Article
Full entry |
PDF
(0.2 MB)
Feedback
PDF
(0.2 MB)
Feedback
Keywords:
Riemannian manifold; naturally reductive Riemannian homogeneous space; D’Atri space; flag manifold
Riemannian manifold; naturally reductive Riemannian homogeneous space; D’Atri space; flag manifold
Summary:
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 $M^{12}$.
References:
[1] Arias-Marco T.: The classification of 4-dimensional homogeneous D’Atri spaces revisited. Differential Geometry and its Applications 25 (2007), 29–34. MR 2293639 | Zbl 1121.53026
[2] Arias-Marco T., Kowalski O.: The classification of 4-dimensional homogeneous D’Atri spaces. to appear in Czechoslovak Math. J. MR 2402535
[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 1063.53042
[4] Bueken P., Vanhecke L.: Three- and Four-dimensional Einstein-like manifolds and homogeneity. Geom. Dedicata 75 (1999), 123–136. MR 1686754 | Zbl 0944.53026
[5] D’Atri J. E.: Geodesic spheres and symmetries in naturally reductive homogeneous spaces. Michigan Math. J. 22 (1975), 71–76. MR 0372786
[6] D’Atri J. E., Nickerson H. K.: Divergence preserving geodesic symmetries. J. Differential Geom. 3 (1969), 467–476. MR 0262969 | Zbl 0195.23604
[7] D’Atri J. E., Nickerson H. K.: Geodesic symmetries in spaces with special curvature tensors. J. Differenatial Geom. 9 (1974), 251–262. MR 0394520 | Zbl 0285.53019
[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 0581924 | Zbl 0444.53032
[9] Kobayashi S., Nomizu K.: Foundations of Differential Geometry, Vols. I and II. Interscience, New York, 1963 and 1969. MR 0152974
[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 0829002 | Zbl 0631.53033
[11] Kowalski O., Prüfer F., Vanhecke L.: D’Atri Spaces. Progr. Nonlinear Differential Equations Appl. 20 (1996), 241–284. MR 1390318 | Zbl 0862.53039
[12] Podestà F., Spiro A.: Four-dimensional Einstein-like manifolds and curvature homogeneity. Geom. Dedicata 54 (1995), 225–243. MR 1326728 | Zbl 0835.53056
[13] Szabó Z. I.: Spectral theory for operator families on Riemannian manifolds. Proc. Sympos. Pure Math. 54(3) (1993), 615–665. MR 1216651
[14] Wallach N. R.: Compact homogeneous Riemannian manifols with strictly positive curvature. Ann. of Math. 96 (1972), 276–293. MR 0307122
[15] Wolf J., Gray A.: Homogeneous spaces defined by Lie group automorphisms, I. J. Differential Geom. 2 (1968), 77–114, 115–159. MR 0236328 | Zbl 0169.24103
Wolf J., Gray A.: Homogeneous spaces defined by Lie group automorphisms, II. J. Differential Geom. 2 (1968), 115–159. MR 0236329 | Zbl 0182.24702
Search
Partner of
