On the projective Randers metrics

Rafie-Rad, Mehdi; Rezaei, Bahman

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

Rafie-Rad, Mehdi ^1,² ; Rezaei, Bahman 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, Tome 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 :

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

@article{CRMATH_2012__350_5-6_281_0,
     author = {Rafie-Rad, Mehdi and Rezaei, Bahman},
     title = {On the projective {Randers} metrics},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {281--283},
     publisher = {Elsevier},
     volume = {350},
     number = {5-6},
     year = {2012},
     doi = {10.1016/j.crma.2012.02.010},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2012.02.010/}
}

TY  - JOUR
AU  - Rafie-Rad, Mehdi
AU  - Rezaei, Bahman
TI  - On the projective Randers metrics
JO  - Comptes Rendus. Mathématique
PY  - 2012
SP  - 281
EP  - 283
VL  - 350
IS  - 5-6
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2012.02.010/
DO  - 10.1016/j.crma.2012.02.010
LA  - en
ID  - CRMATH_2012__350_5-6_281_0
ER  -

%0 Journal Article
%A Rafie-Rad, Mehdi
%A Rezaei, Bahman
%T On the projective Randers metrics
%J Comptes Rendus. Mathématique
%D 2012
%P 281-283
%V 350
%N 5-6
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2012.02.010/
%R 10.1016/j.crma.2012.02.010
%G en
%F CRMATH_2012__350_5-6_281_0

Rafie-Rad, Mehdi; Rezaei, Bahman. On the projective Randers metrics. Comptes Rendus. Mathématique, Tome 350 (2012) no. 5-6, pp. 281-283. doi : 10.1016/j.crma.2012.02.010. http://www.numdam.org/articles/10.1016/j.crma.2012.02.010/

Bibliographie
Cité par

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

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

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

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

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

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

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

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

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

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

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

Cité par Sources :