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Kähler manifolds and their relatives. (English) Zbl 1253.53066
Summary: Let $$M_1$$ and $$M_2$$ be two Kähler manifolds. We call $$M_1$$ and $$M_2$$ relatives if they share a non-trivial Kähler submanifold $$S$$, namely, if there exist two holomorphic and isometric immersions (Kähler immersions) $$h_1: S\to M_1$$ and $$h_2: S\to M_2$$. Moreover, two Kähler manifolds $$M_1$$ and $$M_2$$ are said to be weak relatives if there exist two locally isometric (not necessarily holomorphic) Kähler manifolds $$S_1$$ and $$S_2$$ which admit two Kähler immersions into $$M_1$$ and $$M_2$$, respectively. These notions are not equivalent. Our main results are Theorem 1.2 and Theorem 1.4.
In the first theorem we show that a complex bounded domain $$D\subset\mathbb{C}^n$$ with its Bergman metric and a projective Kähler manifold (i.e. a projective manifold endowed with the restriction of the Fubini-Study metric) are not relatives.
In the second theorem we prove that a Hermitian symmetric space of noncompact type and a projective Kähler manifold are not weak relatives. Notice that the proof of the second result does not follows trivially from the first one. We also remark that the above results are of local nature. i.e. no assumptions are used about the compactness or completeness of the manifolds involved.

MSC:
 53C55 Global differential geometry of Hermitian and Kählerian manifolds 58C25 Differentiable maps on manifolds
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