A long-standing principle in classical surface theory, dating back over 150 years to Pierre Ossian Bonnet, states that the geometry of a compact surface is uniquely determined by its metric and mean curvature. A team of mathematicians from Technische Universität Berlin, the Technical University of Munich (TUM), and North Carolina State University has now shown that this is not always the case.

The researchers constructed two distinct donut-shaped surfaces (tori) that share the same metric and mean curvature but differ in their global geometry—an example that had eluded mathematicians for decades. While theoretical work had suggested that such pairs might exist for tori, no explicit construction had previously been found.

“Discrete differential geometry played a crucial role in our research. This is a modern mathematical discipline with important applications, which we developed as part of the Collaborative Research Centre SFB/TRR 109 ‘Discretization in Geometry and Dynamics,'” says MATH+ member Alexander Bobenko from the Geometry and Mathematical Physics Group at TU Berlin, the coordinating institution for SFB/Transregio 109; TU Berlin and TU Munich were two of the main centers for the SFB.

For more detailed information about the research and its implications, please refer to the original TU Berlin press release.

The surprising discovery and its detailed story are the subject of a recent Quanta article, as well as the film “Solving the Bonnet Problem.”

Publication:

Bobenko, A.I., Hoffmann, T. & Sageman-Furnas, A.O. Compact Bonnet pairs: Isometric tori with the same curvatures. Publications Mathématiques de l’IHÉS, 142, 241–293 (2025): https://doi.org/10.1007/s10240-025-00159-z.

LINKS:

• TU Berlin press release: Researchers Resolve Age-Old Problem of Classical Geometry
• TU Berlin news: Solving the Bonnet Problem: A Breakthrough in Differential Geometry
• Quanta article: “Two Twisty Shapes Resolve a Centuries-Old Topology Puzzle”
• Film “Solving the Bonnet Problem”