A Hopf theorem for ambient spaces of dimensions higher than three
The authors consider surfaces \(M^2\) immersed in \(E^n_c \times\mathbb{R}\), where \(E^n_c\) is a simply connected \(n\)-dimensional complete Riemannian manifold with constant sectional curvature \(c \neq 0\), and assume that the mean curvature vector of the immersion is parallel in the normal bundle. They consider further a Hopf-type complex quadratic form \(Q\) on \(M^2\), where the complex structure of \(M^2\) is compatible with the induced metric. They use the fact that \(Q\) is holomorphic [see, Commun. Anal. Geom. 15, No.~2, 283--298 (2007; Zbl 1134.53031); p. 289] to give a reasonable description of immersed surfaces in \(E_c^n\times\mathbb{R}\) that have parallel mean curvature vector: Theorem 1. Let \(M\) be a surface and \(E^n_c\) be a Riemannian manifold of constant sectional curvature \(c\neq 0\), and let \(x: M \to E^n_c \times\mathbb{R}\) be an immersion with parallel mean curvature vector. Then, one of the following assertions holds: (1)~\(x(M)\) is a minimal surface in a totally umbilical hypersurface of \(E^n_c\); (2)~\(x(M)\) is a surface with constant mean curvature of a 3-dimensional totally umbilic or totally geodesic submanifold of \(E^n_c\); (3)~\(x(M)\) lies in \(E^4_c\times\mathbb{R}\). The authors also consider the cases when \(M\) is diffeomorphic to a sphere (Theorem~2) and \(M\) is complete with non-negative Gaussian curvature (Theorem~3).
scientific article; zbMATH DE number 5724746
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