Comptes Rendus
no. 5-6
Geometry
On a class of Riemannian metrics arising from Finsler structures
[Sur une classe de métriques riemanniennes issues de structures finsleriennes]

Présenté par : Jean-Pierre Demailly

Akbar Tayebi1 ; Esmaeil Peyghan2
1 Department of Mathematics, Faculty of Science, Qom University, Qom, Iran
2 Department of Mathematics, Faculty of Science, University of Arak, Arak, Iran
Comptes Rendus. Mathématique, Volume 349 (2011) no. 5-6, pp. 319-322.

Sur le fibré tangent dʼune variéte finslerienne, nous introduisons une certaines classe de métriques et étudions la relation entre la connexion de Levi-Civita, la connexion de Vaisman, et les espaces de Reinhart. Nous montrons que les connexions de Levi-Civita et de Vaisman induisent les mêmes connexions dans le fibré structurel si seulement si la variété de base est de Landsberg. En outre, toute varété de Reinhart feuilletée se réduit à une variété riemannienne.

On the slit tangent bundle of Finsler manifolds, we introduce a class of metrics and study the relation between Levi-Civita connection, Vaisman connection, vertical foliation, and Reinhart spaces. We show that the Levi-Civita and the Vaisman connections induce the same connections in the structural bundle if and only if the base manifold is Landsbergian. Moreover every foliated Reinhart manifold reduces to a Riemannian manifold.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2011.01.021
Akbar Tayebi 1 ; Esmaeil Peyghan 2

1 Department of Mathematics, Faculty of Science, Qom University, Qom, Iran
2 Department of Mathematics, Faculty of Science, University of Arak, Arak, Iran 
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Akbar Tayebi; Esmaeil Peyghan. On a class of Riemannian metrics arising from Finsler structures. Comptes Rendus. Mathématique, Volume 349 (2011) no. 5-6, pp. 319-322. doi : 10.1016/j.crma.2011.01.021. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2011.01.021/

[1] K.M.T. Abbassi; O. Kowalski On Einstein Riemannian g-natural metrics on unit tangent sphere bundles, Ann. Global Anal. Geom., Volume 38 (2010) no. 1, pp. 11-20

[2] A. Bejancu Finsler Geometry and Applications, Ellis Horwood, New York, 1990

[3] A. Bejancu; H.R. Farran A geometric characterization of Finsler manifolds of constant curvature K=1, Internat. J. Math. Math. Sci., Volume 23 (2000), pp. 399-407

[4] A. Bejancu; H.R. Farran Finsler metrics of positive constant flag curvature on Sasakian space forms, Hokkaido Math. J., Volume 31 (2002), pp. 459-468

[5] A. Bejancu; H.R. Farran Foliation and Geometrics Structures, Springer, 2006

[6] A. Bejancu; H.R. Farran Finsler geometry and natural foliations on the tangent bundle, Rep. Math. Phys., Volume 58 (2006), pp. 131-146

[7] A. Bejancu; H.R. Farran On totally geodesic foliations with bundle-like metric, J. Geom., Volume 85 (2006), pp. 7-14

[8] V. Oproiu; N. Papaghiuc A Kähler structure on the non-zero tangent bundle of a space form, Diff. Geom. Appl., Volume 11 (1999), pp. 1-12

[9] I. Vaisman Cohomology and Differential Forms, Marcel Dekker Inc., New York, 1973

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