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Structure Jacobi Operators of Semi-invarinat Submanifolds in a Complex Space Form II

김수진|기우항
영남수학회 (Youngnam Mathematical Society)|2022.01|East Asian Mathematical Journal|제38권 제1호|pp. 43-63

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키워드

structure Jacobi operatorSemi-invariant submanifoldalmost contact metric structureHopf hypersurfacescalar curvature.
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초록

Let $M$ be a semi-invariant submanifold of codimension 3 with almost contact metric structure$(\phi, \xi, \eta, g)$ in a complex space form $M_{n+1} (c)$. We denote by $R_\xi$ the structureJacobi operator with respect to the structure vector field $\xi$ and by $\bar{r}$ the scalar curvature of $M$. Suppose that $R_\xi$ is $\phi\nabla_\xi \xi$-parallel and at the same time 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 $Y$ on $M$. In this paper, we prove that if it satisfies $R_\xi \phi=\phi R_\xi$, then $M$ is a Hopf hypersurfaceof type (A) in $M_{n+1} (c)$ provided that $\bar{r}-2(n-1)c\leq0$.

Abstract

Let $M$ be a semi-invariant submanifold of codimension 3 with almost contact metric structure$(\phi, \xi, \eta, g)$ in a complex space form $M_{n+1} (c)$. We denote by $R_\xi$ the structureJacobi operator with respect to the structure vector field $\xi$ and by $\bar{r}$ the scalar curvature of $M$. Suppose that $R_\xi$ is $\phi\nabla_\xi \xi$-parallel and at the same time 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 $Y$ on $M$. In this paper, we prove that if it satisfies $R_\xi \phi=\phi R_\xi$, then $M$ is a Hopf hypersurfaceof type (A) in $M_{n+1} (c)$ provided that $\bar{r}-2(n-1)c\leq0$.

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관련 논문AI추천

Structure Jacobi Operator of Semi-invarinat Submanifolds in Complex Space Forms기우항, 김수진영남수학회 · East Asian Mathematical Journal2020 · 제36권 제3호Semi-invariant submanifolds of codimension 3 in a complex space form in terms of the structure Jacobi operatorHiroyuki Kurihara, U-Hang Ki대한수학회 · 대한수학회논문집2022 · 제37권 제1호SEMI-INVARIANT SUBMANIFOLDS OF CODIMENSION 3 IN A COMPLEX SPACE FORM WITH ξ-PARALLEL STRUCTURE JACOBI OPERATOR기우항, 송현정영남수학회 · East Asian Mathematical Journal2024 · 제40권 제1호Commuting Structure Jacobi Operator for Semi-Invariant Submanifolds of Codimension 3 in Complex Space Forms기우항, 송현정영남수학회 · East Asian Mathematical Journal2022 · 제38권 제5호SEMI-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 제1호

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