On a 2-dimensional Einstein Kaehler submanifold of a complex space form

Abstract

In this paper we consider when a Kaehler submanifold of a complex space form is Einstein with respect to the induced metric. Then we shall show that (1) a 2 2 -dimensional complete Kaehler submanifold M M of a 4 4 -dimensional complex projective space P 4 ( C ) {P^4}\left ( C \right ) is Einstein if and only if M M is holomorphically isometric to P 2 ( C ) {P^2}\left ( C \right ) which is totally geodesic in P 4 ( C ) {P^4}\left ( C \right ) or a hyperquadric Q 2 ( C ) {Q^2}\left ( C \right ) in P 3 ( C ) {P^3}\left ( C \right ) which is totally geodesic in P 4 ( C ) {P^4}\left ( C \right ) , and that (2) if M M is a 2 2 -dimensional Einstein Kaehler submanifold of a 4 4 -dimensional complex space form M ~ 4 ( c ~ ) {\tilde M^4}\left ( {\tilde c} \right ) of nonpositive constant holomorphic sectional curvature c ~ \tilde c , then M M is totally geodesic.