  We show that if $Y_j\subset \mathbb{C}^{n_j}$ is a bounded strongly convex
domain with $C^3$-boundary for $j=1,\dots,q$, and $X_j\subset \mathbb{C}^{m_j}$
is a bounded convex domain for $j=1,\ldots,p$, then the product domain
$\prod_{j=1}^p X_j\subset \mathbb{C}^m$ cannot be isometrically embedded into
$\prod_{j=1}^q Y_j\subset \mathbb{C}^n$ under the Kobayashi distance, if $p>q$.
This result generalises Poincar\'e's theorem which says that there is no
biholomorphic map from the polydisc onto the Euclidean ball in $\mathbb{C}^n$
for $n\geq 2$.
  The method of proof only relies on the metric geometry of the spaces and will
be derived from a result for products of proper geodesic metric spaces with the
sup-metric. In fact, the main goal of the paper is to establish a general
criterion, in terms of certain asymptotic geometric properties of the
individual metric spaces, that yields an obstruction for the existence of an
isometric embedding between product metric spaces.
