Related papers: The surface of Gauss double points
A generalized quadrangle is a point-line incidence geometry such that any two points lie on at most one line and, given a line $\ell$ and a point $P$ not incident with $\ell$, there is a unique point of $\ell$ collinear with $P$. We study…
In this paper, we find all constant slope surfaces in the Euclidean 3-space, namely those surfaces for which the position vector of a point of the surface makes constant angle with the normal at the surface in that point. These surfaces…
In this article, we introduce an important class of surfaces, namely, quadrics in the Euclidean 3-space $\mathbb{E}^{3}$. We prove that planes, spheres and circular cylinders are the only quadric surfaces whose Gauss map $\boldsymbol{G}$…
We consider the Laplace normal vector field of relatively normalized ruled surfaces with non-vanishing Gaussian curvature in the three-dimensional Euclidean space $\mathbb{R}^{3}$. We determine all ruled surfaces and all relative…
We embark in a program of studying the problem of better approximating surfaces by triangulations(triangular meshes) by considering the approximating triangulations as finite metric spaces and the target smooth surface as their…
We apply the invariant theory of surfaces in the four-dimensional Euclidean space to the class of general rotational surfaces with meridians lying in two-dimensional planes. We find all minimal super-conformal surfaces of this class.
We generalize the Ruh-Vilms problem by characterizing the submanifolds in Euclidean spaces with proper biharmonic Gauss map and we construct examples of such hypersurfaces.
We study generalized symmetries in a simplified arena in which the usual quantum field theories of physics are replaced with topological field theories and the smooth structure with which the symmetry groups of physics are usually endowed…
In the present paper we study two-dimensional maximal surfaces with harmonic level-sets. As a corollary we obtain a new class of one-periodic maximal surfaces.
We construct and investigate smooth orientable surfaces in su(N) algebras. The structural equations of surfaces associated with Grassmannian sigma models on Minkowski space are studied using moving frames adapted to the surfaces. The first…
We study in this work flat surfaces with conical singularities, that is, surfaces provided with a flat structure with conical singular points. Finding good parameters for these surfaces in the general case is an open question. We give an…
In this paper is studied the behavior of principal curvature lines near a curve of umbilic points of a smooth surface.
The notion of quadratic maps between arbitrary groups appeared at several places in the literature on quadratic algebra. Here a unified extensive treatment of their properties is given; the relation with a relative version of Passi's…
Sectors at centre of affine quadrics with point symmetry are investigated over arbitrary fields of characteristic different from two. As an application we demonstrate nice formulas for the area and the volume of such planar and spatial…
We study the geometry of bifurcation sets of generic unfoldings of $D_4^\pm$-functions. Taking blow-ups, we show each of the bifurcation sets of $D_4^\pm$-functions admit a parametrization as a surface in $R^3$. Using this parametrization,…
A number of new terminating series involving $\sin(n^2/k)$ and $\cos(n^2/k)$ are presented and connected to Gauss quadratic sums. Several new closed forms of generic Gauss quadratic sums are obtained and previously known results are…
We classify the singularities of a surface ruled by conics: they are rational double points of type $A_n$ or $D_n$. This is proved by showing that they arise from a precise series of blow-ups of a suitable surface geometrically ruled by…
In this work, we study some classes of rotational surfaces in the pseudo-Euclidean space $\mathbb{E}^4_t$ with profile curves lying in 2-dimensional planes. First, we determine all such surfaces in the Minkowski 4-space $\mathbb{E}^4_1$…
This work aims to extend the existing results on thick points of logarithmic-correlated Gaussian Free Fields to Gaussian random fields that are more singular. To be specific, we adopt a sphere averaging regularization to study…
Topological properties of the gauge field in a two-dimensional Higgs model are investigated. Results of exploratory numerical simulations are presented.