One of differential geometry’s most challenging problems has finally been solved after 150 years, as an international team of researchers has identified a closed surface Bonnet pair for the first time.
Pierre Ossian Bonnet, a consequential French mathematician whose name graces several important theorems in differential geometry, asked a consequential geometry question a century and a half ago, to which these Bonnet pairs form exceptions.
In their new paper published in Publications Mathématiques de l’IHÉS, a trio of mathematicians from the Technical University of Munich (TUM), the Technical University of Berlin, and North Carolina State University demonstrated that the donut shape provides a new answer to an old geometry puzzle.
Bonnet’s Theorem in Differential Geometry
The new work is adjacent to Bonnet’s theorem, one of the most fundamental concepts in differential geometry, but asks a slightly different question.
“Bonnet’s theorem is closely related to the question at hand, but it is not the same,” co-author Tim Hoffman explained to The Debrief. “It states that a surface is uniquely determined (up to rigid motions) by its metric (that is, basically how lengths are measured on the surface) and its so-called second fundamental form (the metric being the first one), which roughly describes how the normals of the surface change.”
“Now, the unsatisfying thing about this result is that the metric and the second fundamental form need to satisfy some intricate compatibility conditions. They are not independent,” Hoffman said. “So Bonnet asked the question if metric and mean curvature together could be such an independent minimal set of information that determines the surface uniquely (again, up to rigid motions of course).”
“Here, the answer is generically yes, but unfortunately, there are exceptions,” Hoffman adds.
Geometry Finds Bonnet Pairs
Bonnet knew at the time that some exceptions to this idea exist, a fact which has since been confirmed by other researchers.
“Bonnet already knew this. If we just look at a small piece of a surface, generically, metric and mean curvature determine it uniquely,” Hoffman explained. “But in some cases, there are two non-congruent surfaces that share metric and mean curvature. These are called Bonnet pairs.”
While other researchers have identified Bonnet pairs in the past, all have been on non-closed surfaces, either with distinct edges or stretching into infinity. Whether a closed surface Bonnet pair could exist remained a lingering question in geometry.
“There are also some exceptions where whole families of surfaces share mean curvature and metric (so in those cases, there are infinitely many surfaces with the same data). However, this is for non-closed surfaces,” Hoffman said.
“You should think of a closed surface as one that has no border. A sheet of paper, for example, does have a border,” Hoffman added. “If an ant walks on it, it might hit the rim at some point. In contrast, the surface of a tennis ball or a donut are closed surfaces. There is no boundary the ant could hit.”
Closed Surface Bonnet Pair
It can be shown that for a closed surface without holes (like a tennis ball), metric and mean curvature do indeed determine the surface uniquely. So the open question was: could this be true for all closed surfaces?
“It had been shown that for tori (donut-shaped closed surfaces), there could be, at most, two surfaces with the same mean curvature and metric,” Hoffman explained. “So they gave an upper bound on how many Bonnet mates a torus could have: a torus is either unique or it might have one Bonnet mate, but there was no statement on whether this could actually happen.”
With their numerically produced tori, the three researchers behind the paper finally found a concrete taurus-shaped Bonnet pair.
“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,” Hoffman concluded. “This allows us to solve a decades-old problem in differential geometry for surfaces.”
The paper, “Compact Bonnet Pairs: Isometric Tori with the Same Curvatures,” appeared in Publications Mathématiques de l’IHÉS on October 14, 2025.
Ryan Whalen covers science and technology for The Debrief. He holds an MA in History and a Master of Library and Information Science with a certificate in Data Science. He can be contacted at ryan@thedebrief.org, and follow him on Twitter @mdntwvlf.