For over 150 years, a rule of thumb dating back to the French mathematician Pierre Ossian Bonnet has been accepted in surface theory: if the metric and mean curvature of a compact surface are known at every point, then the surface can be uniquely determined. However, a team of three mathematicians from the Technical University of Munich (TUM), the Technical University of Berlin and North Carolina State University have now managed to disprove this recognized rule.
Astrid Eckert / TUM To do this, they constructed two compact, self-contained, doughnut-shaped surfaces, known as tori, which have the same metric and mean curvature, even though they are structurally different on a global scale. Researchers had been searching in vain for such an example for decades. The metric describes the distances on the surface, that is, how far two points on the surface are from each other. The mean curvature indicates how strongly the surface curves outward or inward in space.
Bonnets' rule of thumb and its limitations
Exceptions to Bonnet's rule of thumb were known, but they only occurred in non-compact surfaces. Such surfaces either extend infinitely in one direction, like a plane, or have edges where they end abruptly. For compact surfaces such as spheres, however, it has since been shown that the metric and the mean curvature uniquely determine the surface. For tori, it was known that a given metric and a given mean curvature can correspond to at most two different torus surfaces. Although this possibility has been known for decades, a concrete case was lacking.
Missing example for a tori pair found
The three mathematicians now provide precisely this example. "After many years of research, we have succeeded for the first time in finding a concrete case that shows that even for closed, doughnut-like surfaces, local measurement data do not necessarily determine a single global shape," says Tim Hoffmann, Professor of Applied Computational Mathematics at the TUM School of Computation, Information and Technology. "This allows us to solve a decades-old problem in differential geometry for surfaces."
Bobenko, A.I., Hoffmann, T. & Sageman-Furnas, A.O. Compact Bonnet pairs: isometric tori with the same curvatures. Publ.math.IHES 142, 241-293 (2025). https://doi.org/10.1007/s10240-025-00159-z
