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Hypersurfaces with Nonzero Constant Gauss–Kronecker Curvature in M n+1(±1)

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Ukrainian Mathematical Journal Aims and scope

We study hypersurfaces in a unit sphere and in a hyperbolic space with nonzero constant Gauss–Kronecker curvature and two distinct principal curvatures one of which is simple. Denoting by K the nonzero constant Gauss–Kronecker curvature of hypersurfaces, we obtain some characterizations of the Riemannian products \( {S}^{n-1}(a)\times {S}^1\left(\sqrt{1-{a}^2}\right),\kern0.5em {a}^2=1/\left(1+{K}^{{\scriptscriptstyle \frac{2}{n-2}}}\right)\mathrm{or}\kern0.5em {S}^{n-1}(a)\times {H}^1\left(-\sqrt{1+{a}^2}\right),\kern0.5em {a}^2=1/\left({K}^{{\scriptscriptstyle \frac{2}{n-2}}}-1\right). \)

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Authors and Affiliations

  1. School of Mathematics and Information Science, Xianyang Normal University, Xianyang, China

    Sh. Shu

  2. School of Mathematics and Information Science, Weinan Normal University, Weinan, China

    T. Zhu

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  1. Sh. Shu
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  2. T. Zhu
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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 68, No. 11, pp. 1540–1551, November, 2016.

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Shu, S., Zhu, T. Hypersurfaces with Nonzero Constant Gauss–Kronecker Curvature in M n+1(±1). Ukr Math J 68, 1782–1797 (2017). https://doi.org/10.1007/s11253-017-1327-5

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  • DOI: https://doi.org/10.1007/s11253-017-1327-5

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