Communications of the
Korean Mathematical Society CKMS

Let $M$ be a semi-invariant submanifold of codimension $3$ with almost contact metric structure $(phi, xi, eta, g)$ in a complex space form $M_{n +1} (c), c
e 0$. We denote by $A$ and $R_{xi}$ the shape operator in the direction of distinguished normal vector field and the structure Jacobi operator with respect to the structure vector $xi$, respectively. Suppose that the third fundamental form $t$ satisfies $dt (X,Y) =2 heta g (phi X, Y)$ for a scalar $ heta (< 2 c)$ and any vector fields $X$ and $Y$ on $M$. In this paper, we prove that if it satisfies $R_{xi} A =A R_{xi}$ and at the same time $
abla_{xi} R_{xi} =0$ on $M$, then $M$ is a Hopf hypersurface of type ($A$) provided that the scalar curvature $s$ of $M$ holds $s -2(n -1)c leq 0$.

Keywords: Semi-invariant submanifold, almost contact metric structure, the third fundamental form, distingushed normal vector, structure Jacobi operator, Hopf real hypersurface

MSC numbers: Primary 53B25, 53C40, 53C42