Spacelike submanifolds with parallel mean curvature in pseudo-Riemannian space forms
The study of constant mean curvature spacelike hypersurfaces in Lorentzian space forms, or, more generally, spacelike submanifolds with parallel mean curvature vector in pseudo-Riemannian space forms, has been of substantial interest in the last years. Quite recently, the first and third authors developed some integral formulas for compact spacelike submanifolds in de Sitter spaces and applied them to the case of parallel mean curvature vector, obtaining some interesting characterizations of the totally geodesic and the totally umbilical round spheres [\textit{L. J. Alías} and \textit{A. Romero}, Manuscr. Math. 87, 405-416 (1995; Zbl 0838.53043)]. In the paper under review, the authors study compact spacelike \(n\)-submanifolds with (nonzero) parallel mean curvature vector in a pseudo-Riemannian space form \(N^{n+p+1}_{p}(c)\) with constant curvature \(c\) and Lorentzian normal bundle. They use the same approach as above, but the family of ambient spaces is extended in order to consider flat and negatively curved space forms. In particular, they prove that a compact spacelike submanifold \(M^n\) in \(N^{n+p+1}_{p}(c)\) is a totally umbilical and non-totally geodesic hypersurface in the corresponding Riemannian space form \(N^{n+1}(c)\subset N^{n+p+1}_p(c)\) if and only if \(M^n\) is pseudo-umbilical with nonzero spacelike parallel mean curvature vector. As a consequence of this, in the case where \(n=2\), they conclude that the only topological 2-spheres which are spacelike surfaces in \(N^{3+p}_p(c)\) with nonzero spacelike parallel mean curvature vector are the totally umbilical ones in \(N^3(c)\subset N^{3+p}_p(c)\). Finally, the authors also derive some other consequences for the case where the mean curvature vector has another causal character, namely, lightlike or timelike.
scientific article; zbMATH DE number 1102918