On the projective Randers metrics

Mehdi Rafie-Rad; Bahman Rezaei

doi:10.1016/j.crma.2012.02.010

Differential Geometry/Mathematical Physics

On the projective Randers metrics
[Sur les métriques de Randers projectives]

Présenté par : the Editorial Board

Mehdi Rafie-Rad ^1,² ; Bahman Rezaei ³

¹ Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran
² School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5746, Tehran, Iran
³ Department of Mathematics, Faculty of Sciences, Urmia University, Urmia, Iran

Comptes Rendus. Mathématique, Volume 350 (2012) no. 5-6, pp. 281-283.

Résumé
Abstract

On démontre quʼune métrique de Randers $F = α + β$ sur une variété de dimension $n ⩾ 3$ est projective si et seulement si lʼalgèbre de Lie des champs de vecteurs projectifs $p (M, F)$ est (localement) de dimension $n (n + 2)$ . Ceci peut être considéré comme un analogue du résultat correspondant en géométrie riemannienne.

It is proved that a Randers metric $F = α + β$ on a manifold of dimension $n ⩾ 3$ is projective if and only if the Lie algebra of projective vector fields $p (M, F)$ has (locally) dimension $n (n + 2)$ . This can be regarded as an analogue of the corresponding result in Riemannian geometry.

Reçu le : 2011-11-06
Accepté le : 2012-02-28
Publié le : 2012-03-01

DOI : 10.1016/j.crma.2012.02.010

Affiliations des auteurs :

Mehdi Rafie-Rad ^{1, 2} ; Bahman Rezaei ³

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Mehdi Rafie-Rad; Bahman Rezaei. On the projective Randers metrics. Comptes Rendus. Mathématique, Volume 350 (2012) no. 5-6, pp. 281-283. doi : 10.1016/j.crma.2012.02.010. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2012.02.010/

Bibliographie
Cité par

[1] H. Akbar-Zadeh Champs de vecteurs projectifs sur le fibré unitaire, J. Math. Pures Appl., Volume 65 (1986), pp. 47-79

[2] D. Bao; Z. Shen Finsler metrics of constant positive curvature on the Lie group $S^{3}$ , J. Lond. Math. Soc., Volume 66 (2002), pp. 453-467

[3] X. Chen; X. Mo; Z. Shen On the flag curvature of Finsler metrics of scalar curvature, J. Lond. Math. Soc., Volume 68 (2003), pp. 762-780

[4] V.S. Matveev Geometric explanation of Beltrami theorem, Int. J. Geom. Methods Mod. Phys., Volume 3 (2006) no. 3, pp. 623-629

[5] V.S. Matveev On the dimension of the group of projective transformations of closed Randers and Riemannian manifolds, SIGMA, Volume 8 (2012), p. 007 (4 pages)

[6] B. Najafi; A. Tayebi A new quantity in Finsler geometry, C. R. Acad. Sci. Paris, Ser. I, Volume 349 (2011) no. 1–2, pp. 81-83

[7] M. Rafie-Rad Some new characterizations of projective Randers metrics with constant S-curvature, J. Geom. Phys., Volume 6 (2012) no. 2, pp. 272-278

[8] M. Rafie-Rad Special projective algebra Randers metrics of constant S-curvature, Int. J. Geom. Methods Mod. Phys., Volume 9 (2012) no. 4

[9] M. Rafie-Rad; B. Rezaei On the projective algebra of Randers metrics of constant flag curvature, SIGMA, Volume 7 (2011), p. 085 (12 pages)

[10] G. Randers On an asymmetric metric in the four-space of general relativity, Phys. Rev., Volume 59 (1941), pp. 195-199

[11] Z. Shen Differential Geometry of Spray and Finsler Spaces, Kluwer Academic Publishers, 2001

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