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 Tchebychev Vector Field in the Relative Differential Geometry

In this paper we deal with relative normalizations of hypersurfaces in the (n+1)-dimensional Euclidean space $\mathbb{R}^{n+1}$. Considering a relative normalization $\bar{y}$ of an hypersurface $\Phi$ we decompose the corresponding Tchebychev vector $\bar{T}$ in two components, one parallel to the Tchebychev vector $\bar{T}_{EUK}$ of the Euclidean normalization $\bar{\xi}$ and one parallel to the orthogonal projection $\bar{y}_{T}$ of $\bar{y}$ in the tangent hyperplane of $\Phi$. We use this decomposition to investigate some properties of $\Phi$, which concern its Gaussian curvature, the support function, the Tchebychev vector field etc.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 Surfaces of finite Chen-type

We investigate some relations concerning the first and the second Beltrami operators corresponding to the fundamental forms I, II, III of a surface in the three-dimensional Euclidean space and we study surfaces which are of finite type in the sense of B.-Y. Chen with respect to the fundamental forms II and III.Comment: 13 page

Ruled surfaces right normalized

This paper deals with skew ruled surfaces $\varPhi$ in the Euclidean space $\mathbb{E}^{3}$ which are right normalized, that is they are equipped with relative normalizations, whose support function is of the form $q(u,v) = \frac{f(u) + g(u)\, v}{w(u,v)}$, where $w^2(u,v)$ is the discriminant of the first fundamental form of $\varPhi$. This class of relatively normalized ruled surfaces contains surfaces such that their relative image $\varPhi^{*}$ is either a curve or it is as well as $\varPhi$ a ruled surface whose generators are, additionally, parallel to those of $\varPhi$. Moreover we investigate various properties concerning the Tchebychev vector field and the support vector field of such ruled surfaces.Comment: 16 pages, detailed version of the paper On right relative normalizations of ruled surface

Surfaces of revolution satisfying △IIIx=Ax\triangle^{III}\boldsymbol{x}=A\boldsymbol{x}

We consider surfaces of revolution in the three-dimensional Euclidean space which are of coordinate finite type with respect to the third fundamental form. We show that a surface of revolution satisfying the preceding relation is a catenoid or part of a sphere

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