arXiv subjects

Ioannis Kaffas

Publications and source records attributed to Ioannis Kaffas.

5 recordsLinked to original sources

arXiv2017

On the shape operator of relatively parallel hypersurfaces in the $n$-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 $\varPhi_μ$ with position vector field $$\mathbf{x}_μ= \mathbf{x} + μ\, \mathbf{y},$$ where $μ$ is a real constant. We call every hypersurface $\varPhi_μ\in \mathcal{F}$ relatively parallel to $\varPhi$ at the "relative distance" $μ$. 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 $\left( \varPhi_μ,\mathbf{y}\right)$ to $\left(\varPhi,\mathbf{y}\right)$.

math.DG↗

arXiv2017

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 $\varPhi_μ$ with position vector field $\vect{x}_μ= \vect{x} + μ\, \vect{y}$, where $μ$ is a real constant. We call every hypersurface $\varPhi_μ\in \mathcal{F}$ 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 $\varPhi_μ$ relatively parallel to $\varPhi$ by means of the ones of $\varPhi$ and the "relative distance" $μ$. 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.

math.DG↗

arXiv2017

Generalization of two Bonnet's Theorems to the relative Differential Geometry of the 3-dimensional Euclidean space

This paper is devoted to the 3-dimensional relative differential geometry of surfaces. In the Euclidean space $\R{E} ^3 $ we consider a surface $\varPhi %\colon \vect{x} = \vect{x}(u^1,u^2) $ with position vector field $\vect{x}$, which is relatively normalized by a relative normalization $\vect{y}% (u^1,u^2) $. A surface $\varPhi^*% \colon \vect{x}^* = \vect{x}^*(u^1,u^2) $ with position vector field $\vect{x}^* = \vect{x} + μ\, \vect{y}$, where $μ$ is a real constant, is called a relatively parallel surface to $\varPhi$. Then $\vect{y}$ is also a relative normalization of $\varPhi^*$. The aim of this paper is to formulate and prove the relative analogues of two well known theorems of O.~Bonnet which concern the parallel surfaces (see~\cite{oB1853}).

math.DG↗

arXiv2014

Characterizations of Ruled Surfaces in $\mathbb{R}^3$ and of Hyperquadrics in $\mathbb{R}^{n+1}$ 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.

math.DG↗

arXiv2013

Ruled surfaces asymptotically normalized

We consider a skew ruled surface $Φ$ 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 $Φ$. We call such relative normalizations and the resulting relative images of $Φ$ \emph{asymptotic}. We determine all ruled surfaces and the asymptotic normalizations of them, for which $Φ$ is a relative sphere (proper or inproper) or the asymptotic image degenerates into a curve. Moreover we study the sequence of the ruled surfaces ${Ψ_{i}}_{i\in \mathbb{N}}$, where $Ψ_{1}$ is an asymptotic image of $Φ$ and $Ψ_{i}$, for $i\geq2$, is an asymptotic image of $Ψ_{i-1}$. We conclude the paper by the study of various properties concerning some vector fields, which are related with $Φ$.

math.DG↗