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On product affine hyperspheres in ℝn+1

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Abstract

In this paper, we study locally strongly convex affine hyperspheres in the unimodular affine space ℝn+1 which, as Riemannian manifolds, are locally isometric to the Riemannian product of two Riemannian manifolds both possessing constant sectional curvature. As the main result, a complete classification of such affine hyperspheres is established. Moreover, as direct consequences, 3- and 4-dimensional affine hyperspheres with parallel Ricci tensor are also classified.

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Acknowledgements

The first two authors were supported by National Natural Science Foundation of China (Grant No. 11771404). The third author is a postdoctoral fellow of FWO-Flanders, Belgium.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Zhengzhou University, Zhengzhou, 450001, China

    Xiuxiu Cheng & Zejun Hu

  2. Henan Key Laboratory of Financial Engineering, Zhengzhou, 450001, China

    Xiuxiu Cheng & Zejun Hu

  3. Department of Mathematics, KU Leuven, Leuven, 3001, Belgium

    Marilena Moruz & Luc Vrancken

  4. Institut des Sciences et Techniques de Valenciennes (ISTV), Université Polytechnique Hauts de France, Valenciennes, 59313, France

    Luc Vrancken

Authors
  1. Xiuxiu Cheng
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  2. Zejun Hu
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  3. Marilena Moruz
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  4. Luc Vrancken
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Correspondence to Zejun Hu.

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Cheng, X., Hu, Z., Moruz, M. et al. On product affine hyperspheres in ℝn+1. Sci. China Math. 63, 2055–2078 (2020). https://doi.org/10.1007/s11425-018-9457-9

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  • DOI: https://doi.org/10.1007/s11425-018-9457-9

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