Abstract
Let \(M^{2n}\) be a compact Riemannian manifold of non-positive (resp. negative) sectional curvature. We call \((M,J,\theta )\) a d(bounded) locally conformally Kähler manifold if the lifted Lee form \({\tilde{\theta }}\) on the universal covering space of M is d(bounded). We show that if \(M^{2n}\) is homeomorphic to a d(bounded) LCK manifold, then its Euler number satisfies the inequality \((-1)^{n}\chi (M^{2n})\ge \) (resp. >) 0.
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09 March 2021
A Correction to this paper has been published: https://doi.org/10.1007/s00209-021-02709-7
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Acknowledgements
We would like to thank Professor H. Y. Wang for drawing our attention to the LCK manifold and generously helpful suggestions about these. We would also like to thank the anonymous referee for careful reading of my manuscript and helpful comments. This work is supported by Nature Science Foundation of China no. 11801539.
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Huang, T. A note on Euler number of locally conformally Kähler manifolds. Math. Z. 296, 1725–1733 (2020). https://doi.org/10.1007/s00209-020-02491-y
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DOI: https://doi.org/10.1007/s00209-020-02491-y