"Rigid" Riemannian metrics What do we mean when we say that a Riemannian metric $g$ is rigid? For example, the Eguchi-Hanson metric is rigid as an Einstein metric. Any help is appreciated!

It means that all other metrics with this property are isometric to the given metric.

For example the Mostow rigidity theorem says that all hyperbolic metrics on a closed n-Manifold are isometric to each other, if $n\ge 3$. So the hyperbolic metric is rigid in this case.

On the other Hand, hyperbolic metrics on 2-manifolds are not rigid. There is a whole modulo space of them (Teichmüller theory).