  The Gromov-Wasserstein distances were proposed a few years ago to compare
distributions which do not lie in the same space. In particular, they offer an
interesting alternative to the Wasserstein distances for comparing probability
measures living on Euclidean spaces of different dimensions. In this paper, we
focus on the Gromov-Wasserstein distance with a ground cost defined as the
squared Euclidean distance and we study the form of the optimal plan between
Gaussian distributions. We show that when the optimal plan is restricted to
Gaussian distributions, the problem has a very simple linear solution, which is
also solution of the linear Gromov-Monge problem. We also study the problem
without restriction on the optimal plan, and provide lower and upper bounds for
the value of the Gromov-Wasserstein distance between Gaussian distributions.
