295 research outputs found
Isoparametric and Dupin Hypersurfaces
A hypersurface in a real space-form , or is
isoparametric if it has constant principal curvatures. For and
, 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 . A hypersurface in a real space-form is proper Dupin if
the number of distinct principal curvatures is constant on , 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
Darboux cyclides and webs from circles
Motivated by potential applications in architecture, we study Darboux
cyclides. These algebraic surfaces of order a most 4 are a superset of Dupin
cyclides and quadrics, and they carry up to six real families of circles.
Revisiting the classical approach to these surfaces based on the spherical
model of 3D Moebius geometry, we provide computational tools for the
identification of circle families on a given cyclide and for the direct design
of those. In particular, we show that certain triples of circle families may be
arranged as so-called hexagonal webs, and we provide a complete classification
of all possible hexagonal webs of circles on Darboux cyclides.Comment: 34 pages, 20 figure
Dupin Cyclides as a Subspace of Darboux Cyclides
Dupin cyclides are interesting algebraic surfaces used in geometric design
and architecture to join canal surfaces smoothly and construct model surfaces.
Dupin cyclides are special cases of Darboux cyclides, which in turn are rather
general surfaces in R^3 of degree 3 or 4. This article derives the algebraic
conditions (on the coefficients of the implicit equation) for recognition of
Dupin cyclides among the general implicit form of Darboux cyclides. We aim at
practicable sets of algebraic equations describing complete intersections
inside the parameter space.Comment: 20 pages, 1 figur
The Power of Poincar\'e: Elucidating the Hidden Symmetries in Focal Conic Domains
Focal conic domains are typically the "smoking gun" by which smectic liquid
crystalline phases are identified. The geometry of the equally-spaced smectic
layers is highly generic but, at the same time, difficult to work with. In this
Letter we develop an approach to the study of focal sets in smectics which
exploits a hidden Poincar\'e symmetry revealed only by viewing the smectic
layers as projections from one-higher dimension. We use this perspective to
shed light upon several classic focal conic textures, including the concentric
cyclides of Dupin, polygonal textures and tilt-grain boundaries.Comment: 4 pages, 3 included figure
Harmonic maps in unfashionable geometries
We describe some general constructions on a real smooth projective 4-quadric
which provide analogues of the Willmore functional and conformal Gauss map in
both Lie sphere and projective differential geometry. Extrema of these
functionals are characterized by harmonicity of this Gauss map.Comment: plain TeX, uses bbmsl for blackboard bold, 20 page
- …