Related papers: Covariant description of isothermic surfaces

Canonical principal parameters are introduced for surfaces in $\mathbb R^3$ without umbilical points. It is proved that in these parameters the surface is determined (up to position in space) by a pair of invariants satisfying a partial…

In this paper, we formulate a supersymmetric extension of the Gauss-Weingarten and Gauss-Codazzi equations for conformally parametrized surfaces immersed in a Grassmann superspace. We perform this analysis using a superspace-superfield…

Channel linear Weingarten surfaces in space forms are investigated in a Lie sphere geometric setting, which allows for a uniform treatment of different ambient geometries. We show that any channel linear Weingarten surface in a space form…

Let $N$ be a Riemannian, neutral or Lorentzian $4$-dimensional space form. In this paper, the expressions of the equations of Gauss, Codazzi and Ricci of a space-like or time-like surface in $N$ given in [7] are naturally understood in…

Surfaces of revolution in three-dimensional Euclidean space are considered. Several new examples of surfaces of revolution associated with well-known solvable cases of the Schoedinger equation (infinite well, harmonic oscillator, Coulomb…

In this work, we consider 2-surfaces in ${\mathbb R}^3$ arising from the modified Korteweg de Vries (mKdV) equation. We give a method for constructing the position vector of the mKdV surface explicitly for a given solution of the mKdV…

For CMC surfaces in $3$-dimensional space forms, we relate the moment class of Korevaar--Kusner--Solomon to a second cohomology class arising from the integrable systems theory of isothermic surfaces. In addition, we show that both classes…

The contribution of this paper is twofold. First, we generalize the definition of discrete isothermic surfaces. Compared with the previous ones, it covers more discrete surfaces, e.g., the associated families of discrete isothermic minimal…

We characterise the maps into the space of $2$-spheres in $S^n$ that are the conformal Gauss maps of conformal immersions of a surface. In particular, we give an invariant formulation and efficient proof of a characterisation, due to…

We address the problem of second order conformal deformation of spacelike surfaces in compactified Minkowski 4-space. We explain the construction of the exterior differential system of conformal deformations and discuss its general and…

We study Noncommutative Electrodynamics using the concept of covariant coordinates. We propose a scheme for interpreting the formalism and construct two basic examples, a constant field and a plane wave. Superposing these two, we find a…

A discrete harmonic surface is a trivalent graph which satisfies the balancing condition in the 3-dimensional Euclidean space and achieves energy minimizing under local deformations. Given a topological trivalent graph, a holomorphic…

In Part I, we develop the notions of a Moebius structure and a conformal Cartan geometry, establish an equivalence between them; we use them in Part II to study submanifolds of conformal manifolds in arbitrary dimension and codimension. We…

A survey of some recent and important results which have to do with integrable equations and their relationship with the theory of surfaces is given. Some new results are also presented. The concept of the moving frame is examined, and it…

We give an account of the classical and integrable geometry of isothermic surfaces in arbitrary co-dimension. We show that the classical transformation theory of Darboux, Bianchi and Calapso goes through unchanged in arbitrary co-dimension…

We develop the geometric description of submanifolds in Newton--Cartan spacetime. This provides the necessary starting point for a covariant spacetime formulation of Galilean-invariant hydrodynamics on curved surfaces. We argue that this is…

We establish cross-ratio invariants for surfaces in 4-space in an analogous way to Uribe-Vargas's work for surfaces in 3-space. We study the geometric locii of local and multi-local singularities of ortogonal projections of the surface. The…

We establish a uniformization result for metric surfaces - metric spaces that are topological surfaces with locally finite Hausdorff 2-measure. Using the geometric definition of quasiconformality, we show that a metric surface that can be…

In this paper, we consider the orthogonal projection of a surface in $\mathbb{R}^3$ for a given view direction. We then introduce and investigate several invariants of the families of the plane curves that locally configure the projection…

In this paper we describe the $\epsilon$-isothermic surfaces in the pseudo-Euclidean 3-space and we obtain the pseudo-Calapso equation. In sequence, we classify the Dupin surfaces in pseudo-Euclidean 3-space having distinct principal…