= Curvature markers distinguishing finite-cover quotients
For a finite isometric group action on a closed <Riemannian surface>, choose a disk in the free locus whose translates are disjoint. Modify the <Riemannian metric> there, and identically on its translates, by small supported changes with three distinguished high <Gaussian curvature> maxima. Give their heights distinct values occurring nowhere else, arrange them as a tight noncollinear cluster, and keep separate clusters far apart. Such markers can be made by local conformal bump functions controlling curvature and its derivatives. Every quotient <isometry> must send an entire labelled cluster to another. After identifying the marked disks with the original one, an <isometry> fixes its three noncollinear points and is therefore the identity locally. If $p_i:M\to U_i\backslash M$ are free finite quotients, any quotient <isometry> $F$ consequently makes $F\circ p_1=p_2\circ t$ on an open disk for some element $t$ of the original group. Two local isometries agreeing on an open set agree on the connected surface, by propagation along geodesics. The equality therefore holds globally and implies $tU_1t^{-1}\subseteq U_2$, hence equality for equal subgroup orders. Nonconjugate equal-order subgroups thus have nonisometric quotients for this marked invariant metric; constant curvature is not required.
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