For more than 150 years, guiding ideas in geometry have shaped the way mathematicians think about surfaces. Developed by the French mathematician Pierre Ossien Bonnet, this principle states that if you know two important properties of a densely curved surface at every point: its distance and its average curvature, you can determine its exact shape. New results from mathematicians at the Technical University of Munich (TUM), the Technical University of Berlin, and North Carolina State University show that this assumption does not always hold true.
To challenge long-held beliefs, researchers constructed two compact, self-contained surfaces shaped like donuts known as tori. These two surfaces share the same values for both metric and average curvature, but their overall structure is not the same. Examples of this type have been sought for decades but have never been found.
Metrics represent distances along a surface. This means how far apart two points are when measured on a surface. Average curvature captures how a surface curves in space, indicating whether and how much it curves inward or outward.
Limits of rules regarding bonnet surface shape
Mathematicians already recognized that Bonnet’s law does not apply in all situations. Known exceptions include non-compact surfaces that extend to infinity, such as planes, or have edges at their ends. In contrast, dense surfaces such as spheres were thought to follow rules, with metric and average curvature completely determining their shape.
For torus-shaped surfaces, previous studies have shown that a single set of metric and average curvature values can accommodate two different shapes. However, no one has been able to provide a clear, concrete example to demonstrate this possibility.
The long-sought counterexample has finally been found.
The new work fills that gap. By constructing pairs of toruses that matched locally but differed globally, the researchers provided the first clear example of this phenomenon.
“After many years of research, we have succeeded for the first time in finding a concrete case showing that local measurement data do not necessarily determine a single global shape, even for a donut-shaped closed surface,” says Tim Hoffmann, Professor of Applied and Computational Topology at the TUM School of Computational and Information Technology. “This allows us to solve decades-old problems in the differential geometry of surfaces.”
The discovery solved a long-standing question in geometry and revealed deeper insights. Even with complete local information, the complete shape of a surface cannot always be uniquely determined.
