We introduce the notion of Killing normal Jacobi operator for real hypersurfaces in the complex quadric Qm = SOm+2/SOmSO2. The Killing normal Jacobi operator implies that the unit normal vector field N becomes 𝔄-principal or 𝔄-isotropic. Then according to each case, we give a complete classification of real hypersurfaces in Qm = SOm+2/SOmSO2 with Killing normal Jacobi operator.

It is known that the minimal 3-spheres of CR type with constant sectional curvature have been classified explicitly, and also that the weakly Lagrangian case has been studied. In this paper, we provide some examples of minimal 3-spheres with constant curvature in the complex projective space, which are neither of CR type nor weakly Lagrangian, and give the adapted frame of a minimal 3-sphere of CR type with constant sectional curvature.

In this paper we prove that minimal 3-spheres of CR type with constant sectional curvature c in the complex projective space CPn are all equivariant and therefore the immersion is rigid. The curvature c of the sphere should be c = 1/(m2-1) for some integer m≥ 2, and the full dimension is n = 2m2-3. An explicit analytic expression for such an immersion is given.

For a parabolic subgroup H of the general linear group G = Gl(n, C), we characterize the Kähler classes of G/H and give a formula for the height of any two-dimensional cohomology class. As an application, we classify the automorphisms of the cohomology ring of G/H when this ring is generated by two-dimensional classes.