# A property of Wallach's flag manifolds

Archivum Mathematicum (2007)

- Volume: 043, Issue: 5, page 307-319
- ISSN: 0044-8753

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topArias-Marco, Teresa. "A property of Wallach's flag manifolds." Archivum Mathematicum 043.5 (2007): 307-319. <http://eudml.org/doc/250156>.

@article{Arias2007,

abstract = {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 $M^\{12\}$.},

author = {Arias-Marco, Teresa},

journal = {Archivum Mathematicum},

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},

language = {eng},

number = {5},

pages = {307-319},

publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},

title = {A property of Wallach's flag manifolds},

url = {http://eudml.org/doc/250156},

volume = {043},

year = {2007},

}

TY - JOUR

AU - Arias-Marco, Teresa

TI - A property of Wallach's flag manifolds

JO - Archivum Mathematicum

PY - 2007

PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno

VL - 043

IS - 5

SP - 307

EP - 319

AB - 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 $M^{12}$.

LA - eng

KW - Riemannian manifold; naturally reductive Riemannian homogeneous space; D’Atri space; flag manifold; Riemannian manifold; naturally reductive Riemannian homogeneous space; D'Atri space; flag manifold

UR - http://eudml.org/doc/250156

ER -

## References

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