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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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