Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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ON THE SYNGE'S THEOREM FOR COMPLEX FINSLER MANIFOLDS

  • Won, Dae-Yeon (Department of Mathematics, Duksung Women's University)
  • Published : 2004.02.01
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Abstract

In [13], we developed a theory of complex Finsler manifolds to investigate the global geometry of complex Finsler manifolds. There we proved a version of Bonnet-Myers' theorem for complex Finsler manifolds with a certain condition on the Finsler metric which is a generalization of the Kahler condition for the Hermitian metric. In this paper, we show that if the holomorphic sectional curvature of M is ${\geq}\;c^2\;>\;0$, then M is simply connected. This is a generalization of the Synge's theorem in the Riemannian geometry and the Tsukamoto's theorem for Kahler manifolds. The main point of the proof lies in how we can circumvent the convex neighborhood theorem in the Riemannian geometry. A second variation formula of arc length for complex Finsler manifolds is also derived.

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References

  1. Lecture Notes in Mathematics v.1591 Finsler metric-A global approach M.Abate;G.Patrizio
  2. Graduate Texts in Mathematics An Introduction to Riemannian-Finsler geometry D.Bao;S.S.Chern;Z.Shen
  3. Comparison theorems in Riemannian geometry J.Cheeger;D.Ebin
  4. Principles of algebraic geometry P.Griffiths;J.Harris
  5. Metric structures for Riemannian and non-Riemannian Spaces M.Gromov
  6. Nagoya Math. J. v.57 Negative vector bundles and complex Finsler structures S.Kobayashi
  7. Die Grundlehren der Mathematischen Wissenschaften v.318 Hyperbolic complex spaces
  8. DMV Seminar v.3 Complex differential geometry S.Kobayashi;H.Wu
  9. Foundations of finsler geometry and special Finsler spaces M.Matsumoto
  10. Die Grundlehren der Mathematischen Wissenschaften v.105 Die innere Geometrie der metrischen Raume W.Rinow
  11. A comprehensive introduction to differential geometry(2nd edition) v.4 M.Spivak
  12. Proc. Japan Acad. Ser. A Math. Sci. v.33 On Kahlerian manifolds with positive holomorphic sectional curvature Y.Tsukamoto
  13. Bull. Korean Math. Soc. v.38 no.2 On the Bonnet's theorem for complex Finsler manifolds D.Y.Won