Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

VANISHING OF PROJECTIVE VECTOR FIELDS ON COMPACT FINSLER MANIFOLDS

  • Shen, Bin (School of Mathematics Southeast University)
  • Received : 2016.09.20
  • Accepted : 2017.10.25
  • Published : 2018.01.01
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we give characteristic differential equations of a kind of projective vector fields on Finsler manifolds. Using these equations, we prove the vanishing theorem of projective vector fields on any compact Finsler manifold with the negative mean Ricci curvature, which is defined in [10]. This result involves the vanishing theorem of Killing vector fields in the Riemannian case and the work of [1, 14].

Keywords

Acknowledgement

Supported by : Natural Science Foundation of Jiangsu Province, NSFC

References

  1. H. Akbar-Zadeh, Initiation to Global Finslerian Geometry, North-Holland Mathematical Library, 68, Elsevier Science B.V., Amsterdam, 2006.
  2. D. Bao, S.-S. Chern, and Z. Shen, An Introduction to Riemann-Finsler Geometry, Graduate Texts in Mathematics, 200, Springer-Verlag, New York, 2000.
  3. G. S. Hall, Symmetries and Curvature Structure in General Relativity, World Scientific Lecture Notes in Physics, 46, World Scientific Publishing Co., Inc., River Edge, NJ, 2004.
  4. B. Li, On the classification of projectively flat Finsler metrics with constant flag curvature, Adv. Math. 257 (2014), 266-284. https://doi.org/10.1016/j.aim.2014.02.022
  5. R. L. Lovas, Affne and projective vector fields on spray manifolds, Period. Math. Hungar. 48 (2004), no. 1-2, 165-179. https://doi.org/10.1023/B:MAHU.0000038973.18653.2e
  6. V. S. Matveev, H. Rademacher, M. Troyanov, and A. Zeghib, Finsler conformal Lichnerowicz-Obata conjecture, Ann. Inst. Fourier (Grenoble) 59 (2009), no. 3, 937-949. https://doi.org/10.5802/aif.2452
  7. D. D. Monticelli, Maximum principles and the method of moving planes for a class of degenerate elliptic linear operators, J. Eur. Math. Soc. (JEMS) 12 (2010), no. 3, 611-654.
  8. P. Petersen, Riemannian Geometry, Second edition, Graduate Texts in Mathematics, 171, Springer, New York, 2006.
  9. M. Schottenloher, A mathematical Introduction to Conformal Field Theory, translated from the German, Lecture Notes in Physics. New Series m: Monographs, 43, Springer-Verlag, Berlin, 1997.
  10. B. Shen, Vanishing of Killing vector fields on Finsler manifolds, Kodai Math. J., to be appeared.
  11. Z. Shen, Lectures on Finsler Geometry, World Scientific Publishing Co., Singapore, 2001.
  12. Z. Shen, Differential Geometry of Spray and Finsler Spaces, Kluwer Academic Publishers, Dordrecht, 2001.
  13. Z. Shen, Projectively flat Finsler metrics of constant flag curvature, Trans. Amer. Math. Soc. 355 (2003), no. 4, 1713-1728. https://doi.org/10.1090/S0002-9947-02-03216-6
  14. H. Tian, Projective vector fields on Finsler manifolds, Appl. Math. J. Chinese Univ. Ser. B 29 (2014), no. 2, 217-229. https://doi.org/10.1007/s11766-014-3130-5
  15. H. H. Wu, The Bochner technique in differential geometry, Math. Rep. 3 (1988), no. 2, i-xii and 289-538.
  16. K. Yano, Integral Formulas in Riemannian Geometry, Pure and Applied Mathematics, No. 1, Marcel Dekker, Inc., New York, 1970.