Related papers: On the Geometry of Constant Angle Surfaces in $Sol…
A surface in a three-dimensional metric Lie group $G$ is said invariant if it is invariant with respect to a one-dimensional subgroup $\Gamma$ of the isometry group of $G$. Is this work we focus on unimodular metric Lie groups $G$ that can…
We prove that finite area isolated singularities of surfaces with constant positive curvature in R^3 are removable singularities, branch points or immersed conical singularities. We describe the space of immersed conical singularities of…
The sl_2-triples play a fundamental role for the structure theory of Lie algebras, and representation theory in general. Here we investigate sl_2-triples of global vector fields on schemes X in positive characteristics p>0, and develop a…
Compact flat surfaces of homogeneous Riemannian 3-manifolds with isometry group of dimension 4 are classified. Non-existence results for compact constant Gauss curvature surfaces in these 3-manifolds are established.
Consider the Euclidean space $\mathbb{R}^3$ endowed with a canonical semi-symmetric non-metric connection determined by a vector field $\mathsf{C}\in\mathfrak{X}(\mathbb{R}^3)$. We study surfaces when the sectional curvature with respect to…
A number of results for C$^2$-smooth surfaces of constant width in Euclidean 3-space ${\mathbb{E}}^3$ are obtained. In particular, an integral inequality for constant width surfaces is established. This is used to prove that the ratio of…
We classify all surfaces with constant Gaussian curvature $K$ in Euclidean $3$-space that can be expressed as an implicit equation of type $f(x)+g(y)+h(z)=0$, where $f$, $g$ and $h$ are real functions of one variable. If $K=0$, we prove…
In this paper, our aim is to give surfaces in the Galilean 3-space G3 with the property that there exist four geodesics through each point such that every surface built with the normal lines and the binormal lines along these geodesics is a…
Let G be a n-dimensional Lie group (n>2) with a bi-invariant Riemannian metric. We prove that if a surface of constant Gaussian curvature in G can be expressed as the product of two curves, then it must be flat. In particular, we can…
We shall determine the uniquely existing extension of the alternating group of degree 6 (being normal in the group) by a cyclic group of order 4, which can act on a complex K3 surface.
We study geometric properties of compact stable minimal surfaces with boundary in homogeneous 3-manifolds $X$ that can be expressed as a semidirect product of $\mathbb{R}^2$ with $\mathbb{R}$ endowed with a left invariant metric. For any…
In this article we develop some elementary aspects of a theory of symmetry in sub-Lorentzian geometry. First of all we construct invariants characterizing isometric classes of sub-Lorentzian contact 3 manifolds. Next we characterize vector…
In this paper, on the first, we prove $\Delta r=2H$ where $\Delta $ is the Laplacian operator, $r=\left( r_{1},r_{2},r_{3}\right) $ the position vector field and $H$ is the mean curvature vector field of a surface $\mathcal{S}$ in the…
In the homogeneous space Sol$_3$, a translation surface is parameterized by $x(s,t)=\alpha(s)\ast\beta(t)$, where $\alpha$ and $\beta$ are curves contained in coordinate planes and $\ast$ denotes the group operation of Sol$_3$. In this…
Hypersurface simple K3 singularities defined by nondegenerate quasi-homogeneous polynomials are classified into ninety five classes in term of the weight of the polynomial by T. Yonemura. We consider versal deformations of them. It has been…
We study algebras defined by identities in symmetric monoidal categories. Our focus is on Lie algebras. Besides usual Lie algebras, there are examples appearing in the study of knot invariants and Rozansky-Witten invariants. Our main result…
The 2-parameter family of certain homogeneous Lorentzian 3-manifolds which includes Minkowski 3-space and anti-de Sitter 3-space is considered. Each homogeneous Lorentzian 3-manifold in the 2-parameter family has a solvable Lie group…
We classify six-dimensional Lie groups which admit a left-invariant half-flat SU(3)-structure and which split in a direct product of three-dimensional factors. Moreover, a complete list of those direct products is obtained which admit a…
In this paper we proof that the Holomorphic angle for compact minimal surfaces in the sphere $S^5$ with constant Contact angle and with a parallel normal vector field must be constant.
We enumerate the number of surfaces of degree $d$ in $P^3$ having a singular line of order $k$, passing through $\delta$ generic points (where $\delta$ is the dimension of moduli space of such surfaces).