REAL HYPERSURFACES WITH SEMI-PARALLEL NORMAL JACOBI OPERATOR IN THE REAL GRASSMANNIANS OF RANK TWO

In this paper, we introduce the notion of a semi-parallel normal Jacobi operator for a real hypersurface in the real Grassmannian of rank two, denoted by Q(m) (e), where e = +/- 1. Here, Q(m) (e) represents the complex quadric Q(m) (1) = SOm+2/SOmSO2 for e = 1 and Q(m) (-1) = SOm,20/SOmSO2 for e = -1, respectively. In general, the notion of semi- parallel is weaker than the notion of parallel normal Jacobi operator. In this paper we prove that the unit normal vector field of a Hopf real hypersurface in Q(m) (e), m > 3, with semi-parallel normal Jacobi operator is singular. Moreover, the singularity of the normal vector field gives a nonexistence result for Hopf real hypersurfaces in Q(m) (e), m > 3, admitting a semi- parallel normal Jacobi operator.