Abstract
Let (E, F) be a complex Finsler vector bundle over a compact Kähler manifold (M, g) with Kähler form Φ. We prove that if (E, F) is a weakly complex Einstein-Finsler vector bundle in the sense of Aikou (1997), then it is modeled on a complex Minkowski space. Consequently, a complex Einstein-Finsler vector bundle (E, F) over a compact Kähler manifold (M, g) is necessarily Φ-semistable and (E, F) = (E1, F1) ⨁ · · · ⨁ (Ek; Fk); where F j := F |E j , and each (E j , F j ) is modeled on a complex Minkowski space whose associated Hermitian vector bundle is a Φ-stable Einstein-Hermitian vector bundle with the same factor c as (E, F).
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Acknowledgements
This work was supported by National Natural Science Foundation of China (Grant Nos. 11671330 and 11271304), 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. The authors are grateful to the referees for their careful reading of the article and helpful suggestions.
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Dedicated to Professor Tongde Zhong for his 90th birthday
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Sun, L., Zhong, C. Weakly complex Einstein-Finsler vector bundle. Sci. China Math. 61, 1079–1088 (2018). https://doi.org/10.1007/s11425-016-9085-2
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DOI: https://doi.org/10.1007/s11425-016-9085-2
Keywords
- complex Finsler vector bundle
- complex Einstein-Finsler vector bundle
- Chern-Finsler connection
- mean curvature