Comptes Rendus
no. 1-2
Differential Geometry
The constant osculating rank of the Wilking manifold V3
[Le rang osculateur constant de la variété de Wilking V3]

Présenté par : Thierry Aubin

Enrique Macías-Virgós1 ; Antonio M. Naveira2 ; Ana Tarrío3
1 Dpto. de Xeometría e Topoloxía, Facultade de Matemáticas, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Spain
2 Dpto. de Geometría y Topología, Facultad de Matemáticas, Avda. A. Estellés, No 1. 46100 Burjassot, Valencia, Spain
3 E. U. Arquitectura Técnica, Universidade de A Coruña, Campus A Zapateira, 15192 A Coruña, Spain
Comptes Rendus. Mathématique, Volume 346 (2008) no. 1-2, pp. 67-70.

Nous prouvons que le rang osculateur de la variété de Wilking V3=(SO(3)×SU(3))/U•(2) vaut 2, lorsque on considère la metrique g˜1. La connaissance du rang osculateur nous permet de resoudre l'équation différentielle des champs de vecteurs de Jacobi. Ces résultats peuvent être appliqués pour déterminer l'aire et le volume des sphères et boules géodesiques.

We prove that the osculating rank of the Wilking manifold V3=(SO(3)×SU(3))/U•(2), endowed with the metric g˜1, equals 2. The knowledge of the osculating rank allows us to solve the differential equation of the Jacobi vector fields. These results can be applied to determine the area and the volume of geodesic spheres and balls.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2007.11.009
Enrique Macías-Virgós 1 ; Antonio M. Naveira 2 ; Ana Tarrío 3

1 Dpto. de Xeometría e Topoloxía, Facultade de Matemáticas, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Spain
2 Dpto. de Geometría y Topología, Facultad de Matemáticas, Avda. A. Estellés, No 1. 46100 Burjassot, Valencia, Spain
3 E. U. Arquitectura Técnica, Universidade de A Coruña, Campus A Zapateira, 15192 A Coruña, Spain 
@article{CRMATH_2008__346_1-2_67_0,
     author = {Enrique Mac{\'\i}as-Virg\'os and Antonio M. Naveira and Ana Tarr{\'\i}o},
     title = {The constant osculating rank of the {Wilking} manifold $ {V}_{3}$},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {67--70},
     publisher = {Elsevier},
     volume = {346},
     number = {1-2},
     year = {2008},
     doi = {10.1016/j.crma.2007.11.009},
     language = {en},
}
TY  - JOUR
AU  - Enrique Macías-Virgós
AU  - Antonio M. Naveira
AU  - Ana Tarrío
TI  - The constant osculating rank of the Wilking manifold $ {V}_{3}$
JO  - Comptes Rendus. Mathématique
PY  - 2008
SP  - 67
EP  - 70
VL  - 346
IS  - 1-2
PB  - Elsevier
DO  - 10.1016/j.crma.2007.11.009
LA  - en
ID  - CRMATH_2008__346_1-2_67_0
ER  - 
%0 Journal Article
%A Enrique Macías-Virgós
%A Antonio M. Naveira
%A Ana Tarrío
%T The constant osculating rank of the Wilking manifold $ {V}_{3}$
%J Comptes Rendus. Mathématique
%D 2008
%P 67-70
%V 346
%N 1-2
%I Elsevier
%R 10.1016/j.crma.2007.11.009
%G en
%F CRMATH_2008__346_1-2_67_0
Enrique Macías-Virgós; Antonio M. Naveira; Ana Tarrío. The constant osculating rank of the Wilking manifold $ {V}_{3}$. Comptes Rendus. Mathématique, Volume 346 (2008) no. 1-2, pp. 67-70. doi : 10.1016/j.crma.2007.11.009. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.11.009/

[1] T. Arias-Marco, A.M. Naveira, The osculating rank of the Jacobi operator on g.o. spaces. A method for solving the Jacobi equation, preprint

[2] S. Aloff; N. Wallach An infinite family of 7-manifolds admitting positively curved Riemannian structures, Bull. Amer. Math. Soc., Volume 81 (1975), pp. 93-97

[3] M. Berger Les variétés riemanniennes homogènes normales simplement connexes à courbure strictement positive, Ann. Scuola Norm. Sup. Pisa, Volume 15 (1961), pp. 179-246

[4] M. Berger; P. Gauduchon; E. Mazet Le Spectre d'une Variété Riemannienne, Lecture Notes in Mathematics, vol. 194, Springer-Verlag, 1971

[5] I. Chavel On normal Riemannian homogeneous spaces of rank one, Bull. Amer. Math. Soc., Volume 73 (1967), pp. 477-481

[6] I. Chavel Isotropic Jacobi fields and Jacobi's Equations on Riemannian Homogeneous Spaces, Comment. Math. Helvet., Volume 42 (1967), pp. 237-248

[7] C. Gónzalez-Dávila, A.M. Naveira, Isotropic Jacobi fields on normal homogeneous spaces with positive sectional curvature, preprint

[8] A. Gray Tubes, Addison-Wesley, 1990

[9] A. Gray; L. Vanhecke Riemannian geometry as determined by the volumes of small geodesic balls, Acta Math., Volume 142 (1979), pp. 157-198

[10] S. Kobayashi; K. Nomizu Foundations of Differential Geometry II, Interscience, New York, 1969

[11] A.M. Naveira, A. Tarrío, A method for the resolution of the Jacobi equation Y″+RY=0 on the manifold Sp(2)/SU(2), preprint

[12] T. Püttmann Optimal pinching constants of odd dimensional homogeneous spaces, Invent. Math., Volume 138 (1999) no. 3, pp. 631-684

[13] K. Tojo Totally geodesic submanifolds of naturally reductive homogeneous spaces, Tsukuba J. Math., Volume 20 (1996), pp. 181-190

[14] K. Tsukada Totally geodesic submanifolds of Riemannian manifolds and curvature-invariant subspaces, Kodai Math. J., Volume 19 (1996), pp. 395-437

[15] B. Wilking The normal homogeneous space (SU(3)×SO(3))/U•(2) has positive sectional curvature, Proc. Amer. Math. Soc., Volume 127 (1999), pp. 1191-1194

Cité par Sources :

⁎ Work partially supported by Research Projects MTM2004-05082 (first author), MTM-2007-65852 (second author) and PGIDIT05PXIB16601PR (third author).

Commentaires - Politique

Ces articles pourraient vous intéresser

Non-unimodular Lie foliations

Enrique Macias-Virgós; P. Martín-Méndez

C. R. Math (2005)