W-surfaces having some properties

We investigate some characteristic properties of specific Weingarten surfaces in the three-dimensional Euclidean space using the nets of the lines of curvature resp. the asymptotic lines on both central surfaces of them.Comment: 8 page

Characterizations of special skew ruled surfaces by the normal curvature of some distinguished families of curves

We consider skew ruled surfaces in the three-dimensional Euclidean space and some geometrically distinguished families of curves on them whose normal curvature has a concrete form. The aim of this paper is to find and classify all ruled surfaces with the mentioned property.Comment: 10 page

On the shape operator of relatively parallel hypersurfaces in the nn-dimensional relative differential geometry

We deal with hypersurfaces in the framework of the $n$-dimensional relative differential geometry. We consider a hypersurface $\varPhi$ of $\mathbb{R}^{n+1}$ with position vector field $\mathbf{x}$, which is relatively normalized by a relative normalization $\mathbf{y}$. Then $\mathbf{y}$ is also a relative normalization of every member of the one-parameter family $\mathcal{F}$ of hypersurfaces $\varPhi_\mu$ with position vector field $$\mathbf{x}_\mu = \mathbf{x} + \mu \, \mathbf{y},$$ where $\mu$ is a real constant. We call every hypersurface $\varPhi_\mu \in \mathcal{F}$ relatively parallel to $\varPhi$ at the "relative distance" $\mu$. In this paper we study (a) the shape (or Weingarten) operator, (b) the relative principal curvatures, (c) the relative mean curvature functions and (d) the affine normalization of a relatively parallel hypersurface $\left( \varPhi_\mu,\mathbf{y}\right)$ to $\left(\varPhi,\mathbf{y}\right)$.Comment: 13 pages. arXiv admin note: text overlap with arXiv:1707.0754

Bonnet's type theorems in the relative differential geometry of the 4-dimensional space

We deal with hypersurfaces in the framework of the relative differential geometry in $\mathbb{R}^4$. We consider a hypersurface $\varPhi$ in $\mathbb{R}^4$ with position vector field \vect{x} which is relatively normalized by a relative normalization \vect{y}. Then \vect{y} is also a relative normalization of every member of the one-parameter family $\mathcal{F}$ of hypersurfaces $\varPhi_\mu$ with position vector field \vect{x}_\mu = \vect{x} + \mu \, \vect{y}, where $\mu$ is a real constant. We call every hypersurface $\varPhi_\mu \in \mathcal{F}$ relatively parallel to $\varPhi$. This consideration includes both Euclidean and Blaschke hypersurfaces of the affine differential geometry. In this paper we express the relative mean curvature's functions of a hypersurface $\varPhi_\mu$ relatively parallel to $\varPhi$ by means of the ones of $\varPhi$ and the "relative distance" $\mu$. Then we prove several Bonnet's type theorems. More precisely, we show that if two relative mean curvature's functions of $\varPhi$ are constant, then there exists at least one relatively parallel hypersurface with a constant relative mean curvature's function.Comment: 13 pages, Key Words: relative and equiaffine differential geometry, hypersurfaces in the Euclidean space, Blaschke hypersurfaces in affine differential geometry, Peterson correspondence, relative mean curvature functions, Bonnet's Theorem

Ruled surfaces asymptotically normalized

We consider a skew ruled surface $\Phi$ in the Euclidean space $E^{3}$ and relative normalizations of it, so that the relative normals at each point lie in the corresponding asymptotic plane of $\Phi$. We call such relative normalizations and the resulting relative images of $\Phi$ \emph{asymptotic}. We determine all ruled surfaces and the asymptotic normalizations of them, for which $\Phi$ is a relative sphere (proper or inproper) or the asymptotic image degenerates into a curve. Moreover we study the sequence of the ruled surfaces ${\Psi_{i}}_{i\in \mathbb{N}}$, where $\Psi_{1}$ is an asymptotic image of $\Phi$ and $\Psi_{i}$, for $i\geq2$, is an asymptotic image of $\Psi_{i-1}$. We conclude the paper by the study of various properties concerning some vector fields, which are related with $\Phi$.Comment: 15 page

On polar relative normalizations of ruled surfaces

This paper deals with skew ruled surfaces in the Euclidean space $\mathbb{E}^{3}$ which are equipped with polar normalizations, that is, relative normalizations such that the relative normal at each point of the ruled surface lies on the corresponding polar plane. We determine the invariants of a such normalized ruled surface and we study some properties of the Tchebychev vector field and the support vector field of a polar normalization. Furthermore, we study a special polar normalization, the relative image of which degenerates into a curve.Comment: 10 page

On the Laplace Normal Vector Field of Skew Ruled Surfaces

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 normalizations for which the Laplace normal image degenerates into a point or into a curve. Moreover, we study the Laplace normal image of a non-conoidal ruled surface whose relative normals lie on the asymptotic plane.Comment: appears in the Proceedings of the 1st International Workshop on Line Geometry and Kinematics (IW-LGK-11), April 26-30, 2011, Paphos, Cypru

A relative-geometric treatment of ruled surfaces

We consider relative normalizations of ruled surfaces with non-vanishing Gaussian curvature $K$ in the Euclidean space $\mathbb{R} ^{3}$, which are characterized by the support functions $^{\left( \alpha \right) }q=\left \vert K\right \vert ^{\alpha}$ for $\alpha \in \mathbb{R}$ (Manhart's relative normalizations). All ruled surfaces for which the relative normals, the Pick invariant or the Tchebychev vector field have some specific properties are determined. We conclude the paper by the study of the affine normal image of a non-conoidal ruled surface.Comment: 12 page

Ruled and quadric surfaces of finite Chen-type

In this paper, we study ruled surfaces and quadrics in the 3-dimensional Euclidean space which are of finite $III$-type, that is, they are of finite type, in the sense of B.-Y. Chen, with respect to the third fundamental form. We show that helicoids and spheres are the only ruled and quadric surfaces of finite $III$-type, respectively.Comment: 9 pages, Keywords: Surfaces in the Euclidean 3-space, Surfaces of finite Chen-type, Beltrami operato

Characterizations of Ruled Surfaces in R3\mathbb{R}^3 and of Hyperquadrics in Rn+1\mathbb{R}^{n+1} via Relative Geometric Invariants

We consider hypersurfaces in the real Euclidean space $\mathbb{R}^{n+1}$ ($n\geq2$) which are relatively normalized. We give necessary and sufficient conditions a) for a surface of negative Gaussian curvature in $\mathbb{R}^3$ to be ruled, b) for a hypersurface of positive Gaussian curvature in $\mathbb{R}^{n+1}$ to be a hyperquadric and c) for a relative normalization to be constantly proportional to the equiaffine normalization.Comment: 8 page