Geodesic
On Some Complex Submanifolds in Kaehler Manifolds
Canadian journal of mathematics, Tome 26 (1974) no. 6, pp. 1442-1449

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The purpose of this paper is to give some conditions for complex submanifolds in a Kaehler manifold of constant holomorphic sectional curvature to be Einstein.For a complex hypersurface which is Einstein, Smyth [8] has obtained its classification and Chern [2] has proved the corresponding local result. Moreover, Takahashi [9] and Nomizu-Smyth [3] generalized this to a complex hypersurface with parallel Ricci tensor. We shall consider a condition weaker than the requirement that the Ricci tensor be parallel, that is we shall consider a complex submanifold with commuting curvature and Ricci operator, which condition was treated by Bishop-Goldberg [1]. 
Kon, Masahiro. On Some Complex Submanifolds in Kaehler Manifolds. Canadian journal of mathematics, Tome 26 (1974) no. 6, pp. 1442-1449. doi: 10.4153/CJM-1974-138-5
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