Hypersurface of the hyperbolic space with constant scalar curvature
Let \(\mathbb{L}^{n+2}\) be the \((n+2)\)-dimensional Lorentz-Minkowski space, that is the real vector space \(\mathbb{R}^{n+2}\) endowed with the Lorentzian metric \(\langle v,w\rangle:=-v_{0}w_{0}+\sum_{i=1}^{n+1} v_{i}w_{i}\), for \(v,w \in \mathbb{R}^{n+2}.\) Then, for \(c<0\), the \((n+1)\)-dimensional hyperbolic space \(\mathbb{H}^{n+1}(c)\) of constant sectional curvature \(c\), can be defined as the following hyperquadric of \(\mathbb{L}^{n+2}: \mathbb{H}^{n+1}(c)=\{x\in \mathbb{L}^{n+2}: \langle x,x\rangle=\frac{1}{c}\}\). In this paper the authors classify hypersurfaces of the hyperbolic space \(\mathbb{H}^{n+1}(c)\) with constant scalar curvature and with two distinct principal curvatures. Moreover, they prove that if \(M^{n}\) is a complete hypersurfaces with constant scalar curvature \(n(n-1)R\) and with two distinct principal curvatures such that the multiplicity of one of the principal curvatures is \((n-1)\), then \(R\geq c\). They prove additionally for such hypersurfaces two rigidity theorems, where \(R\) denotes the normalized scalar curvature. Reviewer's remarks: 1. The calculations in Example 1.1 of this paper for the cases \(k=1\) and \(k=n-1\) show that \textit{X. Lui}'s and \textit{W. Su}'s claim is not correct [Balkan J. Geom. Appl. 7, No. 1, 121--132 (2002; Zbl 1023.53045)]. 2. By \textit{M. P. do Carmo} and \textit{M. Djaczer}'s theorem a hypersurface of \(\mathbb{H}^{n+1}(c)\) with constant scalar curvature and two distinct principal curvatures, one which is simple, can be characterized as a rotational hypersurface with two distinct principal curvatures [Trans. Am. Math. Soc. 277, 685--709 (1983; Zbl 0518.53059)]. In [Manuscr. Math. 67, No. 3, 285--304 (1990; Zbl 0695.53040)], \textit{M. L. Leite}, has established the structure classification for rotational hypersurfaces of \(\mathbb{H}^{n+1}(-1)\) with constant scalar curvature; however, she did not establish a rigidity theorem.
scientific article; zbMATH DE number 5009755
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