Geodesic
Einstein-Kaehler Manifolds Immersed in a Complex Projective Space
Canadian journal of mathematics, Tome 28 (1976) no. 1, pp. 1-8

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A Kaehler manifold of constant holomorphic curvature is called a complex space form. By a Kaehler submanifold we mean a complex submanifold with the induced Kaehler metric. B. Smyth [5] has studied a complete Einstein- Kaehler hypersurface in a complete and simply connected complex space form and classified completely the hypersurface. The local version of this result has been shown to be true by S. S. Chern [1], and partially by T. Takahashi [6] independently. 
Nakaga, Hisao. Einstein-Kaehler Manifolds Immersed in a Complex Projective Space. Canadian journal of mathematics, Tome 28 (1976) no. 1, pp. 1-8. doi: 10.4153/CJM-1976-001-9
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[1] 1. Chern, S. S., Einstein hyper surfaces in a Kaehlerian manifold of constant holomorphic curvature, J. Differential Geometry 1 (1967), 21–31. Google Scholar

[2] 2. Nakagawa, H. and Takagi, R., On symmetric Kaehler submanifolds in a complex projective space (to appear). Google Scholar

[3] 3. Nomizu, K. and Smyth, B., Differential geometry of complex hyper surf aces II, J. Math. Soc. Japan 20 (1968), 498–527. Google Scholar

[4] 4. Ogiue, K., Differential geometry of Kaehler submanifolds, Advances in Math. 13 (1974), 73–114. Google Scholar

[5] 5. Smyth, B., Differential geometry of complex hyper surf aces, Ann. of Math. 85 (1967), 246–266. Google Scholar

[6] 6. Takahashi, T., Hyper surfaces with parallel Ricci tensor in a space of constant holomorphic sectional curvature, J. Math. Soc. Japan 19 (1967), 199–204. Google Scholar

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