2-type surfaces in a hypersphere
The authors study surfaces in the unit hypersphere \(S^m\) which are of 2-type in the sense of [the reviewer, Total mean curvature and submanifolds of finite type (Series in Pure Mathematics 1, World Scientific, Singapore) (1984; Zbl 0537.53049)]. 2-type spherical surfaces have been studied by many authors. In particular, in a paper by \textit{M. Barros} and the reviewer [J. Math. Soc. Japan 39, 627-648 (1987; Zbl 0613.53023)], it was shown that a compact stationary mass-symmetric 2-type surface in \(S^m\) is flat and lies fully in \(S^5\) or \(S^7\). Furthermore, they proved that there exists no mass-symmetric 2-type surface lying fully in \(S^4\). \textit{Y. Miyata} [Tokyo J. Math. 11, No. 1, 157-204 (1988; Zbl 0654.53054)] studied mass-symmetric 2-type surfaces \(M^2(c)\) in \(S^m\) with constant Gauss curvature \(c\), showing that for \(c < 0\) there exist no such surfaces, and for \(c\geq 0\), \(m\) is odd (assuming the surface lies fully in \(S^m)\). In addition, a result of M. Kotani states that for a compact mass-symmetric 2-type surface of genus zero lying fully in \(S^m\), \(m\) is necessarily odd. In conjunction with these results, the reviewer posed the following question: Do there exist mass-symmetric 2-type surfaces which lie fully in an even-dimensional hypersphere? In this paper this question is further investigated. The main results are the following: (1) There exists no mass-symmetric 2-type surface which lies fully in \(S^6\). (2) If \(M\) is a full mass-symmetric 2-type surface in \(S^m\) with dimension of the first normal space \(\leq 2\), then \(m\) is odd unless \(M\) is pseudo-umbilical. (3) If \(M\) is a complete mass-symmetric 2-type surface in \(S^m\) with nonnegative Gauss curvature, then \(M\) is flat and \(m\) is odd unless \(M\) is pseudo-umbilical.
scientific article; zbMATH DE number 895928
scientific article