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Cheeger–Gromoll splitting theorem for the Bakry–Emery Ricci tensor

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Abstract

In this paper, we obtain a new Cheeger–Gromoll splitting theorem on a complete Riemannian manifold admitting a smooth vector field such that its Bakry–Emery Ricci tensor is non-negative and the vector field tends to zero at infinity. The result generalizes the classical Cheeger–Gromoll splitting theorem and the splitting type results of Lichnerowicz, Wei–Wylie, Fang–Li–Zhang, Wylie, Khuri–Woolgar–Wylie, Lim, and more.

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Acknowledgements

The authors thank the referee for making valuable suggestions and pointing out many errors which helped to improve the exposition of the paper. Especially the referee’s comment on Theorem 1.1 leads to our improvement of the conditions on V. Originally, we required a stronger degeneration on V. The first author was partially supported by the Innovation training program of the Shanghai University (X202110280019). The second author was partially supported by the Natural Science Foundation of Shanghai (17ZR1412800).

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  1. Department of Mathematics, Shanghai University, Shanghai, 200444, China

    Junhan Tang & Jia-Yong Wu

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  1. Junhan Tang
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  2. Jia-Yong Wu
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Correspondence to Jia-Yong Wu.

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Tang, J., Wu, JY. Cheeger–Gromoll splitting theorem for the Bakry–Emery Ricci tensor. Arch. Math. 117, 697–708 (2021). https://doi.org/10.1007/s00013-021-01658-1

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  • DOI: https://doi.org/10.1007/s00013-021-01658-1

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