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!
2026-04-02 03:45:38.1775101538
"Rigid" Riemannian metrics
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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).