’Atri spaces, to appear in Czechoslovak Math. J. | MR 2402535
A note on a family of reductive Riemannian homogeneous spaces whose geodesic symmetries fail to be divergence-preserving, Proceedings of the XI Fall Workshop on Geometry and Physics. Publicaciones de la RSME 6 (2004), 35–45. | Zbl 1063.53042
Three- and Four-dimensional Einstein-like manifolds and homogeneity, Geom. Dedicata 75 (1999), 123–136. (1999) | MR 1686754 | Zbl 0944.53026
Geodesic spheres and symmetries in naturally reductive homogeneous spaces, Michigan Math. J. 22 (1975), 71–76. (1975) | MR 0372786
Divergence preserving geodesic symmetries, J. Differential Geom. 3 (1969), 467–476. (1969) | MR 0262969 | Zbl 0195.23604
Geodesic symmetries in spaces with special curvature tensors, J. Differenatial Geom. 9 (1974), 251–262. (1974) | MR 0394520 | Zbl 0285.53019
The sixteen classes of almost Hermitian manifolds and their linear invariants, Ann. Mat. Pura Appl. 123(4) (1980), 35–58. (1980) | MR 0581924 | Zbl 0444.53032
Foundations of Differential Geometry, Vols. I and II, Interscience, New York, 1963 and 1969. (1963) | MR 0152974
Spaces with volume-preserving symmetries and related classes of Riemannian manifolds, Rend. Sem. Mat. Univ. Politec. Torino, Fascicolo Speciale, (1983), 131–158. (1983) | MR 0829002 | Zbl 0631.53033
D’Atri Spaces, Progr. Nonlinear Differential Equations Appl. 20 (1996), 241–284. (1996) | MR 1390318 | Zbl 0862.53039
Four-dimensional Einstein-like manifolds and curvature homogeneity, Geom. Dedicata 54 (1995), 225–243. (1995) | MR 1326728 | Zbl 0835.53056
Spectral theory for operator families on Riemannian manifolds, Proc. Sympos. Pure Math. 54(3) (1993), 615–665. (1993) | MR 1216651
Compact homogeneous Riemannian manifols with strictly positive curvature, Ann. of Math. 96 (1972), 276–293. (1972) | MR 0307122
Homogeneous spaces defined by Lie group automorphisms, I, J. Differential Geom. 2 (1968), 77–114, 115–159. (1968) | MR 0236328 | Zbl 0169.24103
Homogeneous spaces defined by Lie group automorphisms, II, J. Differential Geom. 2 (1968), 115–159. (1968) | MR 0236329 | Zbl 0182.24702