Finite type submanifolds in pseudo-Euclidean spaces and applications
Let \({\mathcal E}^ m\) be the \(m\)-dimensional pseudo-Euclidean space of signature \((s,m-s)\) and let \(x\) be a proper immersion of a compact space-like submanifold \(M\) into \({\mathcal E}^ m\). With \(x\) two numbers \(p\) and \(q\) are associated such that \(p\) (resp. \(q\)) is a positive integer (resp. an integer \(\geq p\) or \(+\infty)\). The pair \([p;q]\) is called the order of the submanifold [cf. the author, Total mean curvature and submanifolds of finite type. Series in Pure Mathematics. 1. Singapore: World Scientific (1984; Zbl 0537.53049)] and \(M\) is said to be of finite type if \(q\) is finite. It is proved that the submanifold \(M\) is of finite type if and only if there is a nontrivial polynomial \(P\) such that \(P(\Delta)H=0\), where \(\Delta\) is the Laplacian \(\Delta\) on \(M\) and \(H\) the mean curvature associated with \(x\). Several theorems in connection with these concepts are formulated. We quote here the following: If \(M\) is a compact space-like hypersurface of de Sitter space-time, then \(M\) has nonzero constant mean curvature and constant scalar curvature if and only if \(M\) is mass-symmetric and of 2-type in the Lorentz-Minkowski world.
scientific article; zbMATH DE number 3940163