Localized volume perturbation ($d{V}_{(1+t\eta )g}=(1+t\eta {)}^{m/2}d{V}_g$)

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.