Hypersurfaces with nonzero constant Gauss – Kronecker curvature in $M^{n+1}(±1)$

  • Shichang Shu
  • Tianmin Zhu

Abstract

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 $S^{n-1}(a) \times S^1(\sqrt{1 - a^2}),\quad$ $a^2 = 1/\left(1 + K^{\frac{2}{n - 2}}\right)$ or $S^{n-1}(a) \times H^1(- \sqrt{1 + a^2}),\quad$ $a^2 = 1/\left(K^{\frac{2}{n - 2}} - 1\right)$.
Published
25.11.2016
How to Cite
Shu, S., and T. Zhu. “Hypersurfaces With Nonzero Constant Gauss – Kronecker Curvature in $M^{n+1}(±1)$”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 68, no. 11, Nov. 2016, pp. 1540-51, http://umj.imath.kiev.ua/index.php/umj/article/view/1940.
Section
Research articles