arXiv research

A multi-neck spacetime wormhole is constructed with a simple metric tensor.

Geometrically revisits Dupin cyclidic systems using evolving circles and cyclides.

Lie sphere geometry helps classify Dupin hypersurfaces.

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 the framework of conformal geometry. Further we classify the spacelike Dupin hype…

We introduce the notion of weak reduciblity for Dupin submanifolds with arbitrary codimension. We give a complete characterization of all weakly reducible Dupin submanifolds, as a consequence of a general result on a broader class of Euclidean submanifolds. As a main application, we derive an explicit recursive procedu…

Survey of Dupin hypersurfaces in Lie sphere geometry.

Survey on Dupin hypersurfaces and isoparametric hypersurfaces in spheres.

In this paper we show that a Dupin hypersurface with constant Möbius curvatures is Möbius equivalent to either an isoparametric hypersurface in the sphere or a cone over an isoparametric hypersurface in a sphere. We also show that a Dupin hypersurface with constant Laguerre curvatures is Laguerre equivalent to a flat L…

The method of moving frames in Lie sphere geometry has produced significant results in the classification of Dupin hypersurfaces in spheres. What is the secret of its effectiveness? The answer emerges in the classification of nonumbilic isoparametric surfaces in the space form geometries. Using the method of moving fra…

New findings on Chern's conjecture for Dupin hypersurfaces.

The paper updates methods for studying proper Dupin hypersurfaces in Lie sphere geometry.

Compact Dupin hypersurfaces without constant Lie curvatures found.

This article is devoted to the study of cyclides osculating general surfaces. We show that generically, at any point of a surface, one has a one-parameter family of cyclides tangent to a surface curve of order three and among them just one is tangent to this curve of order four. This one will be called the osculating c…

New examples of austere submanifolds and hypersurfaces with specific curvature properties.

We show that every spherical 2-Dupin submanifold that is not a hypersurface is conformally congruent to the standard embedding of the real, complex, quaternionic or octonionic projective plane. We also classify 2-CPC, 2-umbilical and weakly 2-umbilical submanifolds in space forms.

Cyclidic nets are introduced as discrete analogs of curvature line parametrized surfaces and orthogonal coordinate systems. A 2-dimensional cyclidic net is a piecewise smooth $C^1$-surface built from surface patches of Dupin cyclides, each patch being bounded by curvature lines of the supporting cyclide. An explicit de…

The paper studies special surfaces in pseudo-Euclidean space.

We prove that any connected proper Dupin hypersurface in $\R^n$ is analytic algebraic and is an open subset of a connected component of an irreducible algebraic set. We prove the same result for any connected non-proper Dupin hypersurface in $\R^n$ that satisfies a certain finiteness condition. Hence any taut submanifo…

Study integrable discretizations of cyclic systems with circular coordinate lines.

If $M$ is an isoparametric hypersurface in a sphere $S^n$ with four distrinct principal curvatures, then the principal curvatures $κ_1,...,κ_4$ can be ordered so that their multiplicities satisfy $m_1=m_2$ and $m_3=m_4$, and the cross-ratio $r$ of the principal curvatures (the Lie curvature) equals -1. In this paper, w…

In this paper we develop the vectorial Ribaucour transformation for Euclidean submanifolds. We prove a general decomposition theorem showing that under {appropriate} conditions the composition of two or more vectorial Ribaucour transformations is again a vectorial Ribaucour transformation. An immediate consequence of t…

The works of William Rowan Hamilton in Geometrical Optics are presented, with emphasis on the Malus-Dupin theorem. According to that theorem, a family of light rays depending on two parameters can be focused to a single point by an optical instrument made of reflecting or refracting surfaces if and only if, before ente…

We prove some new rigidity results for proper biharmonic immersions in ${\mathbb S}^n$ of the following types: Dupin hypersurfaces; hypersurfaces, both compact and non-compact, with bounded norm of the second fundamental form; hypersurfaces satisfying intrinsic properties; PMC submanifolds; parallel submanifolds.

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

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 subje…

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 isoparametric hypersurfaces in Euclidean spaces. On the other hand, by an example of local…

The paper studies Einstein-Hilbert action on complex manifolds.

Supercyclides are surfaces with a characteristic conjugate parametrization consisting of two families of conics. Patches of supercyclides can be adapted to a Q-net (a discrete quadrilateral net with planar faces) such that neighboring surface patches share tangent planes along common boundary curves. We call the result…

The paper explores formulas and applications for mixed scalar curvature in multi-product manifolds.

Of all real Lagrangian--Grassmannians $LG(n,2n)$, only $LG(2,4)$ admits a distinguished (Lorentzian) conformal structure and hence is identified with the indefinite M\"obius space $S^{1,2}$. Using Cartan's method of moving frames, we study hyperbolic (timelike) surfaces in $LG(2,4)$ modulo the conformal symplectic gro…

Square Clifford torus uniquely determined by isoperimetric ratio, rectangular torus not.

We discuss the problem of $R$-separability (separability of variables with a factor $R$) in the stationary Schrödinger equation on $n$-dimensional Riemann space. We follow the approach of Gaston Darboux who was the first to give the first general treatment of $R$-separability in PDE (Laplace equation on ${\mathbb E}^3$…