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Einstein Kähler submanifolds with codimension 2 in a complex space form. (English) Zbl 0592.53046
Einstein-Kähler hypersurfaces in complex space forms are completely classified by B. Smyth [Ann. Math., II. Ser. 85, 246-266 (1967; Zbl 0168.196)] and S. S. Chern [J. Differ. Geom. 1, 21-31 (1967; Zbl 0168.195)]. The author proves that the same classification holds for Einstein-Kähler submanifolds of codimension 2 in complex space forms, that is, he proves the following: Let M be an n-dimensional Einstein- Kähler submanifold immersed in $$P_{n+2}(C)$$, $$C^{n+2}$$ or $$D^{n+2}$$. Then M is either totally geodesic or locally a complex hyperquadric in some $$P_{n+1}(C)$$ in $$P_{n+2}(C)$$.
Reviewer: K.Ogiue

##### MSC:
 53C40 Global submanifolds 53C55 Global differential geometry of Hermitian and Kählerian manifolds
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##### References:
 [1] Chern, S.S.: On Einstein hypersurfaces in a Kähler manifold of constant holomorphic sectional curvature. J. Differ. Geom.1, 21-31 (1967) · Zbl 0168.19505 [2] Matsuyama, Y.: On a 2-dimensional Einstein Kähler submanifold of a complex space form. Proc. Am. Math. Soc.95, 595-603 (1985) · Zbl 0536.53023 [3] Nakagawa, H., Takagi, R.: On locally symmetric Kähler submanifolds in a complex projective space. J. Math. Soc. Japan28, 638-667 (1976) · Zbl 0328.53009 [4] Ogiue, K.: Differential geometry of Kähler submanifolds. Adv. Math.13, 73-114 (1974) · Zbl 0275.53035 [5] Smyth, B.: Differential geometry of complex hypersurfaces. Ann. Math.85, 246-266 (1967) · Zbl 0168.19601
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