On a variational problem for curves in Lie sphere geometry

A hypersurface M in Rn or Sn is said to be Dupin if along each curvature surface, the corresponding principal curvature is constant. A Dupin hypersurface is said to be proper Dupin if each principal curvature has constant multiplicity on M, i.e., the number of distinct principal curvatures is constant on M. The notions of Dupin and proper Dupin hypersurfaces in Rn or Sn can be generalized to the setting of Lie sphere geometry, and these properties are easily seen to be invariant under Lie sphere transformations. This makes Lie sphere geometry an effective setting for the study of Dupin hypersurfaces, and many classifications of proper Dupin hypersurfaces have been obtained up to Lie sphere transformations. In these notes, we give a detailed introduction to this method for studying Dupin hypersurfaces in Rn or Sn, including proofs of several fundamental results. We also give a survey of the results in the field that have been obtained using this approach.

This is a survey of local and global classification results concerning Dupin hypersurfaces in S n (or R n ) that have been obtained in the context of Lie sphere geometry. The emphasis is on results that relate Dupin hypersurfaces to isoparametric hypersurfaces in spheres. Along with these classification results, many important concepts from Lie sphere geometry, such as curvature spheres, Lie curvatures, and Legendre lifts of submanifolds of S n (or R n ), are described in detail. The paper also contains several important constructions of Dupin hypersurfaces with certain special properties.

Although electrons (fermions)and photons (bosons) produce the same interference patterns in the two-slit experiments, known in optics for photons since the 17th Century, the description of these patterns for electrons and photons thus far was markedly different. Photons are spin one, relativistic and massless particles while electrons are spin half massive particles producing the same interference patterns irrespective to their speed. Experiments with other massive particles demonstrate the same kind of interference patterns. In spite of these differences, in the early 1930s of the 20th Century, the isomorphism between the source-free Maxwell and Dirac equations was established. In this work, we were permitted replace the Born probabilistic interpretation of quantum mechanics with the optical. In 1925, Rainich combined source-free Maxwell equations with Einstein’s equations for gravity. His results were rediscovered in the late 1950s by Misner and Wheeler, who introduced the word "geometrodynamics” as a description of the unified field theory of gravity and electromagnetism. An absence of sources remained a problem in this unified theory until Ranada’s work of the late 1980s. However, his results required the existence of null electromagnetic fields. These were absent in Rainich–Misner–Wheeler’s geometrodynamics. They were added to it in the 1960s by Geroch. Ranada’s solutions of source-free Maxwell’s equations came out as knots and links. In this work, we establish that, due to their topology, these knots/links acquire masses and charges. They live on the Dupin cyclides—the invariants of Lie sphere geometry. Symmetries of Minkowski space-time also belong to this geometry. Using these symmetries, Varlamov recently demonstrated group-theoretically that the experimentally known mass spectrum for all mesons and baryons is obtainable with one formula, containing electron mass as an input. In this work, using some facts from polymer physics and differential geometry, a new proof of the knotty nature of the electron is established. The obtained result perfectly blends with the description of a rotating and charged black hole.

We investigate curved flats in Lie sphere geometry. We show that in this setting curved flats are in one-to-one correspondence with pairs of Demoulin families of Lie applicable surfaces related by Darboux transformation.

Lie sphere geometry in nuclear scattering processes

In this paper we propose to use Lie sphere geometry as a new tool to systematically construct time-symmetric initial data for a wide variety of generalised black-hole configurations in lattice cosmology. These configurations are iteratively constructed analytically and may have any degree of geometric irregularity. We show that for negligible amounts of dust these solutions are similar to the swiss-cheese models at the moment of maximal expansion. As Lie sphere geometry has so far not received much attention in cosmology, we will devote a large part of this paper to explain its geometric background in a language familiar to general relativists.

We present a definition of discrete channel surfaces in Lie sphere geometry, which reflects several properties for smooth channel surfaces. Various sets of data, defined at vertices, on edges or on faces, are associated with a discrete channel surface that may be used to reconstruct the underlying particular discrete Legendre map. As an application we investigate isothermic discrete channel surfaces and prove a discrete version of Vessiot’s Theorem.

The conditions for a cuspidal edge, swallowtail and other fundamental singularities are given in the context of Lie sphere geometry. We then use these conditions to study the Lie sphere transformations of a surface.

This article describes an entirely algebraic construction for developing conformal geometries, which provide models for, among others, the Euclidean, spherical and hyperbolic geometries. On one hand, their relationship is usually shown analytically, through a framework comparing the measurement of distances and angles in Cayley–Klein geometries, including Lorentzian geometries, as done by F. Bachmann and later R. Struve. On the other hand, such a relationship may also be expressed in a purely linear algebraic manner, as explained by D. Hestens, H. Li and A. Rockwood. The model described in this article unifies these approaches via a generalization of Lie sphere geometry. Like the work of N. Wildberger, it is a purely algebraic construction, and as such it works over any field of odd characteristic. It is shown that measurement of distances and angles is an inherent property of the model that is easy to identify, and the possible models are classified over the real, complex and finite fields, and partially in characteristic 2, revealing a striking analogy between the real and finite geometries. This is an abbreviated version of a previous manuscript, with certain expository parts removed for the sake brevity. The original manuscript is available on the website arXiv, with more motivation, examples, properties. Several definitions and theorems can be extended to include the characteristic 2 case, which are also omitted here.

We discuss channel surfaces in the context of Lie sphere geometry and characterise them as certain Omega _{0}-surfaces. Since Omega _{0}-surfaces possess a rich transformation theory, we study the behaviour of channel surfaces under these transformations. Furthermore, by using certain Dupin cyclide congruences, we characterise Ribaucour pairs of channel surfaces.

Huygens triviality of the time-independent Schrödinger equation. Applications to atomic and high energy physics

Pythagorean-normal (PN) surfaces, defined as rational surfaces admitting rational offsets, are important for industrial Computer-aided Design. Traditionally PN surfaces are considered from the point of view of Laguerre geometry, using three main models: cyclographic model, Blaschke cylinder or isotropic model. We propose a unifying formalism to deal with PN surfaces: all these models are embedded into one ambient pseudo-Euclidean space \({{\mathbb R}^{4, 2}}\), that is known as a model for Lie sphere geometry. Various relations between different models are described in terms of closed formulas in the geometric algebra \({{\mathcal Cl}(4, 2)}\) and illustrated by examples of applications.

In Lie sphere geometry, a cycle in $\RR^n$ is either a point or an oriented sphere or plane of codimension $1$, and it is represented by a point on a projective surface $\Omega\subset \PP^{n+2}$. The Lie product, a bilinear form on the space of homogeneous coordinates $\RR^{n+3}$, provides an algebraic description of geometric properties of cycles and their mutual position in $\RR^n$. In this paper, we discuss geometric objects which correspond to the intersection of $\Omega$ with projective subspaces of $\PP^{n+2}$. Examples of such objects are spheres and planes of codimension~$2$ or more, cones and tori. The algebraic framework which Lie geometry provides gives rise to simple and efficient computation of invariants of these objects, their properties and their mutual position in $\RR^n$.

Following Burstall and Hertrich-Jeromin we study the Ribaucour transformation of Legendre submanifolds in Lie sphere geometry. We give an explicit parametrization of the resulted Legendre submanifold $\hat{F}$ of a Ribaucour transformation, via a single real function $\tau$ which represents the regular Ribaucour sphere congruence $s$ enveloped by the original Legendre submanifold $F$.

In this paper we give a brief review of the pseudo-Riemannian geometry of the five-dimensional homogeneous space for the conformal group O(4,2). Its topology is described and its relation to the conformally compactified Minkowski space is described. Its metric is calculated using a generalized half-space representation. Compactification via Lie-sphere geometry is outlined. Possible applications to Jaime Keller's START theory may follow by using its predecessor - the 5-optics of Yu. B. Rumer. The point of view of Rumer is given extensively in the last section of the paper. Keywords. Kaluza,Klein, Rumer, conformal symmetry, hyperbolic space, START, fifth dimension, action coordinate, 5-optics

Geometry and Shape of Minkowski's Space Conformal Infinity

In this paper, we study surfaces of S3 in the context of Lie sphere geometry. We construct invariants with respect to Lie sphere transformations on the surfaces, which determine the surfaces up to a Lie sphere transformation. Finally we classify completely the homogeneous surfaces in S3 with respect to the Lie sphere transformation group of S3.

A hypersurface M n 1 in a real space-form R n , S n or H n is isoparametric if it has constant principal curvatures. For R n and H n , the classification of isoparametric hypersurfaces is complete and relatively simple, but as ´ Elie Cartan showed in a series of four papers in 1938-1940, the subject is much deeper and more complex for hypersurfaces in the sphere S n . A hypersurface M n 1 in a real space-form is proper Dupin if the number g of distinct principal curvatures is constant on M n 1 , and each principal curvature function is constant along each leaf of its corresponding principal foliation. This is an important genera- lization of the isoparametric property that has its roots in nineteenth century differential geometry and has been studied effectively in the context of Lie sphere geometry. This paper is a survey of the known results in these fields with emphasis on results that have been obtained in more recent years and discussion of important open problems in the field.

The theory of surfaces in Euclidean space can be naturally formulated in the more general context of Legendre surfaces into the space of contact elements. We address the question of deformability of Legendre surfaces with respect to the symmetry group of Lie sphere contact transformations from the point of view of the deformation theory of submanifolds in homogeneous spaces. Necessary and sufficient conditions are provided for a Legendre surface to admit non-trivial deformations, and the corresponding existence problem is discussed.