Weakly complex Einstein-Finsler vector bundle
type: Item
instance of: scholarly article
title: Weakly complex Einstein-Finsler vector bundle
zbMATH Open document ID: 1394.53075
author: Liling Sun
author: Chunping Zhong
published in: Science China. Mathematics
publication date: 2018-07-05T00:00:00Z
review text: In [Manuscr. Math. 42, 245--257 (1983; Zbl 0558.53037)], \textit{M. Lübke} proved the following result: ``Lübke's theorem. Let \((E,h)\) be an Einstein-Hermitian vector bundle over a compact Kähler manifold \((M,g)\) with Kähler form \(\Phi\). Then \(E\) is \(\Phi\)-semistable and \((E,h)\) is a direct sum \((E,h) = (E_1,h_1)\oplus\cdots\oplus (E_k,h_k)\) of \(\Phi\)-stable Einstein-Hermitian vector bundles \((E_1,h_1),\cdots,(E_k,h_k)\) with the same factor \(c\) as \((E,h)\)''. In [Publ. Math. 51, No. 3--4, 363--384 (1997; Zbl 0905.53015)], \textit{T. Aikou} generalized Lübke's theorem to complex Einstein-Finsler vector bundles \((E,F)\) under the assumption that \((E,F)\) is modeled on a complex Minkowski space. In this paper, the authors prove the following main result: ``If \((E,F)\) is a weakly complex Einstein-Finsler vector bundle over a Hermitian manifold \((M,g)\) in the sense of Aikou, then \((E,F)\) is modeled on a complex Minkowski space''. Using the above result, the authors are able to generalize Lübke's theorem to the complex Einstein-Finsler case without the assumption of Aikou: ``Let \((E,F)\) be a complex Einstein-Finsler vector bundle over a compact Kähler manifold \((M,g)\) with Kähler form \(\Phi\). Then \(E\) is \(\Phi\)-semistable and \((E,F)\) is a direct sum \((E, F) = (E_1,F_1)\oplus\cdots\oplus(E_k,F_k)\), 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 \(\Phi\)-stable Einstein-Hermitian vector bundle with the same factor \(c\) as \((E,F)\)''. The paper is concluded with an example of a complex Einstein-Finsler vector bundle.
Mathematics Subject Classification ID: 53C60
Mathematics Subject Classification ID: 53B40
Mathematics Subject Classification ID: 53C40
zbMATH DE Number: 6898602
zbMATH Keywords: complex Finsler vector bundle
zbMATH Keywords: complex Einstein-Finsler vector bundle
zbMATH Keywords: Chern-Finsler connection
zbMATH Keywords: mean curvature
zbMATH Keywords: \(\Phi\)-stable
zbMATH Keywords: \(\Phi\)-semistable
reviewed by: Nabil L. Youssef
MaRDI profile type: Publication
full work available at URL: s11425-016-9085-2
OpenAlex ID: W2768420116
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DOI: 10.1007/S11425-016-9085-2
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description: scientific article; zbMATH DE number 6898602