Search

In recent decades, there has been an increase in the number of publications related to the hypersurfaces of real space forms with two principal curvatures. The works focus mainly on the case when one of the two principal curvatures is simple. The purpose of this paper is to study a slightly more general class of complete minimal hypersurfaces in real space forms of constant curvature c, namely those with $$\mathrm{n}-1$$ principal curvatures having the same sign everywhere. From assumptions on the scalar curvature R and the Gauss–Kronecker curvature K we characterize Clifford tori if $$c > 0$$ and prove that K is identically zero if $$c \le 0$$ .

Let $${{\mathbb{Q}^4}(c)}$$ be a four-dimensional space form of constant curvature c. In this paper we show that the infimum of the absolute value of the Gauss–Kronecker curvature of a complete minimal hypersurface in $${\mathbb{Q}^4(c), c\leq 0}$$ , whose Ricci curvature is bounded from below, is equal to zero. Further, we study the connected minimal hypersurfaces M 3 of a space form $${{\mathbb{Q}^4}(c)}$$ with constant Gauss–Kronecker curvature K. For the case c ≤ 0, we prove, by a local argument, that if K is constant, then K must be equal to zero. We also present a classification of complete minimal hypersurfaces of $${\mathbb{Q}^4(c)}$$ with K constant.

In this paper we give a partially affirmative answer to the following question posed by Haizhong Li: is a complete spacelike hypersurface in De Sitter space S 1 n + 1 ( c ) , n ⩾ 3 , with constant normalized scalar curvature R satisfying n − 2 n c ⩽ R ⩽ c totally umbilical?

In this work we obtain a Simons' type inequality for a suitable tensor and apply it in order to obtain some results characterizing umbilical submanifolds and a product of submanifolds in a semi-Riemannian space form.

In this paper we study n-dimensional complete spacelike submanifolds with constant normalized scalar curvature immersed in semi-Riemannian space forms. By extending Cheng-Yau's technique to these ambients, we obtain results to such submanifolds satisfying certain conditions on both the squared norm of the second fundamental form and the mean curvature. We also characterize compact non-negatively curved submanifolds in De Sitter space of index p.

Let H, K and R be, respectively, the mean curvature, the Gauss–Kronecker curvature and the scalar curvature of a closed hypersurface immersed in the unit sphere \(\mathbb{S}^4\). In this paper we completely classify the closed hypersurfaces of \(\mathbb{S}^4\) where any two of H, K and R are constant.

In this paper we study complete maximal spacelike hypersurfaces in anti-de Sitter space H 1 n + 1 with either constant scalar curvature or constant non-zero Gauss–Kronecker curvature. We characterize the hyperbolic cylinders H m ( c 1 ) × H n − m ( c 2 ) , 1 ≤ m ≤ n − 1 , as the only such hypersurfaces with ( n − 1 ) principal curvatures with the same sign everywhere. In particular we prove that a complete maximal spacelike hypersurface in H 1 5 with negative constant Gauss–Kronecker curvature is isometric to H 1 ( c 1 ) × H 3 ( c 2 ) .