  Let $M_1$ and $M_2$ be two K\"ahler manifolds. We call $M_1$ and $M_2$ {\em
relatives} if they share a non-trivial K\"ahler submanifold $S$, namely, if
there exist two holomorphic and isometric immersions (K\"ahler immersions)
$h_1: S\to M_1$ and $h_2: S\to M_2$. Moreover, two K\"ahler manifolds $M_1$ and
$M_2$ are said to be {\em weakly relatives} if there exist two locally
isometric (not necessarily holomorphic) K\"ahler manifolds $S_1$ and $S_2$
which admit two K\"ahler immersions into $M_1$ and $M_2$ respectively. The
notions introduced are not equivalent. Our main results in this paper are
Theorem 1.2 and Theorem 1.4. In the first theorem we show that a complex
bounded domain with its Bergman metric and a projective K\"ahler 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 K\"ahler manifold are not weakly
relatives. 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. As a corollary we get that Hermitian symmetric spaces of different
types are not weakly relatives.
