Geodesic
A property of Wallach's flag manifolds
Archivum mathematicum, Tome 43 (2007) no. 5, pp. 307-319 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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 $M^{12}$.
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 $M^{12}$.
Classification : 53B21, 53C21, 53C25, 53C30, 53Cxx
Keywords: Riemannian manifold; naturally reductive Riemannian homogeneous space; D’Atri space; flag manifold 
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     title = {A property of {Wallach's} flag manifolds},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ARM_2007_43_5_a1/}
}
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Arias-Marco, Teresa. A property of Wallach's flag manifolds. Archivum mathematicum, Tome 43 (2007) no. 5, pp. 307-319. http://geodesic.mathdoc.fr/item/ARM_2007_43_5_a1/

[1] Arias-Marco T.: The classification of 4-dimensional homogeneous D’Atri spaces revisited. Differential Geometry and its Applications 25 (2007), 29–34. | MR | Zbl

[2] Arias-Marco T., Kowalski O.: The classification of 4-dimensional homogeneous D’Atri spaces. to appear in Czechoslovak Math. J. | MR

[3] 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. | Zbl

[4] Bueken P., Vanhecke L.: Three- and Four-dimensional Einstein-like manifolds and homogeneity. Geom. Dedicata 75 (1999), 123–136. | MR | Zbl

[5] D’Atri J. E.: Geodesic spheres and symmetries in naturally reductive homogeneous spaces. Michigan Math. J. 22 (1975), 71–76. | MR

[6] D’Atri J. E., Nickerson H. K.: Divergence preserving geodesic symmetries. J. Differential Geom. 3 (1969), 467–476. | MR | Zbl

[7] D’Atri J. E., Nickerson H. K.: Geodesic symmetries in spaces with special curvature tensors. J. Differenatial Geom. 9 (1974), 251–262. | MR | Zbl

[8] Gray A., Hervella L. M.: The sixteen classes of almost Hermitian manifolds and their linear invariants. Ann. Mat. Pura Appl. 123(4) (1980), 35–58. | MR | Zbl

[9] Kobayashi S., Nomizu K.: Foundations of Differential Geometry, Vols. I and II. Interscience, New York, 1963 and 1969. | MR

[10] Kowalski O.: Spaces with volume-preserving symmetries and related classes of Riemannian manifolds. Rend. Sem. Mat. Univ. Politec. Torino, Fascicolo Speciale, (1983), 131–158. | MR | Zbl

[11] Kowalski O., Prüfer F., Vanhecke L.: D’Atri Spaces. Progr. Nonlinear Differential Equations Appl. 20 (1996), 241–284. | MR | Zbl

[12] Podestà F., Spiro A.: Four-dimensional Einstein-like manifolds and curvature homogeneity. Geom. Dedicata 54 (1995), 225–243. | MR | Zbl

[13] Szabó Z. I.: Spectral theory for operator families on Riemannian manifolds. Proc. Sympos. Pure Math. 54(3) (1993), 615–665. | MR

[14] Wallach N. R.: Compact homogeneous Riemannian manifols with strictly positive curvature. Ann. of Math. 96 (1972), 276–293. | MR

[15] Wolf J., Gray A.: Homogeneous spaces defined by Lie group automorphisms, I. J. Differential Geom. 2 (1968), 77–114, 115–159. | MR | Zbl

Wolf J., Gray A.: Homogeneous spaces defined by Lie group automorphisms, II. J. Differential Geom. 2 (1968), 115–159. | MR | Zbl