Compact space-like \(m\)-submanifolds in a pseudo-Riemannian sphere \(S_ p^{m+p} (c)\)
The author deals with the problem whether or not there exists a compact spacelike \(m\)-dimensional submanifold in an \((m+ p)\)-dimensional pseudo-Riemannian sphere \(S^{m+ p}_p(c)\) with index \(p\) and sectional curvature \(c\in \mathbb{R}_+\), with parallel mean curvature vector which is not totally umbilical. She proves the following theorem: Let \(M\) be a compact spacelike \(m\)-dimensional submanifold in \(S^{m+ p}_p(c)\) with parallel mean curvature vector. If the normal connection is flat (in particular, this holds when \(p\leq 2\)), then \(M\) is totally umbilical. For \(m= 2\), she obtains that only compact spacelike surfaces in \(S^{2+ p}_p(c)\) with parallel mean curvature vector are totally umbilical ones. So, the answer to the problem in the case \(m= 2\) is nonexistence. This last result was independently obtained, using another method, by \textit{L. J. Alias} and the reviewer [Manuscr. Math. 87, 405-416 (1995)]. Finally, she also proves, as a consequence of the previous theorem, that given a compact spacelike \(m\)-dimensional submanifold \(M\) in \(S^{m+ p}_p(c)\) with parallel mean curvature vector then \(M\) is totally umbilical, if the sectional curvature of \(M\) is nonnegative. Thus, it seems that the answer to the problem could be nonexistence, although in general it remains open.
scientific article; zbMATH DE number 821711