This seems like it could be false but I can't think of any counter examples. I'd appreciate some help.
2026-03-26 17:53:33.1774547613
If a (smooth) deformation of a curve on a surface $S$ preserves geodesic curvature, is it necessarily an isometry of $S$?
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No. Consider an arbitrary invertible linear map of the plane. It carries lines through the origin (which have $\kappa_g=0$) to other lines through the origin. Yet it's an isometry of the plane only when it's given by an orthogonal matrix (with respect to the standard basis).