2-type surfaces in a hypersphere
type: Item
instance of: scholarly article
title: 2-type surfaces in a hypersphere
zbMATH Open document ID: 0856.53019
author: T. Vlachos
author: Thomas Hasanis
published in: Kodai Mathematical Journal
publication date: 1997-02-24T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 53B25
Mathematics Subject Classification ID: 53C40
zbMATH DE Number: 895928
zbMATH Keywords: submanifolds of finite type
zbMATH Keywords: 2-type surfaces
zbMATH Keywords: mass-symmetric 2-type surface
reviewed by: Bang-Yen Chen
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2996/kmj/1138043544
OpenAlex ID: W2008291190
cites work: Stationary 2-type surfaces in a hypersphere
cites work: Minimal Surfaces of Constant Curvature in S n
cites work: Q5678153
cites work: Q3322795
cites work: Q3987661
cites work: Reduction of the codimension of an isometric immersion
cites work: Spherical Chen surfaces which are mass-symmetric and of 2-type
cites work: A Local Classification of 2-Type Surfaces in S 3
cites work: Spherical 2-type hypersurfaces
cites work: On subharmonic functions and differential geometry in the large
cites work: A decomposition theorem of 2-type immersions
cites work: 2-type surfaces of constant curvature in \(S^n\)
cites work: Codazzi-Tensoren und Kennzeichnungen sphärischer Immersionen
cites work: Minimal immersions of Riemannian manifolds
DOI: 10.2996/KMJ/1138043544
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description: scientific article; zbMATH DE number 895928