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Einstein Kaehler submanifolds of a complex linear or hyperbolic space. (English) Zbl 0648.53031
The author proves that every Einstein Kaehler submanifold of $${\mathbb{C}}^ n$$ or $${\mathbb{C}}H^ n$$ (the complex hyperbolic space of negative constant holomorphic sectional curvature) is always totally geodesic. He uses in his proof some previous results (by himself) on the so-called “diastasis” associated to a metric, a notion introduced by E. Calabi in Ann. Math., II. Ser. 58, 1-23 (1953; Zbl 0051.131)].
Reviewer: J.Girbau

##### MSC:
 53C40 Global submanifolds
##### Keywords:
Einstein Kaehler submanifold; totally geodesic
Full Text:
##### References:
 [1] E. CALABI, Isometric imbedding of complex manifolds, Ann. of Math. 58 (1953), 1-23. · Zbl 0051.13103 [2] S. S. CHERN, On Einstein hypersurfaces in a Kaehlerian manifold of constant holomorphi sectional curvature, J. Differential Geom. 1 (1967), 21-31. · Zbl 0168.19505 [3] S. KOBAYASHI AND K. NOMIZU, Foundation of Differential Geometry, vol. 2, Wiley Interscience, New York, 1969. · Zbl 0526.53001 [4] B. SMYTH, Differential geometry of complex hypersurface, Ann. of Math. 85 (1967), 246-266. · Zbl 0168.19601 [5] T. TAKAHASHI, Hypersurface with parallel Ricci tensor in a space of constant holomorphi sectional curvature, J. Math. Soc. Japan 19 (1967), 199-204. · Zbl 0147.40603 [6] K. TSUKADA, Einstein Kaehler submanifolds with codimension 2 in a complex spac form, Math. Ann. 274 (1986), 503-516. · Zbl 0592.53046 [7] M. UMEHARA, Kaehler submanifolds of complex space forms, to appear in Tokyo J. Math · Zbl 0679.53016 [8] M. UMEHARA, Diastases and real analytic functions on complex manifolds, to appear i J. Math. Soc. Japan. · Zbl 0651.53046
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