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Riemannian submersions from compact four manifolds

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Abstract

We show that under certain conditions, a nontrivial Riemannian submersion from positively curved four manifolds does not exist. This gives a partial answer to a conjecture due to Fred Wilhelm. We also prove a rigidity theorem for Riemannian submersions with totally geodesic fibers from compact four-dimensional Einstein manifolds.

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Acknowledgments

This paper is a part of my Ph.D thesis at University of Notre Dame [5]. The author would like to express gratitude to his advisor, Professor Karsten Grove, for many helpful discussions. He also thanks Professor Anton Petrunin for discussing the proof of Theorem 3.1. The author benefits a lot from his “Exercises in orthodox geometry” [18]. He also would like to thank the referee for his helpful suggestions concerning the presentation of the paper.

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Authors and Affiliations

  1. Department of Mathematics, University of Notre Dame, Notre Dame, IN, 46637, USA

    Xiaoyang Chen

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  1. Xiaoyang Chen
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Correspondence to Xiaoyang Chen.

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The author is supported in part by NSF DMS-1209387.

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Chen, X. Riemannian submersions from compact four manifolds. Math. Z. 282, 165–175 (2016). https://doi.org/10.1007/s00209-015-1536-2

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  • DOI: https://doi.org/10.1007/s00209-015-1536-2

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