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


SIGMA 5 (2009), 045, 7 pages      arXiv:0904.2170      http://dx.doi.org/10.3842/SIGMA.2009.045
Contribution to the Special Issue “Élie Cartan and Differential Geometry”

The Explicit Construction of Einstein Finsler Metrics with Non-Constant Flag Curvature

Enli Guo a, Xiaohuan Mo b and Xianqiang Zhang c
a) College of Applied Science, Beijing University of Technology, Beijing 100022, China
b) Key Laboratory of Pure and Applied Mathematics, School of Mathematical Sciences, Peking University, Beijing 100871, China
c) Tianfu College, Southwestern University of Finance and Economics, Mianyang 621000, China

Received December 08, 2008, in final form April 09, 2009; Published online April 14, 2009

Abstract
By using the Hawking Taub-NUT metric, this note gives an explicit construction of a 3-parameter family of Einstein Finsler metrics of non-constant flag curvature in terms of navigation representation.

Key words: Finsler manifold; Einstein Randers metric; Ricci curvature.

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References

  1. Bao D., Robles C., Ricci and flag curvatures in Finsler geometry, in A Sampler of Riemann-Finsler Geometry, Math. Sci. Res. Inst. Publ., Vol. 50, Cambridge Univ. Press, Cambridge, 2004, 197-259.
  2. Bao D., Robles C., Shen Z., Zermelo navigation on Riemannian manifolds, J. Differential Geom. 66 (2004), 377-435, math.DG/0311233.
  3. Hamel G., Über die Geometrieen in denen die Geraden die Kürzesten sind, Math. Ann. 57 (1903), 231-264.
  4. Hawking S.W., Gravitational Instantons, Phys. Lett. A 60 (1977), 81-83.
  5. LeBrun C., Complete Ricci-flat Kähler metrics on Cn need not be flat, in Several Complex Variables and Complex Geometry, Part 2 (Santa Cruz, CA, 1989), Proc. Sympos. Pure Math., Vol. 52, Part 2, Amer. Math. Soc., Providence, RI, 1991, 297-304.
  6. Mo X., An introduction to Finsler geometry, Peking University Series in Mathematics, Vol. 1, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2006.
  7. Pantilie R., Harmonic morphisms with 1-dimensional fibres on 4-dimensional Einstein manifolds, Comm. Anal. Geom. 10 (2002), 779-814.
  8. Randers G., On an asymmetric metric in the four-space of general relativity, Phys. Rev. 59 (1941), 195-199.
  9. Robles C., Einstein metrics of Randers type, Ph.D. Thesis, British Columbia University, Canada, 2003.
  10. Shen Z., Projectively flat Randers metrics of constant curvature, Math. Ann. 325 (2003), 19-30.
  11. Wood J.C., Harmonic morphisms between Riemannian manifolds, in Modern Trends in Geometry and Topology, Cluj Univ. Press, Cluj-Napoca, 2006, 397-414.

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