Related papers: Dupin hypersurfaces in Lie sphere geometry
This is a survey of local and global classification results concerning Dupin hypersurfaces in $S^n$ (or ${\bf R}^n$) that have been obtained in the context of Lie sphere geometry. The emphasis is on results that relate Dupin hypersurfaces…
A hypersurface $M^{n-1}$ in Euclidean space $E^n$ is proper Dupin if the number of distinct principal curvatures is constant on $M^{n-1}$, and each principal curvature function is constant along each leaf of its principal foliation. This…
A hypersurface $M$ in ${\bf R}^n$ or $S^n$ 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…
In this paper we show that a Dupin hypersurface with constant M\"{o}bius curvatures is M\"{o}bius equivalent to either an isoparametric hypersurface in the sphere or a cone over an isoparametric hypersurface in a sphere. We also show that a…
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…
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…
Similar to the definition of Dupin hypersurface in Riemannian space forms, we define the spacelike Dupin hypersurface in Lorentzian space forms. As conformal invariant objects, spacelike Dupin hypersurfaces are studied in this paper using…
A hypersurface $M$ in the unit sphere $S^n \subset {\bf R}^{n+1}$ is Dupin if along each curvature surface of $M$, the corresponding principal curvature is constant. If the number $g$ of distinct principal curvatures is constant on $M$,…
Dupin cyclides are interesting algebraic surfaces used in geometric design and architecture to join canal surfaces smoothly and to construct model surfaces. Dupin cyclides are special cases of Darboux cyclides, which in turn are rather…
In this article we study the Hamiltonian non-displaceability of Gauss images of isoparametric hypersurfaces in the spheres as Lagrangian submanifolds embedded in complex hyperquadrics.
A Lie hypersurface in the complex hyperbolic space is a homogeneous real hypersurface without focal submanifolds. The set of all Lie hypersurfaces in the complex hyperbolic space is bijective to a closed interval, which gives a deformation…
If $M$ is an isoparametric hypersurface in a sphere $S^n$ with four distrinct principal curvatures, then the principal curvatures $\kappa_1,...,\kappa_4$ can be ordered so that their multiplicities satisfy $m_1=m_2$ and $m_3=m_4$, and the…
The Lie sphere geometry is a natural extension of the M\"obius geometry, where the latter is very important in string theory and the AdS/CFT correspondence. The extension to Lie sphere geometry is applied in the following to a sequence of…
We classify real two-dimensional orbits of conformal subgroups such that the orbits contain two circular arcs through a point. Such surfaces must be toric and admit a M\"obius automorphism group of dimension at least two. Our theorem…
A hypersurface $M$ in ${\bf R}^n$ 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 the number of distinct principal curvatures is…
I give a theory of Moebius-flat hypersurfaces in n-dimensional projective space, analogous to that in conformal geometry. This unifies the classes of hypersurfaces with flat induced conformal structure (n > 3) and a classically studied…
The consideration of the so-called rotation minimizing frames allows for a simple and elegant characterization of plane and spherical curves in Euclidean space via a linear equation relating the coefficients that dictate the frame motion.…
The classification of isoparametric hypersurfaces in spheres with four or six different principal curvatures is still not complete. In this paper we develop a structural approach that may be helpful for a classification. Instead of working…
In this talk I will discuss an example of the use of fully nonlinear parabolic flows to prove geometric results. I will emphasise the fact that there is a wide variety of geometric parabolic equations to choose from, and to get the best…
In this note, we give a classification of complete anisotropic isoparametric hypersurfaces, i.e., hypersurfaces with constant anisotropic principal curvatures, in Euclidean spaces, which is in analogue with the classical case for…