SEMI-INVARIANT SUBMANIFOLDS OF CODIMENSION 3 IN A COMPLEX SPACE FORM IN TERMS OF THE STRUCTURE JACOBI OPERATOR

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 not equal 6 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(< 2c) and any vector fields X and Y on M. In this paper, we prove that if it satisfies R(xi)A = AR(xi) and at the same time del R-xi(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 <= 0.