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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

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


Author: Yoshio Matsuyama
Journal: Proc. Amer. Math. Soc. 95 (1985), 595-603
MSC: Primary 53C25; Secondary 53C40
MathSciNet review: 810170
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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$-dimensional complete Kaehler submanifold $ M$ of a $ 4$-dimensional complex projective space $ {P^4}\left( C \right)$ is Einstein if and only if $ M$ is holomorphically isometric to $ {P^2}\left( C \right)$ which is totally geodesic in $ {P^4}\left( C \right)$ or a hyperquadric $ {Q^2}\left( C \right)$ in $ {P^3}\left( C \right)$ which is totally geodesic in $ {P^4}\left( C \right)$, and that (2) if $ M$ is a $ 2$-dimensional Einstein Kaehler submanifold of a $ 4$-dimensional complex space form $ {\tilde M^4}\left( {\tilde c} \right)$ of nonpositive constant holomorphic sectional curvature $ \tilde c$, then $ M$ is totally geodesic.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1985-0810170-0
PII: S 0002-9939(1985)0810170-0
Keywords: Einstein Kaehler submanifolds, complex space forms, second fundamental forms
Article copyright: © Copyright 1985 American Mathematical Society