Science China Mathematics

, Volume 61, Issue 3, pp 535–544 | Cite as

Strongly convex weakly complex Berwald metrics and real Landsberg metrics

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Abstract

Under the assumption that F is a strongly convex weakly Kähler Finsler metric on a complex manifold M, we prove that F is a weakly complex Berwald metric if and only if F is a real Landsberg metric. This result together with Zhong (2011) implies that among the strongly convex weakly Kähler Finsler metrics there does not exist unicorn metric in the sense of Bao (2007). We also give an explicit example of strongly convex Kähler Finsler metric which is simultaneously a complex Berwald metric, a complex Landsberg metric, a real Berwald metric, and a real Landsberg metric.

Keywords

strongly convex Kähler Finsler metric Kähler Finsler metric real Landsberg metric real Berwald metric 

MSC(2010)

53C60 53C40 

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Notes

Acknowledgements

This work was supported by Program for New Century Excellent Talents in University (Grant No. NCET-13-0510), National Natural Science Foundation of China (Grant Nos. 11271304, 10971170, 11171277, 11571288, 11461064 and 11671330), the Fujian Province Natural Science Funds for Distinguished Young Scholar (Grant No. 2013J06001) and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry.

References

  1. 1.
    Abate M, Patrizio G. Finsler Metrics—A Global Approach with Applications to Geometric Function Theory. Berlin-Heidelberg: Springer-Verlag, 1994CrossRefMATHGoogle Scholar
  2. 2.
    Aikou T. On complex Finsler manifolds. Rep Fac Sci Kagoshima Univ, 1991, 24: 9–25MathSciNetMATHGoogle Scholar
  3. 3.
    Aikou T. Complex manifolds modeled on a complex Minkowski space. J Math Kyoto Univ, 1995, 35: 85–103MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Aikou T. Some remarks on locally conformal complex Berwald spaces. In: Finsler Geometry. Contemporary Mathematics, vol. 196. Seattle-Providence: Amer Math Soc, 1996: 109–120CrossRefMATHGoogle Scholar
  5. 5.
    Aldea N, Munteanu G. On complex Landsberg and Berwald spaces. J Geom Phys, 2012, 62: 368–380MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Bao D. On two curvature-driven problems in Riemann-Finsler geometry. In: Finsler Geometry, Sapporo 2005—In Memory of Makoto Matsumoto. Advanced Studies in Pure Mathematics, vol. 48. Tokyo: Math Soc Japan, 2007: 19–71Google Scholar
  7. 7.
    Bao D, Chern S S, Shen Z. An Introduction to Riemann-Finsler Geometry. New York: Springer-Verlag, 2000CrossRefMATHGoogle Scholar
  8. 8.
    Berwald L. Untersuchung der Krümmung allgemeiner metrischer Räume auf Grund des in ihnen herrschenden Parallelismus. Math Z, 1926, 25: 40–73MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Berwald L. über zweidimensionale allgemeine metrische Räume. I, II. J Reine Angew Math, 1927, 156: 191–222MathSciNetMATHGoogle Scholar
  10. 10.
    Berwald L. Parallelübertragung in allgemeinen Räumen. Atti Congr Intern Mat Bologna, 1928, 4: 263–270MATHGoogle Scholar
  11. 11.
    Chen B, Shen B. Kähler Finsler metrics are actually strongly Kähler. Chin Ann Math Ser B, 2009, 30: 173–178MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Ichijy¯o Y. Finsler manifolds modeled on a Minkowski space. J Math Kyoto Univ, 1976, 16: 639–652MathSciNetCrossRefGoogle Scholar
  13. 13.
    Matsumoto M. Foundations of Finsler Geometry and Special Finsler Spaces. Shigaken: Kaiseisha Press, 1986MATHGoogle Scholar
  14. 14.
    Matsumoto M. Remarks on Berwald and Landsberg spaces. In: Finsler Geometry. Contemporary Mathematics, vol. 196. Seattle-Providence: Amer Math Soc, 1996: 79–82Google Scholar
  15. 15.
    Munteanu G. Complex Spaces in Finsler, Lagrange and Hamilton Geometries. Dordrecht: Kluwer Academic, 2004CrossRefMATHGoogle Scholar
  16. 16.
    Shen Z. On a class of Landsberg metrics in Finsler geometry. Canad J Math, 2009, 61: 1357–1374MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Szabó Z I. All regular Landsberg metrics are Berwald. Ann Global Anal Geom, 2008, 34: 381–386MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Szabó Z I. Correction to “All regular Landsberg metrics are Berwald”. Ann Global Anal Geom, 2009, 35: 227–230MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Xia H, Zhong C. On strongly convex weakly Kähler-Finsler metrics of constant flag curvature. J Math Anal Appl, 2016, 443: 891–912MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Zhong C. On real and complex Berwald connections associated to strongly convex weakly Kähler-Finsler metric. Differential Geom Appl, 2011, 29: 388–408MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Zhong C. On unitary invariant strongly pseudoconvex complex Finsler metrics. Differential Geom Appl, 2015, 40: 159–186MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.School of Mathematical SciencesXiamen UniversityXiamenChina
  2. 2.School of Mathematical SciencesXinjiang Normal UniversityUrumqiChina

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