Real hypersurfaces and complex submanifolds in complex projective space

Abstract

Let M M be a real hypersurface in P n ( C ) {P^n}({\mathbf {C}}) be the complex structure and ξ \xi denote a unit normal vector field on M M . We show that M M is (an open subset of) a homogeneous hypersurface if and only if M M has constant principal curvatures and J ξ J\xi is principal. We also obtain a characterization of certain complex submanifolds in a complex projective space. Specifically, P m ( C ) {P^m}({\mathbf {C}}) (totally geodesic), Q n , P 1 ( C ) × P n ( C ) , S U ( 5 ) / S ( U ( 2 ) × U ( 3 ) ) {Q^n},{P^1}({\mathbf {C}}) \times {P^n}({\mathbf {C}}),SU(5)/S(U(2) \times U(3)) and S O ( 10 ) / U ( 5 ) 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.