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On a 2-dimensional Einstein Kaehler submanifold of a complex space form. (English) Zbl 0536.53023
In this paper the author considers when a Kaehler submanifold of a complex space form is Einstein with respect to the induced metric. Then he shows that (1) a 2-dimensional complete Kaehler submanifold M of a 4- dimensional complex projective space \(P^ 4(C)\) is Einstein if and only if M is holomorphically isometric to \(P^ 2(C)\) which is totally geodesic in \(P^ 4(C)\) or a hyper-quadric \(Q^ 2(C)\) in \(P^ 3(C)\) which is totally geodesic in \(P^ 4(C)\), and that (2) if M is a 2- dimensional Einstein Kaehler submanifold of a 4-dimensional complex space form \(\tilde M^ 4(\tilde c)\) of non-positive constant holomorphic sectional curvature \(\tilde c\), then M is totally geodesic.

MSC:
53B25 Local submanifolds
53C40 Global submanifolds
53B35 Local differential geometry of Hermitian and Kählerian structures
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