Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
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Submanifolds of Codimension 3 in a Complex Space Form with Commuting Structure Jacobi Operator

  • Ki, U-Hang (The National Academy of Sciences) ;
  • Song, Hyunjung (Department of Mathematics Hankuk University of Foreign Studies)
  • Received : 2019.11.12
  • Accepted : 2021.02.24
  • Published : 2022.03.31
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Abstract

Let M be a semi-invariant submanifold with almost contact metric structure (𝜙, 𝜉, 𝜂, g) of codimension 3 in a complex space form Mn+1(c) for c ≠ 0. We denote by S and R𝜉 be the Ricci tensor of M and the structure Jacobi operator in the direction of the structure vector 𝜉, respectively. Suppose that the third fundamental form t satisfies dt(X, Y) = 2𝜃g(𝜙X, Y) for a certain scalar 𝜃 ≠ 2c and any vector fields X and Y on M. In this paper, we prove that if it satisfies R𝜉𝜙 = 𝜙R𝜉 and at the same time S𝜉 = g(S𝜉, 𝜉)𝜉, then M is a real hypersurface in Mn(c) (⊂ Mn+1(c)) provided that $\bar{r}-2(n-1)c{\leq}0$, where $\bar{r}$ denotes the scalar curvature of M.

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Acknowledgement

The authors with to express their sincere thanks to the referee who gave us valuable suggestions and comments to improve the paper.

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