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http://dx.doi.org/10.4134/CKMS.c210004

SEMI-INVARIANT SUBMANIFOLDS OF CODIMENSION 3 IN A COMPLEX SPACE FORM IN TERMS OF THE STRUCTURE JACOBI OPERATOR  

Ki, U-Hang (The National Academy of Sciences)
Kurihara, Hiroyuki (The College of Education Ibaraki University)
Publication Information
Communications of the Korean Mathematical Society / v.37, no.1, 2022 , pp. 229-257 More about this Journal
Abstract
Let M be a semi-invariant submanifold of codimension 3 with almost contact metric structure (𝜙, 𝜉, 𝜂, g) in a complex space form Mn+1(c), c ≠ 0. We denote by A and R𝜉 the shape operator in the direction of distinguished normal vector field and the structure Jacobi operator with respect to the structure vector 𝜉, respectively. Suppose that the third fundamental form t satisfies dt(X, Y) = 2𝜃g(𝜙X, Y) for a scalar 𝜃(< 2c) and any vector fields X and Y on M. In this paper, we prove that if it satisfies R𝜉A = AR𝜉 and at the same time ∇𝜉R𝜉 = 0 on M, then M is a Hopf hypersurface of type (A) provided that the scalar curvature s of M holds s - 2(n - 1)c ≤ 0.
Keywords
Semi-invariant submanifold; almost contact metric structure; the third fundamental form; distingushed normal vector; structure Jacobi operator; Hopf real hypersurface;
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