Semi-invariant submanifolds of codimension $3$ in a complex space forms in terms of the structure Jacobi operator

Commun. Korean Math. Soc. Published online July 7, 2021

hiroyuki kurihara and u-hang ki
Ibaraki University; The National Academy of Science

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 $M_{n +1} (c), c\ne 0$. 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 $dt (X,Y) =2 \theta g (\phi X, Y)$ for a scalar $\theta (< 2 c)$ 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 $\nabla_{\xi} R_{\xi} =0$ 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