A 150-year-old geometry rule has been overturned after mathematicians found two different torus surfaces with identical metric and curvature.
For more than 150 years, a principle attributed to French mathematician Pierre Ossian Bonnet has guided surface theory. It states that if the metric and mean curvature of a compact surface are known at every point, the surface can be uniquely identified. A team of three mathematicians from the Technical University of Munich (TUM), the Technical University of Berlin, and North Carolina State University has now shown that this widely accepted idea is not always true.
After years of work, the researchers identified a clear example demonstrating that even closed, donut-shaped surfaces cannot always be uniquely determined from local measurements alone. Their findings are published in the journal Publications mathématiques de l’IHÉS.
To reach this result, they created two compact, self-contained surfaces shaped like donuts, known as tori. These surfaces share identical metric and mean curvature values, yet differ in their structure. For decades, mathematicians had searched unsuccessfully for such a case.
Understanding Metric and Mean Curvature
The metric defines distances along a surface, describing how far apart two points are. Mean curvature measures how much the surface bends, either outward or inward, within space.
Previously known exceptions to Bonnet’s rule applied only to non-compact surfaces. These include shapes that extend infinitely, such as planes, or surfaces with edges where they end. In contrast, compact surfaces like spheres were believed to be fully determined by their metric and mean curvature.
For tori, it had long been understood that a single set of metric and mean curvature values could correspond to no more than two distinct shapes. However, no explicit example had ever been found.
Solving a Long-Standing Mathematical Problem
The researchers have now provided that missing 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 and Computational Topology at the TUM School of Computation, Information, and Technology.
“This allows us to solve a decades-old problem in differential geometry for surfaces.”
Reference: “Compact Bonnet pairs: isometric tori with the same curvatures” by Alexander I. Bobenko, Tim Hoffmann and Andrew O. Sageman-Furnas, 14 October 2025, Publications mathématiques de l’IHÉS.
DOI: 10.1007/s10240-025-00159-z
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