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Kaehler submanifolds of complex space forms. (English) Zbl 0679.53016
Under the notion “diastasis” introduced by E. Calabi [Ann. Math., II. Ser. 58, 1-23 (1953; Zbl 0051.131)] the author proves the following theorem: Any two complex space forms of different types have no Kaehler submanifold in common, that is, (1) a Kaehler submanifold of $${\mathbb{C}}^ N$$ cannot be a Kaehler submanifold of any complex hyperbolic space, (2) a Kaehler submanifold of $${\mathbb{C}}^ N$$ cannot be a Kaehler submanifold of any complex projective space, (3) a Kaehler submanifold of a complex hyperbolic space cannot be a Kaehler submanifold of any complex projective space.
Reviewer: S.S.Singh

##### MSC:
 53B25 Local submanifolds 53B35 Local differential geometry of Hermitian and Kählerian structures 32Q45 Hyperbolic and Kobayashi hyperbolic manifolds
##### Keywords:
complex space forms; Kaehler submanifold
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