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On higher-order Codazzi tensors on complete Riemannian manifolds

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Abstract

We prove several Liouville-type nonexistence theorems for higher-order Codazzi tensors and classical Codazzi tensors on complete and compact Riemannian manifolds, in particular. These results will be obtained by using theorems of the connections between the geometry of a complete smooth manifold and the global behavior of its subharmonic functions. In conclusion, we show applications of this method for global geometry of a complete locally conformally flat Riemannian manifold with constant scalar curvature because, its Ricci tensor is a Codazzi tensor and for global geometry of a complete hypersurface in a standard sphere because its second fundamental form is also a Codazzi tensor.

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Acknowledgements

Our work was supported by the Internal Grant Agency of the Faculty of Science of Palacky University, Olomouc (Grant No. 2019018 “Mathematical Structures”). The authors would like to express their thanks to Professor Ivor Hall (BA Hons) for her editorial suggestions, that contributed to the improvement in this paper as well as to the referee for his careful reading of the previous version of the manuscript and suggestions about this paper which led to various improvements.

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Authors and Affiliations

  1. Department of Data Analysis and Financial Technology, Finance University, 49-55, Leningradsky Prospect, Moscow, Russia, 125468

    Igor G. Shandra & Sergey E. Stepanov

  2. Department of Mathematics, Russian Institute for Scientific and Technical Information of the Russian Academy of Sciences, 20, Usievicha street, Moscow, Russia, 125190

    Sergey E. Stepanov

  3. Department of Algebra and Geometry, Palacky University, 12, 17. listopadu, 77146, Olomouc, Czech Republic

    Josef Mikeš

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  1. Igor G. Shandra
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Shandra, I.G., Stepanov, S.E. & Mikeš, J. On higher-order Codazzi tensors on complete Riemannian manifolds. Ann Glob Anal Geom 56, 429–442 (2019). https://doi.org/10.1007/s10455-019-09673-w

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