Real Hypersurfaces and Complex Submanifolds in Complex Projective Space

Let M be a real hypersurface in Pⁿ(C) be the complex structure and ξ denote a unit normal vector field on M. We show that M is (an open subset of) a homogeneous hypersurface if and only if M has constant principal curvatures and Jξ is principal. We also obtain a characterization of certain complex submanifolds in a complex projective space. Specifically, Pᵐ(C) (totally geodesic), Qⁿ,P¹(C) × Pⁿ(C),SU(5)/S(U(2) × U(3)) and $SO(10)/U(5)$ are the only complex submanifolds whose principal curvatures are constant in the sense that they depend neither on the point of the submanifold nor on the normal vector.