= Localized volume perturbation
{title2=$dV_{(1+t\eta)g}=(1+t\eta)^{m/2}dV_g$}
For a nonnegative smooth <bump function> supported in an open set, multiplying a <Riemannian metric> by $1+t\eta$ increases volume there for $t>0$ while leaving the metric elsewhere fixed. For distinct regular compact domains a perturbation supported in the interior of one outside the other breaks equal volume, and therefore breaks any <isometry> between the domains. The perturbations tend to the original metric in the smooth topology.
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