Isoparametric and Dupin Hypersurfaces

A hypersurface $M^{n-1}$ in a real space-form ${\bf R}^n$, $S^n$ or $H^n$ is isoparametric if it has constant principal curvatures. For ${\bf R}^n$ 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 generalization 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.Comment: This is a contribution to the Special Issue "Elie Cartan and Differential Geometry", published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGM

Geometry and Topology of the Minkowski Product

The Minkowski product can be viewed as a higher dimensional version of interval arithmetic. We discuss a collection of geometric constructions based on the Minkowski product and on one of its natural generalizations, the quaternion action. We also will present some topological facts about these products, and discuss the applications of these constructions to computer aided geometric design

Isoparametric and Dupin Hypersurfaces

A hypersurface Mn−1 in a real space-form Rn, Sn or Hn is isoparametric if it has constant principal curvatures. For Rn and Hn, the classification of isoparametric hypersurfaces is complete and relatively simple, but as Élie Cartan showed in a series of four papers in 1938–1940, the subject is much deeper and more complex for hypersurfaces in the sphere Sn. A hypersurface Mn−1 in a real space-form is proper Dupin if the number g of distinct principal curvatures is constant on Mn−1, and each principal curvature function is constant along each leaf of its corresponding principal foliation. This is an important generalization 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

Maxwell-Dirac isomorphism revisited: from foundations of quantum mechanics to geometrodynamics and cosmology

Although electrons (fermions) and photons(bosons) produce the same interference patterns in the two-slit experiments, the description of these patterns is markedly different. Photons are spin one, relativistic and massless while electrons are spin 1/2 massive particles producing the same interference patterns. In spite of these differences, already in early 30ies of 20ieth century the isomorphism between the source-free Maxwell and Dirac equations was established. It permitted us to replace 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 by Misner and Wheeler in their "geometrodynamics". Absence of sources in it remained a problem until Ranada's work. His results required the existence of null electromagnetic fields absent in geometrodynamics. They were added later on by Geroch. Ranada's solutions of source-free Maxwell's equations came out as knots and links. In this work new proof of knotty nature of the electron is established. The obtained result perfectly blends with the descripion of rotating and charged black hole.Comment: 60 pages , no figures. arXiv admin note: text overlap with arXiv:1703.0467

Disclinations, dislocations and continuous defects: a reappraisal

Disclinations, first observed in mesomorphic phases, are relevant to a number of ill-ordered condensed matter media, with continuous symmetries or frustrated order. They also appear in polycrystals at the edges of grain boundaries. They are of limited interest in solid single crystals, where, owing to their large elastic stresses, they mostly appear in close pairs of opposite signs. The relaxation mechanisms associated with a disclination in its creation, motion, change of shape, involve an interplay with continuous or quantized dislocations and/or continuous disclinations. These are attached to the disclinations or are akin to Nye's dislocation densities, well suited here. The notion of 'extended Volterra process' takes these relaxation processes into account and covers different situations where this interplay takes place. These concepts are illustrated by applications in amorphous solids, mesomorphic phases and frustrated media in their curved habit space. The powerful topological theory of line defects only considers defects stable against relaxation processes compatible with the structure considered. It can be seen as a simplified case of the approach considered here, well suited for media of high plasticity or/and complex structures. Topological stability cannot guarantee energetic stability and sometimes cannot distinguish finer details of structure of defects.Comment: 72 pages, 36 figure