Non-rigidity and symmetry breaking

The principal aim of the project is to gain a better understanding of "symmetry breaking" in geometry. Properties or equations used to specify geometric objects possess certain symmetries, such as the specification of a triangle in terms of three angles, which determines a triangle up to similarity, or in terms of three edge lengths, which determines it up to Euclidean motion: these two specifications have different symmetry groups. Various theorems in geometry describe a situation, whereby a "conserved quantity", naturally associated with a geometric object, reduces the symmetry group of its defining properties: the original symmetry is broken. Vessiot`s theorem yields a classical example of symmetry breaking in differential geometry: if a surface can be deformed while preserving all properties relating to angle measurement and, at the same time, envelops a 1-parameter family of spheres, then it is piece of a cone, a cylinder or a surface of revolution. The first two properties only depend on an angle measurement, while being a cone, cylinder or surface of revolution depends on a length measurement. Thus symmetry breaking has occurred. Several other theorems of a similar flavour are known for surfaces in the sphere geometries of Lie, Laguerre and Möbius, in particular, for deformable or "non-rigid" surfaces in these geometries. These surfaces are of interest for other reasons also: for example, they admit an integrable systems approach, another notion known from physics, hence can be "transformed" in various ways, without changing their key properties. More specifically, certain classes of non-rigid surfaces admit "Weierstrass representations", or can be otherwise specified in an explicit way, which allows to exploit experimental methods for their investigation: to detect properties or to substantiate claims before a rigorous proof is descried. We will investigate relations between non-rigidity and symmetry breaking, in particular, whether deformability in more than one way invariably leads to symmetry breaking, thus generalizing Vessiot`s theorem. However, our main concern will be to detect causes for symmetry breaking rather than just its occurrence, for example, by studying the appearance of the aforementioned "conserved quantities". Again, experimental methods will help to generate material or examples for inspection. In this way, we do not only hope and expect to establish "geometric symmetry breaking" as a novel area of research in geometry, which resonates well with corresponding research in theoretical physics and with potential implications for design and engineering, but we will also gain new perspectives and a more structural understanding of a variety of thus far unrelated results in geometry, that will facilitate exploitation of "geometric symmetry breaking" as a new tool in geometry.

The mission of this project was to gain deeper insight of a phenomenon in geometry that can be observed in a wide variety of situations: we termed this phenomenon as "symmetry breaking", reminiscent of a similar phenomenon observable in physics. For example, the equations describing the drop of a drop of liquid onto a horizontal surface of the same liquid have a rotational symmetry (about the path of the drop as an axis); any solution (path of a drop) at first displays the same symmetry, during fall and formation of a rotationally symmetric splash, but then the splash forms a (fairly symmetric) coronet - hence the (high) rotational symmetry "breaks" (reduces) to a (lower) symmetry of only finitely many rotations by a certain angle. As there is no apparent reason for this loss of (smooth rotational) symmetry this sort of symmetry breaking is referred to as "spontaneous". A similar mechanism in geometry has been described in F Klein's visionary "Erlanger Programm" of 1893, where a symmetry (group) based approach to geometry was constituted. Subsequently, various occurrences of "symmetry breaking" in geometry have been observed - and successfully employed - but the sources of this phenomenon remain obscure to date. As a model scenario, to study said phenomenon, we consider surfaces in a geometry of very high symmetry (large symmetry group), where points, spheres and planes become indistinguishable (Lie sphere geometry): for example, any parallel surface of a given surface "looks the same" in this geometry; however, features such as touching (contact) of surfaces are preserved in this (contact) geometry. The "non-rigid" (deformable) surfaces of this geometry provide a rich source of sample material, as a wide range of methods and results are known for (some of) those surfaces, e.g., for soap films/bubbles or flat surfaces in various ambient geometries. In particular, we were interested in the question of whether symmetry breaking were related to the possibility of deformability in multiple ways. A main body of project work consisted of the preparation or adaptation of the rich methodology in the various branches of our model scenario, including a suitably adapted description of various classes of "non-rigid surfaces" in terms of certain "conserved quantities" (that induce geometric symmetry breaking) and of general formulas or transformations to describe the generation of certain surfaces (solutions). In the process we collected further evidence on "spontaneous symmetry breaking" in geometry, including some evidence regarding the above question for its relation to (multiple) non-rigidity. This facilitated a first attempt at a general description of the phenomenon of "symmetry breaking" in geometry, including a comparison with a corresponding notion in physics.