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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?

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 ) .