Two tori can share the same local geometry and still be different shapes
A construction in Publications mathematiques de l'IHES gives the first compact Bonnet pairs: two smooth, real-analytic tori in three-dimensional space that have the same intrinsic metric and the same mean curvature at corresponding points, yet are not the same surface up to a rigid motion. The result closes old uniqueness questions in surface geometry, and it is a precise counterexample to a tempting idea: that enough local measurements must identify a compact shape.