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 \(\mathbb Q^{m}(\varepsilon)\), where \(\varepsilon=\pm 1\). Here, \(\mathbb Q^{m}(\varepsilon)\) represents the complex quadric \(\mathbb Q^{m}(1)=SO_{m+2}/SO_{m}SO_{2}\) for \(\varepsilon=1\) and \(\mathbb Q^{m}(-1)=SO_{m,2}^{0}/SO_{m}SO_{2}\) for \(\varepsilon =-1\), 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 \({\mathbb Q^{m}(\varepsilon)}\), \(m \geq 3\), 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 \(\mathbb Q^{m}(\varepsilon)\), \(m \geq 3\), admitting a semi-parallel normal Jacobi operator.