Asian Journal of Mathematics

Volume 19 (2015)

A characterization of quadric constant Gauss-Kronecker curvature hypersurfaces of spheres

Pages: 251 – 264

DOI: https://dx.doi.org/10.4310/AJM.2015.v19.n2.a3

Authors

Oscar M. Perdomo (Department of Mathematics, Central Connecticut State University, New Britain, Conn., U.S.A.)

Guoxin Wei (School of Mathematical Sciences, South China Normal University, Guangzhou, China)

Abstract

Let $M \subset S^{n+1}$ be a complete orientable hypersurface with constant Gauss-Kronecker curvature $G$. For any $\upsilon \in \mathbf{R}^{n+2}$, let us define the following two real functions $l_\upsilon , f_\upsilon : M \to \mathbf{R}$ on $M$ by $l_\upsilon (x) = \langle x, \upsilon \rangle$ and $f_\upsilon (x) = \langle \nu (x), \upsilon \rangle$ with $\nu : M \to S^{n+1}$ a Gauss map of $M$. In this paper, we show that if $n = 3, l_\upsilon = \lambda f_\upsilon$ for some nonzero vector $\upsilon \in \mathbb{R}^5$ and some real number $\lambda$, then $M$ is either totally umbilical (a Euclidean sphere) or $M$ is a cartesian product of Euclidean spheres. We will also show with an example that the completeness condition is needed in the result we just mentioned. We also show that if $n = 4, l_\upsilon = \lambda f_\upsilon$ for some nonzero vector $\upsilon \in \mathbb{R}^6$ and some real number $\lambda$ and $(\lambda^2 - 1)^2 + (G - 1)^2 \neq 0$, then $M$ is either totally umbilical (a Euclidean sphere) or $M$ is a cartesian product of Euclidean spheres. Moreover, we will give an example of a complete hypersurface in $S^5$ with constant Gauss-Kronecker curvature that satisfies the condition $l_\upsilon = \lambda f_\upsilon$ for some non zero $\upsilon$, which is neither a totally umbilical hypersurface nor a cartesian product of Euclidean spheres.

Keywords

Clifford hypersurfaces, Gauss-Kronecker curvature, spheres

2010 Mathematics Subject Classification

53A10, 53C42

Published 25 March 2015