A property of Wallach's flag manifolds

Teresa Arias-Marco

Archivum Mathematicum (2007)

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

Abstract

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In this note we study the Ledger conditions on the families of flag manifold ( M 6 = S U ( 3 ) / S U ( 1 ) × S U ( 1 ) × S U ( 1 ) , g ( c 1 , c 2 , c 3 ) ) , ( M 12 = S p ( 3 ) / S U ( 2 ) × S U ( 2 ) × S U ( 2 ) , g ( c 1 , c 2 , c 3 ) ) , 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 .

How to cite

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Arias-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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  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. Zbl1063.53042
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