Related papers: Lie applicable surfaces and curved flats
For a surface in the 3-dimensional real projective space, we define a Gauss map, which is a quadric in $\mathbb R^{4}$ and called the first-order Gauss map. It will be shown that the surface is a Demoulin surface if and only if the…
In the present paper we study the Lie sphere geometry of Legendre surfaces by the method of moving frame and we prove an existence theorem for real-analytic Lie-minimal Legendre surfaces.
We present the first steps of a procedure which discretises surface theory in classical projective differential geometry in such a manner that underlying integrable structure is preserved. We propose a canonical frame in terms of which the…
In this paper, we deal with the gluing of two surfaces, where the gluing locus is assumed to be a curve. We consider a moving frame along the gluing locus, and define developable surfaces with respect to the frame. Considering geometric…
We prove that the cohomology sheaves of the relative dualizing complex of a flat family of varieties with semi-log-canonical or Du Bois singularities are flat and commute with base change. This is a local version of our earlier similar…
In this study, we introduce Darboux slant ruled surfaces in the Euclidean 3-space which is defined by the property that the Darboux vector of orthonormel frame of ruled surface makes a constant angle with a fixed, non-zero direction. We…
In this paper, we investigate families of singular holomorphic Lie algebroids on complex analytic spaces. We introduce and study a special type of deformation called unfoldings of Lie algebroids, which generalizes the theory of singular…
We investigate the geometry of holomorphic curves and complex surfaces from the perspective of singularity theory. We show that, with a suitable choice of a complex bilinear symmetric form, the families of functions and mappings that…
We study Christoffel and Darboux transforms of discrete isothermic nets in 4-dimensional Euclidean space: definitions and basic properties are derived. Analogies with the smooth case are discussed and a definition for discrete Ribaucour…
A Lie algebra structure on variation vector fields along an immersed curve in a $2$-dimensional real space form is investigated. This Lie algebra particularized to plane curves is the cornerstone in order to define a Hamiltonian structure…
We define and study the properties of Darboux-type transformations between Sturm--Liouville problems with boundary conditions containing rational Herglotz--Nevanlinna functions of the eigenvalue parameter (including the Dirichlet boundary…
This is a summary of some of the basic facts about flat 2-orbifold groups, otherwise known as 2-dimensional crystallographic groups. We relate the geometric and topological presentations of these groups, and consider structures…
We study surfaces in Euclidean space constructed by the sum of two curves or that are graphs of the product of two functions. We consider the problem to determine all these surfaces with constant Gauss curvature. We extend the results to…
This PhD Thesis consists of two parts. The first part focuses on novel algebraic and geometric approaches to the classification problem of coboundary Lie bialgebras up to Lie algebra automorphisms. More specifically, Grassmann, graded…
In an earlier paper we introduced rectangular diagrams of surfaces and showed that any isotopy class of a surface in the three-sphere can be presented by a rectangular diagram. Here we study transformations of those diagrams and introduce…
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…
A link between first-order ordinary differential equations (ODEs) and 2-dimensional Riemannian manifolds is explored. Given a first-order ODE, an associated Riemannian metric on the variable space is defined, and some properties of the…
In this paper, we consider a regular curve on an oriented surface in Euclidean 3-space with the Darboux frame $\{\mathsf{T},\mathsf{V},\mathsf{U}\}$ along the curve, where $\mathsf{T}$ is the unit tangent vector field of the curve,…
We investigate the Lawson genus $2$ surface by methods from integrable system theory. We prove that the associated family of flat connections comes from a family of flat connections on a $4-$punctured sphere. We describe the symmetries of…
We study the space of conformal immersions of a 2-torus into the 4-sphere. The moduli space of generalized Darboux transforms of such an immersed torus has the structure of a Riemann surface, the spectral curve. This Riemann surface arises…