Complete maximal spacelike submanifolds
The authors obtain some interesting results on complete maximal spacelike \(n\)-submanifolds in de Sitter spaces \({\text S}^{n+p}_{q}(c)\), \(1\leq q\leq p\), where by maximal here it is meant that the mean curvature vector field vanishes everywhere. In a previous paper [\textit{L. J. Alías} and \textit{A. Romero}, Manuscr. Math. 87, 405-416 (1995; Zbl 0838.53043)], the reviewer and Romero obtained some integral formulas for compact spacelike \(n\)-submanifolds in \({\text S}^{n+p}_{q}(c)\), with applications to the case of parallel mean curvature vector field. In particular, we characterized the totally geodesic submanifolds of \({\text S}^{n+p}_{q}(c)\) as the only complete maximal \(n\)-submanifolds satisfying \(\text{Ric}\geq (n-1)c\), and constructed elementary counterexamples for \(\text{Ric}\leq (n-2)c\). On the other hand, some other authors have recently studied the case of parallel mean curvature when \(p=q\) by using the classical Simons' formula for the Laplacian of the squared norm \(S\) of the second fundamental form. In the paper under review, the authors generalize Simons' method to complete spacelike \(n\)-submanifolds in \({\text S}^{n+p}_{q}(c)\). As a consequence, they reprove and extend our previous result when \(q=1\), \(p=2\), including also the case \((n-2)c < \text{Ric} < (n-1)c\). In particular, they obtain that complete maximal spacelike \(n\)-submanifolds in \({\text S}^{n+2}_{1}(c)\) satisfying \(\text{Ric}\geq (n-2)c\) are either totally geodesic or Einstein with \(\text{Ric}= (n-2)c\) and parallel second fundamental form. For general \(p,q\), \(1\leq q\leq p\), they prove that the only compact maximal spacelike \(n\)-submanifolds in \({\text S}^{n+p}_{q}(c)\) with \(S\leq \max\{ nc/(2-1/(p-q)), 2nc/3 \}\) are either totally geodesic or some concrete submanifolds (where the equality must be reached at some point). These concrete submanifolds appear just when \(p=q+1\) (a Clifford torus in the totally geodesic spacelike submanifold \({\text S}^{n+1}(c)\subset{\text S}^{n+q+1}_{q}(c)\)), or when \(p=q+2\) and \(n=2\) (a Veronese surface in the totally geodesic spacelike submanifold \({\text S}^4(c)\subset{\text S}^{4+q}_{q}\)). Moreover, when \(p=q+1\), the authors also obtain that the only compact maximal spacelike submanifolds in \({\text S}^{n+q+1}_{q}(c)\) with positive sectional curvature are the totally geodesic submanifolds.
scientific article; zbMATH DE number 1180763