Stationary 2-type surfaces in a hypersphere
The notions of order of a submanifold and submanifolds of finite type were introduced by the second author in [Bull. Inst. Math., Acad. Sin. 7, 301--311 (1979; Zbl 0442.53050); ibid. 11, 309--328 (1983; Zbl 0498.53039)]. Subsequently [\textit{B. Y. Chen}, Total mean curvature and submanifolds of finite type. Series in Pure Mathematics. 1. Singapore: World Scientific (1984; Zbl 0537.53049)], mass-symmetric, 2-type surfaces in a 3-sphere are completely classified. In the first part of this paper, the authors prove that there exist no mass-symmetric, 2-type surfaces which lie fully in a 4-sphere. In the second part, the authors classify stationary, mass-symmetric, 2-type surfaces in an n-sphere. In particular, they show that if \(M\) is a stationary, mass-symmetric, 2-type surface in an m-sphere, then \(M\) is a flat surface which lies fully in a totally geodesic 5-sphere or in a totally geodesic 7-sphere. Examples of such surfaces are constructed.
scientific article; zbMATH DE number 3992359