2017
Том 69
№ 6

All Issues

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

Shu Shichang, Zhu Tianmin


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 $S^{n-1}(a) \times H^1(- \sqrt{1 + a^2}),\quad$ $a^2 = 1/\left(K^{\frac{2}{n - 2}} - 1\right)$.

Citation Example: Shu Shichang, Zhu Tianmin Hypersurfaces with nonzero constant Gauss – Kronecker curvature in $M^{n+1}(±1)$ // Ukr. Mat. Zh. - 2016. - 68, № 11. - pp. 1540-1551.