Kaehler manifolds and their relatives

Let M_1 and M_2 be two Kaehler manifolds. We call M_1 and M_2 relatives if they share a non-trivial Kaehler submanifold S, namely, if there exist two holomorphic and isometric immersions (Kaehler immersions) h_1: S--> M_1 and h_2:S--> M_2. Moreover, two Kaehler manifolds M_1 and M_2 are said to be weakly relatives if there exist two locally isometric not necessarily holomorphic) Kaehler manifolds S_1 and S_2 which admit two Kaehler immersions into M_1 and M_2 respectively. The notions introduced are not equivalent. Our main results in this paper are Theorem 1 and Theorem 2. In the first theorem we show that a complex bounded domain D\subset C^n with its Bergman metric and a projective Kaehler 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 Kaehler manifold are not weakly 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.