The constant osculating rank of the Wilking manifold $ {V}_{3}$

Enrique Macías-Virgós; Antonio M. Naveira; Ana Tarrío

doi:10.1016/j.crma.2007.11.009

Differential Geometry

The constant osculating rank of the Wilking manifold

V_{3}

Presented by: Thierry Aubin

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

Comptes Rendus. Mathématique, Volume 346 (2008) no. 1-2, pp. 67-70.

Abstract
Résumé

We prove that the osculating rank of the Wilking manifold $V_{3} = (SO (3) \times SU (3)) / U^{•} (2)$ , endowed with the metric ${\tilde{g}}_{1}$ , 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.

Nous prouvons que le rang osculateur de la variété de Wilking $V_{3} = (SO (3) \times SU (3)) / U^{•} (2)$ vaut 2, lorsque on considère la metrique ${\tilde{g}}_{1}$ . 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.

Received: 2007-09-04
Accepted: 2007-11-08
Published online: 2008-01-01

DOI: 10.1016/j.crma.2007.11.009

Author's affiliations:

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

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     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}$},
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     pages = {67--70},
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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/

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Cited by Sources:

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

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