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

Matsuyama, Yoshio

doi:10.1090/S0002-9939-1985-0810170-0

On a $2$-dimensional Einstein Kaehler submanifold of a complex space form
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by Yoshio Matsuyama PDF

Proc. Amer. Math. Soc. 95 (1985), 595-603 Request permission

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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