The Ribaucour families of discrete R-congruences

A note on Ribaucour transformations in Lie sphere geometry

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 $τ$ which represents the regular Ribaucour sphere congruence $s$ enveloped by the original Legendre submanifold $F$.

Orthogonal nets and Clifford algebras

A Clifford algebra model for M"obius geometry is presented. The notion of Ribaucour pairs of orthogonal systems in arbitrary dimensions is introduced, and the structure equations for adapted frames are derived. These equations are discretized and the geometry of the occuring discrete nets and sphere congruences is discussed in a conformal setting. This way, the notions of ``discrete Ribaucour congruences'' and ``discrete Ribaucour pairs of orthogonal systems'' are obtained --- the latter as a generalization of discrete orthogonal systems in Euclidean space. The relation of a Cauchy problem for…

Surfaces with flat normal bundle: an explicit construction

An explicit construction of surfaces with flat normal bundle in the Euclidean space (unit hypersphere) in terms of solutions of certain linear system is proposed. In the case of 3-space our formulae can be viewed as the direct Lie sphere analog of the generalized Weierstrass representation of surfaces in conformal geometry or the Lelieuvre representation of surfaces in the affine space. An explicit parametrization of Ribaucour congruences of spheres by three solutions of the linear system is obtained. In view of the classical Lie correspondence between Ribaucour congruences and surfaces with f…

A connection between flat fronts in hyperbolic space and minimal surfaces in euclidean space

A geometric construction is provided that associates to a given flat front in $\mathbb{H}^3$ a pair of minimal surfaces in $\mathbb{R}^3$ which are related by a Ribaucour transformation. This construction is generalized associating to a given frontal in $\mathbb{H}^3$ , a pair of frontals in $\mathbb{R}^3$ that are envelopes of a smooth congruence of spheres. The theory of Ribaucour transformations for minimal surfaces is reformulated in terms of a complex Riccati ordinary differential equation for a holomorphic function. This enables one to simplify and extend the classical theory, that in pr…

Channel surfaces in Lie sphere geometry

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 characterise Ribaucour pairs of channel surfaces.

维数不超过28的弦流形扭曲符号的模3同余族

Mod 3 Congruence Families for twisted signatures of string manifolds in dimensions up to twenty-eight

本文将Liu和Wang的双参数模形式应用于弦流形，得到4k（1≤k≤7）维下扭曲狄拉克指标和符号的模3同余关系，强化了部分同余，构造了反例并指出32维扩展的阻碍。