Characterization of Finsler spaces of scalar curvature

Document Type : Original Article

Authors

1 Department of Mathematics, Faculty of Science, Benha University, Benha 13518, Egypt. E-mail: amrsoleiman@yahoo.com

2 Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt. E-mail: nlyoussef2003@yahoo.fr

Abstract

In Finsler Geometry, all special spaces are investigated locally (using local coordinates) by many authors. On the other hand, the global (or intrinsic, free from local coordinates) investigation of such spaces is very rare in the literature. The aim of the present paper is to provide an intrinsic investigation of two special Finsler spaces whose defining properties are related to Berwald connection, namely, Finsler space of scalar curvature and of constant curvature. Some characterizations of a Finsler space of scalar curvature are proved. Necessary and sufficient conditions under which a Finsler space of scalar curvature reduces to a Finsler space of constant curvature are investigated. It should finally be noted that the present work is formulated in a prospective modern coordinate-free form. Moreover, the outcome of this work is twofold. Firstly, the local expressions of the obtained results, when calculated, coincide with the existing local results. Secondly, new coordinates-free proofs have been established.

Keywords

References

  • 1. H. Akbar-Zadeh, Initiation to global Finsler geometry, Elsevier, 2006.
  • 2. H. Izumi and T. N. Srivastava, On R3-like Finsler spaces, Tensor, N. S., 32 (1978),
    339–349.
  • 3. H. Izumi and M. Yoshida, On Finsler spaces of perpendicular scalar curvature, Tensor,
    N. S., 32 (1978), 219–224.
  • 4. M. Matsumoto, On Finsler spaces with curvature tensors of some special forms, Tensor,
    N. S., 22 (1971), 201–204.
  • 5. M. Matsumoto, Foundations of Finsler geometry and special Finsler spaces, Kaiseisha
    Press, Otsu, Japan, 1986.
  • 6. R. Miron and M. Anastasiei, The geometry of Lagrange spaces: Theory and applications,
    Kluwer Acad. Publ., 1994.
  • 7. H. Rund, The differential geometry of Finsler spaces, Springer-Verlag, Berlin, 1959.
  • 8. T. Sakaguchi, On Finsler spaces of scalar curvature, Tensor, N. S., 38 (1982), 211–219.
  • 9. A. A. Tamim, Special Finsler manifolds, J. Egypt. Math. Soc., 10(2) (2002), 149–177.
  • 10. M. Yoshida, On Finsler spaces of HP -scalar curvature, Tensor, N. S., 38 (1982), 205–210.
  • 11. Nabil L. Youssef, S. H. Abed and A. Soleiman, Cartan and Berwald connections in the
    pullback formalism, Algebras, Groups and Geometries, 25, 4 (2008), 363–386. arXiv:
    0707.1320 [math. DG].
  • 12. Nabil L. Youssef, S. H. Abed and A. Soleiman, A global approach to the theory of special
    Finsler manifolds, J. Math. Kyoto Univ., 48, 4 (2008), 857–893. arXiv: 0704.0053 [math.
    DG].
  • 13. Nabil L. Youssef, S. H. Abed and A. Soleiman, Geometric objects associated with the
    fundumental connections in Finsler geometry, J. Egypt. Math. Soc., 18, 1 (2010), 67–
    90. arXiv: 0805.2489 [math.DG]. 
Journal of Finsler Geometry and its Applications
Volume 1, Issue 1
July 2020
Pages 15-25

Files

Share

How to cite

Statistics