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SEMI-INVARIANT SUBMANIFOLDS OF CODIMENSION 3 IN A COMPLEX SPACE FORM WITH ξ-PARALLEL STRUCTURE JACOBI OPERATOR

송현정|기우항
영남수학회 (Youngnam Mathematical Society)|2024.01|East Asian Mathematical Journal|제40권 제1호|pp. 1-23

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참고문헌

키워드

scalar curvaturesemi-invariant submanifolddistinguished normalcomplex space formstructure Jacobi operator
주제분류수학

초록

Let $M$ be a semi-invariant submanifold of codimension 3with almost contact metric structure $(\phi, \xi, \eta, g)$ in a complex space form$M_{n+1} (c)$. We denote by $A$, $K$ and $L$ the second fundamental forms with respect to theunit normal vector $C$, $D$ and $E$ respectively, where $C$ is the distinguished normal vector,and by $R_\xi=R(\xi,\cdot)\xi$ the structure Jacobi operator. Suppose that the third fundamental form $t$ satisfies $dt(X,Y)=2\theta g(\phi X,Y)$ for a scalar$\theta(\neq2c)$ and any vector fields $X$ and $Y$, and at the same time $R_\xi K=KR_\xi$ and$\nabla_{\phi\nabla_{\xi}\xi}R_\xi=0$. In this paper, we prove thatif it satisfies $\nabla_\xi R_\xi =0$ on $M$, then $M$ is a real hypersurface of type ($A$) in$M_{n} (c)$ provided that the scalar curvature $\bar{r}$ of $M$ holds $\bar{r}-2(n-1)c\leq0$.

Abstract

Let $M$ be a semi-invariant submanifold of codimension 3with almost contact metric structure $(\phi, \xi, \eta, g)$ in a complex space form$M_{n+1} (c)$. We denote by $A$, $K$ and $L$ the second fundamental forms with respect to theunit normal vector $C$, $D$ and $E$ respectively, where $C$ is the distinguished normal vector,and by $R_\xi=R(\xi,\cdot)\xi$ the structure Jacobi operator. Suppose that the third fundamental form $t$ satisfies $dt(X,Y)=2\theta g(\phi X,Y)$ for a scalar$\theta(\neq2c)$ and any vector fields $X$ and $Y$, and at the same time $R_\xi K=KR_\xi$ and$\nabla_{\phi\nabla_{\xi}\xi}R_\xi=0$. In this paper, we prove thatif it satisfies $\nabla_\xi R_\xi =0$ on $M$, then $M$ is a real hypersurface of type ($A$) in$M_{n} (c)$ provided that the scalar curvature $\bar{r}$ of $M$ holds $\bar{r}-2(n-1)c\leq0$.

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관련 논문AI추천

Semi-invariant submanifolds of codimension 3 in a complex space form in terms of the structure Jacobi operatorHiroyuki Kurihara, U-Hang Ki대한수학회 · 대한수학회논문집2022 · 제37권 제1호SEMI-INVARINAT SUBMANIFOLDS OF CODIMENSION 3 SATISFYING ∇ϕ∇ξξRξ = 0 IN A COMPLEX SPACE FORM기우항영남수학회 · East Asian Mathematical Journal2021 · 제37권 제1호Submanifolds of Codimension 3 in a Complex Space Form with Commuting Structure Jacobi Operator기우항, 송현정경북대학교 자연과학대학 수학과 · Kyungpook Mathematical Journal2022 · 제62권 제1호Structure Jacobi Operator of Semi-invarinat Submanifolds in Complex Space Forms기우항, 김수진영남수학회 · East Asian Mathematical Journal2020 · 제36권 제3호Commuting Structure Jacobi Operator for Semi-Invariant Submanifolds of Codimension 3 in Complex Space Forms기우항, 송현정영남수학회 · East Asian Mathematical Journal2022 · 제38권 제5호Structure Jacobi Operators of Semi-invarinat Submanifolds in a Complex Space Form II기우항, 김수진영남수학회 · East Asian Mathematical Journal2022 · 제38권 제1호

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