Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 7 (2011), 085, 12 pages      arXiv:1108.6127

On the Projective Algebra of Randers Metrics of Constant Flag Curvature

Mehdi Rafie-Rad a, b and Bahman Rezaei c
a) School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5746, Tehran, Iran
b) Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, P.O. Box 47416-1467, Babolsar, Iran
c) Department of Mathematics, Faculty of Sciences, University of Urmia, Urmia, Iran

Received February 26, 2011, in final form August 20, 2011; Published online August 31, 2011

The collection of all projective vector fields on a Finsler space (M,F) is a finite-dimensional Lie algebra with respect to the usual Lie bracket, called the projective algebra denoted by p(M,F) and is the Lie algebra of the projective group P(M,F). The projective algebra p(M,F=α+β) of a Randers space is characterized as a certain Lie subalgebra of the projective algebra p(M,α). Certain subgroups of the projective group P(M,F) and their invariants are studied. The projective algebra of Randers metrics of constant flag curvature is studied and it is proved that the dimension of the projective algebra of Randers metrics constant flag curvature on a compact n-manifold either equals n(n+2) or at most is n(n+1)/2.

Key words: Randers metric; constant flag curvature; projective vector field; projective algebra.

pdf (384 Kb)   tex (17 Kb)


  1. Akbar-Zadeh H., Champs de vecteurs projectifs sur le fibré unitaire, J. Math. Pures Appl. (9) 65 (1986), 47-79.
  2. Akbar-Zadeh H., Sur les espaces de Finsler à courbures sectionelles constantes, Acad. Roy. Belg. Bull. Cl. Sci. (5) 74 (1988), no. 10, 281-322.
  3. Akbar-Zadeh H., Espaces à tenseurs de Ricci parallèle admettant des transformations projectives, Rend. Mat. (6) 11 (1978), 85-96.
  4. Akbar-Zadeh H., Generalized Einstein manifolds, J. Geom. Phys. 17 (1995), 342-380.
  5. Bao D., Robles C., On Randers metrics of constant curvature, Rep. Math. Phys. 51 (2003), 9-42.
  6. Bao D., Shen Z., Finsler metrics of constant positive curvature on the Lie group S3, J. Lond. Math. Soc. 66 (2002), 453-467.
  7. Barnes A., Projective collineations in Einstein spaces, Classical Quantum Gravity 10 (1993), 1139-1145.
  8. Gibbons G.W., Herdeiro C.A.R., Warnick C.M., Werner M.C., Stationary metrics and optical Zermelo-Randers-Finsler geometry, Phys. Rev. D 79 (2009), 044022, 21 pages, arXiv:0811.2877.
  9. Hall G.S., Lonie D.P., The principle of equivalence and cosmological metrics, J. Math. Phys. 49 (2008), 022502, 13 pages.
  10. Hiramato H., Riemannian manifolds admitting a projective vector field, Kodai Math. J. 3 (1980), 397-406.
  11. Israel I., Relativity, astrophysics and cosmology, Dordrecht, Boston, 1973.
  12. Miron R., Finsler-Lagrange spaces with (α,β)-metrics and Ingarden spaces, Rep. Math. Phys. 58 (2006), 417-431.
  13. Mo X., On the non-Riemannian quantity H of a Finsler metric, Differential. Geom. Appl. 27 (2009), 7-14.
  14. Mo X., A global classification result for Randers metrics of scalar curvature on closed manifolds, Nonlinear Anal. 69 (2008), 2996-3004.
  15. Najafi B., Shen Z., Tayebi A., Finsler metrics of scalar flag curvature with special non-Riemannian curvature properties, Geom. Dedicata 131 (2008), 87-97.
  16. Najafi B., Tayebi A., A new quantity in Finsler geometry, C. R. Math. Acad. Sci. Paris 349 (2011), 81-83.
  17. Obata M., Certain conditions for a Riemannian manifold to be isometric with a sphere, J. Math. Soc. Japan. 14 (1962), 333-340.
  18. Randers G., On an asymmetric metric in the four-space of general relativity, Phys. Rev. 59 (1941), 195-199.
  19. Robles C., Einstein metrics of Randers type, Ph.D. thesis, University of British Columbia, Canada, 2003.
  20. Shen Y., Yu Y., On projectively related Randers metric, Internat. J. Math. 19 (2008), 503-520.
  21. Shen Z., Differential geometry of spray and Finsler spaces, Kluwer Academic Publishers, Dordrecht, 2001.
  22. Shen Z., Projectively flat Finsler metrics of constant flag curvature, Trans. Amer. Math. Soc. 355 (2002), 1713-1728.
  23. Shen Z.M., Xing H., On Randers metrics of isotropic S-curvature, Acta Math. Sin. (Engl. Ser.) 24 (2008), 789-796.
  24. Stavrinos P.C., Gravitational and cosmological considerations based on the Finsler and Lagrange metric structures, Nonlinear Anal. 71 (2009), e1380-e1392.
  25. Tanno S., Some differential equations on Riemannian manifolds, J. Math. Soc. Japan 30 (1978), 509-531.
  26. Tashiro Y., Complete Riemannian manifolds and some vector fields, Trans. Amer. Math. Soc. 117 (1965), 251-275.
  27. Yamauchi K., On infinitesimal projective transformations of a Riemannian manifold with constant scalar curvature, Hokkaido. Math. J. 8 (1979), 167-175.
  28. Yano K., The theory of Lie derivatives and its applications, North Holland, Amsterdam, 1957.

Previous article   Next article   Contents of Volume 7 (2011)