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- 학술저널
- 저자정보
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- 2024.1
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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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