\documentclass{mc} %%%%% journal info -- DO NOT CHANGE %%%%%%%%% \setcounter{page}{41} \renewcommand\thisnumber{x} \renewcommand\thisyear {2017} \renewcommand\thismonth{xxx} \renewcommand\thisvolume{22} \renewcommand\datereceived{July 10, 2015} \renewcommand\dateaccepted{May 23, 2016} %%%%%%%% end journal info %%%%%%%%%%%%% \usepackage[displaymath,mathlines,pagewise]{lineno} %Rad zaprimljen dana: 10-7-2015 %Rad prihvacen dana: 23-5-2016 %%%%% author macros %%%%%%%%% % place your own macros HERE %%%%% end %%%%%%%%% %%%%%%%%% already loaded packages -- DO NOT RELOAD %%%%% % \usepackage{amssymb} % \usepackage{amsmath,amsthm} % \usepackage{latexsym} % \usepackage{amsfonts} % \usepackage{amsbsyt} % \usepackage{graphicx} %%%%%% end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{document} %\begin{linenumbers} %%%%% title and author(s): % \markboth{Author(s)}{Short Title} % \title{Title} \markboth{M.\,E.\,Aydin and I.\,Mihai}{On certain surfaces in the isotropic 4-space} \title[On certain surfaces in the isotropic 4-space]{On certain surfaces in the isotropic 4-space} % single author: % \author[AUTHOR]{AUTHOR} % \address{address of AUTHOR} % \email{{\tt email address of AUTHOR} (AUTHOR)} \author[M.\,E.\,Aydin and I.\,Mihai]{Muhittin Evren Aydin\affil{1}\comma\corrauth and Ion Mihai\affil{2}} \address{\affilnum{1}\ Department of Mathematics, Faculty of Science, Firat University, Elazig, 23\,200, Turkey\\ \affilnum{2}\ Department of Mathematics, Faculty of Mathematics and Computer Science, University of Bucharest, Bucharest, 010\,014, Romania} \emails{{\tt meaydin@firat.edu.tr}\,\,(M.\,E.\,Aydin), {\tt imihai@fmi.unibuc.ro} (I.\,Mihai)} %MUHITTIN EVREN AYDIN % multiple authors: % Please mark \corrauth after the name of the corresponding author. % different addresses: %\author[AUTHOR1 and AUTHOR2]{M.E. Aydin\affil{1}\comma\corrauth and I\affil{2}} %\address{\affilnum{1}\ address of AUTHOR1\\ \affilnum{2}\ address of AUTHOR2} % %same address: %\author[AUTHOR1, AUTHOR2 and AUTHOR3]{AUTHOR1, AUTHOR2\corrauth and AUTHOR3} %\address{address of AUTHOR1, AUTHOR2 and AUTHOR3} % %\emails{{\ email of AUTHOR1} (AUTHOR1), {\ email of AUTHOR2} (AUTHOR2), {\tt email of AUTHOR3} (AUTHOR3)} % %%%%% Begin Abstract %%%%%%%%%%% \begin{abstract} The isotropic space is a special ambient space obtained from the Euclidean space by substituting the usual Euclidean distance with the isotropic distance. In the present paper, we establish a method to calculate the second fundamental form of surfaces in the isotropic 4-space. Further, we classify some surfaces (spherical product surfaces and Aminov surfaces) in the isotropic 4-space with vanishing curvatures. \end{abstract} %%%%% end %%%%%%%%%%% %%%%% Keywords %%%%%%%%%%% \keywords{Isotropic 4-space, Aminov surface, spherical product surface, isotropic mean curvature, relative curvature} %%%% AMS subject classifications %%%% \ams{53A35, 53A40, 53B25, 53C42} %%%% maketitle %%%%% \maketitle %%%% Start %%%%%% \section{Preliminaries} Let $\mathbb{R}^{n+1}$ be the Euclidean $\left( n+1\right)$-space, i.e., the Cartesian $\left( n+1\right)$-space endowed with the Euclidean metric. We will denote the Euclidean scalar product and the induced norm on $\mathbb{R}% ^{n+1}$ by $\left\langle \cdot ,\cdot \right\rangle $ and $\left\Vert \cdot\right\Vert ,$ respectively. The isotropic $\left( n+1\right) $-space $\mathbb{I}^{n+1}$ introduced by H. Sachs \cite{22} is the product of $\mathbb{R}^{n}$ and the isotropic line equipped with a degenerate parabolic distance metric. It is derived from $% \mathbb{R}^{n+1}$ by substituting the usual Euclidean distance with the isotropic distance. The group of motions of $\mathbb{I}^{n+1}$ is given by the matrix% \begin{equation*} \begin{bmatrix} A & 0 \\ B & 1% \end{bmatrix}% , \end{equation*}% where $A$ is an orthogonal $\left( n,n\right) -$matrix, $\det A=1,$ $B$ a real $\left( 1,n\right) -$matrix. Consider the points $\mathbf{p}=\left( p,p_{n+1}\right) $ and $\mathbf{q}% =\left( q,q_{n+1}\right) $ in $\mathbb{I}^{n+1}$, with $p=% \left(p_{1}, \ldots ,p_{n}\right) ,$ $q=\left( q_{1}, \ldots ,q_{n}\right) $. Thus the \textit{isotropic distance} (\textit{i-distance}) of two points $\mathbf{p}% =\left( p,p_{n+1}\right) $ and $\mathbf{q}=\left( q,q_{n+1}\right) $ is defined as \begin{equation} \left\Vert \mathbf{p}-\mathbf{q}\right\Vert _{i}=\left\Vert p-q\right\Vert =% \sqrt{\sum_{j=1}^{n}\left( q_{j}-p_{j}\right) ^{2}}. \label{1.1} \end{equation} The $i$-metric\ $\left(\ref{1.1}\right) $ is degenerate along the lines in $% x_{n+1}$-direction, and these lines are called \textit{isotropic} \emph{lines% }. $k$-planes containing an isotropic line are called\textit{\ isotropic }$% k- $\textit{planes. } Other planes are\textit{\ non-isotropic}. A surface $M^{2}$ immersed in $\mathbb{I}^{n+1}$ is called \textit{admissible} if it has no isotropic tangent planes. \textit{Isotropic scalar product} (\textit{i-scalar product}) "$\cdot $" of vectors $\mathbf{u}=\left( u,u_{n+1}\right) $ and $\mathbf{v}=\left( v,v_{n+1}\right) $ in $\mathbb{I}^{n+1}$ for $u=\left( u_{1}, \ldots ,u_{n}\right) $ and $v=\left( v_{1}, \ldots ,v_{n}\right) $ is given by% \begin{equation} \mathbf{u}\cdot \mathbf{v}=\left\{ \begin{array}{ll} \left\langle u,v\right\rangle & \text{, if at least one of }u_{i}\text{ or }% v_{i}\text{ is nonzero, }i=\overline{1,n}, \\ u_{n+1}v_{n+1} & \text{, if }u_{i}=0=v_{i}\text{ for all }i=\overline{1,n}.% \end{array}% \right. \label{1.2} \end{equation} We call vectors of the form $\mathbf{u}=\left( 0,u_{n+1}\right) $ in $% \mathbb{I}^{n+1},$ $0=\left( \underset{n-tuple}{\underbrace{0, \ldots ,0}}\right) ,$ $% u_{n+1}\neq 0,$ \textit{isotropic vectors }and ones of the form $\mathbf{u% }=\left( u\neq 0,u_{n+1}\right) $ \textit{non-isotropic vectors. }With respect to the $i$-scalar product $\left( \ref{1.2}\right) $, all isotropic vectors are orthogonal to non-isotropic ones. Morever, two non-isotropic vectors $\mathbf{u},\mathbf{v}$ in $\mathbb{I}^{n+1}$ are orthogonal if $% \mathbf{u}\cdot \mathbf{v}=0.$ In particular, the isotropic 3-space $\mathbb{I}^{3}$ is a Cayley--Klein space defined from a 3-dimensional projective space $P\left( %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ^{3}\right) $ with the absolute figure which is an ordered triple $\left( \omega ,f_{1},f_{2}\right) $, where $\omega $ is a plane in $P\left( %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ^{3}\right) $ and $f_{1},f_{2}$ are two complex-conjugate straight lines in $% \omega $ (see \cite{23}). The homogeneous coordinates in $P\left( %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ^{3}\right) $ are introduced in such a way that the absolute plane $\omega $ is given by $X_{0}=0$ and the absolute lines $f_{1},f_{2}$ by $% X_{0}=X_{1}+iX_{2}=0,$ $X_{0}=X_{1}-iX_{2}=0.$ The intersection point $% F(0:0:0:1)$ of these two lines is called the absolute point. The group of motions of $\mathbb{I}^{3}$ is a six-parameter group given in the affine coordinates $x_{1}=\frac{X_{1}}{X_{0}},$ $x_{2}=\frac{X_{2}}{X_{0}},$ $x_{3}=% \frac{X_{3}}{X_{0}}$ by \begin{equation} \left( x_{1},x_{2},x_{3}\right) \longmapsto \left( x_{1}^{\prime },x_{2}^{\prime },x_{3}^{\prime }\right) :\left\{ \begin{array}{l} x_{1}^{\prime }=a+x_{1}\cos \phi -x_{2}\sin \phi , \\ x_{2}^{\prime }=b+x_{1}\sin \phi +x_{2}\cos \phi , \\ x_{3}^{\prime }=c+dx_{1}+ex_{2}+x_{3},% \end{array}% \right. \label{1.3} \end{equation}% where $a,b,c,d,e,\phi \in %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion .$ Such affine transformations are called \textit{isotropic congruence transformations }or \textit{i-motions. }It can be easily seen from $\left( \ref{1.3}\right) $ that i-motions are indeed composed of an Euclidean motion in the $x_{1}x_{2}-$plane (i.e. translation and rotation) and an affine shear transformation in $x_{3}-$direction. Consider the points $\mathbf{x}=\left( x_{1},x_{2},x_{3}\right) $ and $% \mathbf{y}=\left( y_{1},y_{2},y_{3}\right) .$ The projection in the $x_{3}-$% direction onto $\mathbb{R}^{2},$ $\left( x_{1},x_{2},x_{3}\right) \longmapsto \left( x_{1},x_{2},0\right) ,$ is called the\textit{\ top view}. In the sequel, many of metric properties in isotropic geometry (invariants under $\left( \ref{1.3}\right) )$ are Euclidean invariants in the top view such as i-distance. \textit{Planes, circles and spheres}. There are two types of planes in $% \mathbb{I}^{3}$ (\cite{17}-\cite{19}). (1) \textit{Non-isotropic planes} are planes non-parallel to the $x_{3}-$% direction. In these planes we basically have a Euclidean metric. This is not the one we are used to, since we have to make the usual Euclidean measurements in the top view. An \textit{i-circle} (of \textit{elliptic type}% ) in a non-isotropic plane $P$ is an ellipse, whose top view is a Euclidean circle. Such an i-circle with center $\mathbf{m}\in P$ and radius $r$ is the set of all points\textbf{\ }$\mathbf{x}\in P$ with $\left\Vert \mathbf{x}-% \mathbf{m}\right\Vert _{i}=r.$ (2) \textit{Isotropic planes }are planes parallel to the $x_{3}-$axis. There, $\mathbb{I}^{3}$ induces an isotropic metric. An \textit{isotropic circle }(of \textit{parabolic type}) is a parabola with an $x_{3}-$parallel axis and thus it lies in an isotropic plane There are also two types of \textit{isotropic spheres}. An \textit{i-sphere of the cylindrical type} is the set of all points $\mathbf{x}\in \mathbb{I}% ^{3}$ with $\left\Vert \mathbf{x}-\mathbf{m}\right\Vert _{i}=r$. Speaking in a Euclidean way, such a sphere is a right circular cylinder with $x_{3}$% \textit{-}parallel rulings; its top view is the Euclidean circle with center $\mathbf{m}$ and radius $r$. A more interesting and important type of spheres are the \textit{i-spheres of parabolic type,}% \begin{equation*} x_{3}=\frac{A}{2}\left( x_{1}^{2}+x_{2}^{2}\right) +Bx_{1}+Cx_{2}+D,\text{ \ \ }A\neq 0. \end{equation*} From an Euclidean perspective, they are paraboloids of revolution with the $x_{3}-$% parallel axis. The intersections of these i-spheres with planes $P$ are i-circles. If $P$ is nonisotropic, then the intersection is an i-circle of elliptic type. If $P$ is isotropic, the intersection curve is an i-circle of parabolic type. For an admissible surface $M^{2}$ the coefficients $g_{11}$, $g_{12}$, $g_{22}$ of its first fundamental form are calculated with respect to the induced metric. The normal field of $M^{2}$ in $\mathbb{I}^{3}$ is always the isotropic vector $\left( 0,0,1\right) $ since it is perpendicular to all tangent vectors to $M^{2}.$ The coefficients $h_{11}$, $h_{12}$, $h_{22}$ of the second fundamental form of $M^{2}$ are calculated with respect to the normal field of $M^{2}$. The \textit{relative curvature } (the so-called \textit{isotropic Gaussian curvature}) and the \textit{isotropic mean curvature} of $M^{2}$ in $\mathbb{I}^{3}$ are defined by% \begin{equation*} K=\frac{\det \left( h_{ij}\right) }{\det \left( g_{ij}\right) },\text{ \ \ }% H=\frac{g_{11}h_{22}-2g_{12}h_{12}+g_{22}h_{11}}{2\det \left( g_{ij}\right) }% . \end{equation*} For the formulas of relative and isotropic mean curvatures of a surface $M^{2}$ with codimension 2 in $\mathbb{I}^{4}$, see Section 2. Also, more details on $\mathbb{I}^{n+1}$ can be found in \cite{1,10,11}, \cite{14}-\cite{16}, \cite{20}-\cite{24}. On the other hand, isotropic geometry naturally appears when properties of functions shall be geometrically visualized and interpreted via their graph surfaces \cite{17}. One of the remarkable applications of isotropic geometry is pertinent to Image Processing and has been presented in \cite{12}% . Another one by H. Pottmann and Y. Liu is the study of discrete surfaces in isotropic geometry with applications in architectural design \cite{18}. More recently, B.Y. Chen et al. \cite{8,9} studied production models in microeconomics via isotropic geometry. One of the present authors \cite{2,3} classified the translation and homothetical hypersurfaces in $\mathbb{I}^{n+1}$ with constant curvature. In this paper, we introduce a method to calculate the second fundamental form of the surfaces with codimension 2 in $\mathbb{I}^{4}$. Moreover, we classify spherical product surfaces and Aminov surfaces in $\mathbb{I}% ^{4}$ with vanishing curvatures. \section{Surfaces in isotropic 4-space} Let $M^{2}$ be a surface immersed in $\mathbb{I}^{4}$ and $D\subseteq \mathbb{R}^{2}$ an open domain. Then we parametrize the surface $M^{2}$ by mapping% \begin{align*} \mathbf{x}:D\subseteq \mathbb{R}^{2}\longrightarrow \mathbb{I}^{4},\text{ }% \left( u_{1},u_{2}\right) \longmapsto \mathbf{x}\left( u_{1},u_{2}\right) :=\left( \right.&\left.x_{1}\left( u_{1},u_{2}\right) ,x_{2}\left( u_{1},u_{2}\right) ,\right.\\ &\left. x_{3}\left( u_{1},u_{2}\right) ,x_{4}\left( u_{1},u_{2}\right) \right) , \end{align*}% where $x_{i},$ $i=\overline{1,4},$ are smooth real-valued functions on $D $. Throughout this paper, we only consider admissible surfaces. As usual, the pair $\left\{ \mathbf{x}_{u_{1}}:=\frac{\partial \mathbf{x}}{% u_{1}},\right. $ $\left. \mathbf{x}_{u_{2}}:=\frac{\partial \mathbf{x}}{u_{2}% }\right\} $ is a basis of $T_{p}M^{2},$ $p\in M^{2}.$ Hence we have% \begin{equation*} \mathfrak{g}:=\sum_{i,j=1}^{2}\mathfrak{g}_{ij}du_{i}du_{j},\text{ }% \mathfrak{g}_{ij}:=\mathbf{x}_{u_{i}}\cdot \mathbf{x}_{u_{j}},\text{ }% i,j=1,2, \end{equation*} where $\mathfrak{g}$ is the metric tensor on $T_{p}M^{2}$ induced from the $% i $-scalar product on $\mathbb{I}^{4}.$ Denote $W_{1}:=\sqrt{\det \left( \mathfrak{g}_{ij}\right) }.$ Now, let $\alpha =\left( \alpha _{i}\right) ,$ $\beta =\left( \beta _{i}\right) ,$ $\gamma =\left( \gamma _{i}\right) $ be vectors in $\mathbb{I}% ^{4}$. Then we can define a cross product on $\mathbb{I}^{4}$ by% \begin{equation*} \alpha \times \beta \times \gamma :=% \begin{vmatrix} e_{1} & e_{2} & e_{3} & 0 \\ \alpha _{1} & \alpha _{2} & \alpha _{3} & \alpha _{4} \\ \beta _{1} & \beta _{2} & \beta _{3} & \beta _{4} \\ \gamma _{1} & \gamma _{2} & \gamma _{3} & \gamma _{4}% \end{vmatrix}% , \end{equation*}% for $e_{i}=\left( \delta _{i1},\delta _{i2},\delta _{i3},\delta _{i4}\right)$, $i=1, \ldots ,4,$ where $ \delta $ is Kronocker delta. It is easy to check that% \begin{equation*} (\alpha \times \beta \times \gamma )\cdot \xi =\det \left( \alpha ,\beta ,\gamma ,\bar{\xi}\right) , \end{equation*}% where $\bar{\xi}$ denotes the projection of $\xi$ on the Euclidean $\left( x_{1},x_{2},x_{3}\right) $-space. Therefore the normal space of $M^{2}$ in $\mathbb{I}^{4}$ is spanned by the vectors $\left\{ N_{1},N_{2}\right\} ,$% \begin{equation*} N_{1}:=\left( 0,0,0,1\right), %\text{ }(\text{which is completely unit isotropic vector}) \end{equation*}% which is completely a unit isotropic vector, and% \begin{equation*} N_{2}:=\frac{1}{W_{1}}\mathbf{x}_{u_{1}}\times \mathbf{x}_{u_{2}}\times N_{1}. \end{equation*}% The second fundamental form of $M^{2}$ in $\mathbb{I}^{4}$ has the components \begin{equation*} h_{ij}^{1}:=\det \left( \mathbf{x}_{u_{i}u_{j}},\mathbf{x}_{u_{1}},\mathbf{x}% _{u_{2}},N_{2}\right) ,\text{ }h_{ij}^{2}:=\mathbf{x}_{u_{i}u_{j}}\cdot N_{2}. \end{equation*} A surface in $\mathbb{I}^{4}$ for which the second fundamental form vanishes is called \textit{totally geodesic.} The \textit{relative curvature} (a counterpart to Gaussian curvature) of $% M^{2} $ in $\mathbb{I}^{4}$ is defined by% \begin{equation} G:=\frac{1}{W_{1}^{2}}\sum_{r=1}^{2}\left[ h_{11}^{r}h_{22}^{r}-\left( h_{12}^{r}\right) ^{2}\right] \label{2.1} \end{equation}% and the \textit{isotropic mean curvature field} by% \begin{equation} \overrightarrow{H}:=\frac{1}{2W_{1}^{2}}\sum_{r=1}^{2}\left[ \mathfrak{g}% _{11}h_{22}^{r}-2\mathfrak{g}_{12}h_{12}^{r}+\mathfrak{g}_{22}h_{11}^{r}% \right] N_{r}. \label{2.2} \end{equation} A surface $M^{2}$ in $\mathbb{I}^{4}$ is called \textit{isotropic} \textit{% minimal} (resp. \textit{isotropic} \textit{flat}) if $\overrightarrow{H}% \equiv 0$ (resp. $G\equiv 0$). \section{Aminov surfaces in isotropic 4-space} Let $r$ be a nonzero smooth real-valued function on an open interval $I\subset \mathbb{R}.$ Then we consider a surface $M^{2}$ in $\mathbb{I}^{4}$ given by% \begin{equation*} \mathbf{x}:I\times \left[ 0,2\pi \right) \longrightarrow \mathbb{I}^{4},% \text{ }\left( u,v\right) \longmapsto \mathbf{x}\left( u,v\right) =\left( u,v,r\left( u\right) \cos v,r\left( u\right) \sin v\right) . \end{equation*}% Such surfaces are called \textit{Aminov surfaces }\cite{7}\textit{.} The basis $\left\{ \mathbf{x}_{u},\mathbf{x}_{v}\right\} $ of the tangent space of $M^{2}$ is% \begin{equation} \mathbf{x}_{u}=\left( 1,0,r^{\prime }\cos v,r^{\prime }\sin v\right) \text{ and }\mathbf{x}_{v}=\left( 0,1,-r\sin v,r\cos v\right) , \label{3.1} \end{equation}% where $r^{\prime }=\frac{dr}{du}.$ From $\left( \ref{3.1}\right) ,$ we have% \begin{equation} \mathfrak{g}_{11}=1+\left( r^{\prime }\cos v\right) ^{2},\text{ }\mathfrak{g}% _{12}=-rr^{\prime }\cos v\sin v,\text{ }\mathfrak{g}_{22}=1+\left( r\sin v\right) ^{2} \label{3.2} \end{equation}% and $W_{1}^{2}=1+\left( r^{\prime }\cos v\right) ^{2}+\left( r\sin v\right) ^{2}.$ The basis vectors of the normal space of $M^{2}$ are% \begin{equation*} N_{1}=\left( 0,0,0,1\right) \text{ and }N_{2}=\frac{1}{W_{1}}\left( -r^{\prime }\cos v,r\sin v,1,0\right) . \end{equation*} The components of the second fundamental form are% \begin{equation} \left\{ \begin{array}{l} h_{11}^{1}=-\frac{r^{\prime \prime }\sin v}{W_{1}}\left( 1+r^{2}\right) ,% \text{ }h_{12}^{1}=-\frac{r^{\prime }\cos v}{W_{1}}\left( 1+\left( r^{\prime }\right) ^{2}\right) ,\text{ }h_{22}^{1}=\frac{r\sin v}{W_{1}}\left( 1+r^{2}\right) \\[2ex] h_{11}^{2}=\frac{r^{\prime \prime }}{W_{1}}\cos v,\text{ }h_{12}^{2}=-\frac{1% }{W_{1}}\left( r^{\prime }\sin v\right) ,\text{ }h_{22}^{2}=-\frac{1}{W_{1}}% \left( r\cos v\right) % \end{array}.% \right. \label{3.3} \end{equation} \begin{theorem} The isotropic flat Aminov surfaces in $\mathbb{I}^{4}$ are only generalized cylinders over circular helices from the Euclidean perspective. \label{theorem1} \end{theorem} \begin{proof} Let $M^{2}$ be a flat isotropic Aminov surface. Then, from $% (\ref{2.1})$, it follows that \begin{equation} h_{11}^{1}h_{22}^{1}-\left( h_{12}^{1}\right) ^{2}+h_{11}^{2}h_{22}^{2}-\left( h_{12}^{2}\right) ^{2}=0. \label{3.4} \end{equation}% Substituting $(\ref{3.3})$ into $(\ref{3.4})$ we have% \begin{equation*} \left( rr^{\prime \prime }\left( 1+r^{2}\right) ^{2}+\left( r^{\prime }\right) ^{2}\right) \sin ^{2}v+\left( \left( r^{\prime }\left( 1+\left( r^{\prime }\right) ^{2}\right) \right) ^{2}+rr^{\prime \prime }\right) \cos ^{2}v=0, \end{equation*}% which implies that% \begin{equation} \left\{ \begin{array}{l} rr^{\prime \prime }\left( 1+r^{2}\right) ^{2}+\left( r^{\prime }\right) ^{2}=0, \\ rr^{\prime \prime }+\left( r^{\prime }\left( 1+\left( r^{\prime }\right) ^{2}\right) \right) ^{2}=0.% \end{array}\right. \label{3.5} \end{equation} If $r$ is not a constant, from $\left( \ref{3.5}\right) $ we get \begin{equation*} \left( r^{\prime }\right) ^{2}\left( \left( 1+\left( r^{\prime }\right) ^{2}\right) ^{2}-\frac{1}{\left( 1+r^{2}\right) ^{2}}\right) =0, \label{3.6} \end{equation*}% or equivalently,% \begin{equation*} r^{2}+\left( r^{\prime }\right) ^{2}+\left( rr^{\prime }\right) ^{2}=0. \label{3.6} \end{equation*}% This is a contradiction and thus we derive $r$ is a constant. Therefore we obtain% \begin{equation*} \mathbf{x}\left( u,v\right) =\left( u,0,0,0\right) +\left( 0,v,\lambda \cos v,\lambda \sin v\right) , \label{3.7} \end{equation*}% which completes the proof. \end{proof} \begin{theorem} There does not exist an isotropic minimal Aminov surface in $\mathbb{I}^{4}.$ \label{theorem2} \end{theorem} \begin{proof} Consider an isotropic minimal Aminov surface in $\mathbb{I}% ^{4}$. It follows from $\left( \ref{2.2}\right) $ that% \begin{equation} \mathfrak{g}_{11}h_{22}^{l}-2\mathfrak{g}_{12}h_{12}^{l}+\mathfrak{g}% _{22}h_{11}^{l}=0,\text{ }l=1,2. \label{3.8} \end{equation}% Taking $l=2$ in $\left( \ref{3.8}\right) $ and using $\left( \ref{3.3}\right) ,$ we have% \begin{equation} -\left( 1+\left( r^{\prime }\cos v\right) ^{2}\right) \left( r\cos v\right) -2r\left( r^{\prime }\right) ^{2}\cos v\sin ^{2}v+\left( 1+\left( r\sin v\right) ^{2}\right) r^{\prime \prime }\cos v=0. \label{3.9} \end{equation}% For $v=0$ in $\left( \ref{3.9}\right) $ we have \begin{equation} -r\left( 1+\left( r^{\prime }\right) ^{2}\right) +r^{\prime \prime }=0. \label{3.10} \end{equation} Now dividing $\left( \ref{3.9}\right) $ by $\cos v$ and taking a partial derivative of $\left( \ref{3.9}\right) $ with respect to $v$ gives that% \begin{equation*} \left(- \left( r^{\prime }\right) ^{2}+rr^{\prime \prime }\right) \sin 2v=0 \end{equation*}% or% \begin{equation} -\left( r^{\prime }\right) ^{2}+rr^{\prime \prime }=0. \label{3.11} \end{equation} On the other hand, for $l=1$ in $\left( \ref{3.8}\right) $, we get% \begin{equation} \begin{split} \left( 1+\left( r^{\prime }\cos v\right) ^{2}\right) \left( 1+\left( r^{2}\right) \right) r&-2r\left( r^{\prime }\right) ^{2}\left( 1+\left( r^{\prime }\right) ^{2}\right) \cos ^{2}v\\ &-\left( 1+\left( r\sin v\right) ^{2}\right) r^{\prime \prime }\left( 1+r^{2}\right) =0.% \end{split}% \label{3.12} \end{equation}% Taking partial derivative of $\left( \ref{3.12}\right) $ with respect to $v$ and dividing by $\sin 2v$ gives% \begin{equation} \left( r^{\prime }\right) ^{2}r\left( 1+r^{2}\right) -2r\left( r^{\prime }\right) ^{2}\left( 1+\left( r^{\prime }\right) ^{2}\right) +r^{2}r^{\prime \prime }\left( 1+r^{2}\right) =0. \label{3.13} \end{equation}% By substituting $rr^{\prime \prime }=\left( r^{\prime }\right) ^{2}$ in $% \left( \ref{3.13}\right) ,$ we derive% \begin{equation} 2r\left( r^{\prime }\right) ^{2}\left\{ r^{2}-\left( r^{\prime }\right) ^{2}\right\} =0. \label{3.14} \end{equation}% If $r^{\prime }=0$, then by (\ref{3.10}) we have $r=0$. This is not possible and thus by $\left( \ref{3.14}\right) $ we conclude% \begin{equation} r^{2}=\left( r^{\prime }\right) ^{2}. \label{3.15} \end{equation}% Substituting $\left( \ref{3.15}\right) $ in $\left( \ref{3.11}\right) $ one gets $% r=r^{\prime \prime }$ and from $\left( \ref{3.10}\right) $ we obtain $% r^{\prime }=0.$ This yields a contradiction and thereby the proof is completed. \end{proof} \section{Spherical product surfaces in isotropic 4-space} The tight embeddings of product spaces were investigated by N.\,H.\,Kuiper (see \cite{13}) who introduced a different tight embedding in the $\left( n_{1}+n_{2}-1\right) -$dimensional Euclidean space $\mathbb{R}% ^{n_{1}+n_{2}-1}$ as follows. Let \begin{eqnarray*} c_{1} &:&M^{m}\longrightarrow %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ^{n_{1}}, \\ c_{1}\left( u_{1}, \ldots ,u_{m}\right) &=&\left( f_{1}\left( u_{1}, \ldots ,u_{m}\right) , \ldots ,f_{n_{1}}\left( u_{1}, \ldots ,u_{m}\right) \right) \end{eqnarray*}% be a tight embedding of an $m-$dimensional manifold $M^{m}$ satisfying the Morse equality and \begin{eqnarray*} c_{2} &:&\mathbb{S}^{n_{2}-1}\longrightarrow %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ^{n_{2}}, \\ c_{1}\left( v_{1}, \ldots ,v_{n_{2}-1}\right) &=&\left( g_{1}\left( v_{1}, \ldots ,v_{n_{2}-1}\right) , \ldots ,g_{n_{2}}\left( v_{1}, \ldots ,v_{n_{2}-1}\right) \right) \end{eqnarray*} the standard embedding of the $\left( n_{2}-1\right) -$sphere in $% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ^{n_{2}}$, where $u=\left( u_{1}, \ldots ,u_{m}\right) $ and $v=\left( v_{1}, \ldots ,v_{n_{2}-1}\right) $ are the local coordinate systems on $M^{m}$ and $\mathbb{S}^{n_{2}-1}$, respectively. Then a new \textit{tight embedding} is given by \begin{gather*} \mathbf{x}=c_{1}\otimes c_{2}:M^{m}\times \mathbb{S}^{n_{2}-1}% \longrightarrow %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ^{n_{1}+n_{2}-1}, \\ \left( u,v\right) \longmapsto \left( f_{1}\left( u\right) , \ldots ,f_{n_{1}-1}\left( u\right) ,f_{n_{1}}\left( u\right) g_{1}\left( v\right) , \ldots ,f_{n_{1}}\left( u\right) g_{n_{2}}\left( v\right) \right). \end{gather*} Such embeddings are obtained from $c_{1}$ by rotating $% \mathbb{R}^{n_{1}}$ about $\mathbb{R}^{n_{1}-1}$ in $\mathbb{R}% ^{n_{1}+n_{2}-1}.$ B. Bulca et al. \cite{5,6} called such embeddings \textit{rotational embeddings }and considered spherical product surfaces in Euclidean spaces, which are a special type of rotational embeddings as taking $m=1, n_{1}=2,3$ and $n_{2}=2$ in the above definition. The surfaces of revolution in $\mathbb{R}^{3}$ can be considered as simplest models of spherical product surfaces as well as the quadrics and the superquadrics \cite{4}. Now, let us consider an isotropic 3-space curve and an isotropic plane curve, respectively, \begin{equation*} c_{1}\left( u\right) =\left( u,f_{1}\left( u\right) ,f_{2}\left( u\right) \right) \text{ and }c_{2}\left( v\right) =\left( v,g\left( v\right) \right) \end{equation*} for nonzero smooth functions $f_{1},f_{2}$ and $g$. Then the \textit{spherical product surface }$\left( M^{2}\mathit{,}% c_{1}\otimes c_{2}\right) $ of two curves $c_{1}$ and $c_{2}$ in $\mathbb{I}% ^{4}$ is defined by% \begin{equation} \mathbf{x:}=c_{1}\otimes c_{2}:\mathbb{R}^{2}\longrightarrow \mathbb{I}^{4},% \text{ }\left( u,v\right) \longmapsto \left( u,f_{1}\left( u\right) ,f_{2}\left( u\right) v,f_{2}\left( u\right) g\left( v\right) \right) . \label{4.1} \end{equation}% We call the curves $c_{1}$ and $c_{2}$ the \textit{generating curves} of the surface. Note that such surfaces given by $\left( \ref{4.1}\right) $ are always admissible. The tangent space of $\left( M^{2}\mathit{,}c_{1}\otimes c_{2}\right) $ is spanned by% \begin{equation*} \mathbf{x}_{u}=\left( 1,f_{1}^{\prime },f_{2}^{\prime }v,f_{2}^{\prime }g\right) \text{ and }\mathbf{x}_{v}=f_{2}\left( 0,0,1,g^{\prime }\right) , \end{equation*}% where $f_{i}^{\prime }=\frac{\partial f_{i}}{\partial u},$ $i=1,2$ and $% g^{\prime }=\frac{\partial g}{\partial v}.$ The induced metric $g$ on $M^{2}$ from $\mathbb{I}^{4}$ has the components% \begin{equation} \mathfrak{g}_{11}=1+\left( f_{1}^{\prime }\right) ^{2}+\left( f_{2}^{\prime }v\right) ^{2},\text{ }\mathfrak{g}_{12}=f_{2}f_{2}^{\prime }v,\text{ }% \mathfrak{g}_{22}=\left( f_{2}\right) ^{2} \label{4.2} \end{equation}% and $W_{1}^{2}=\det \left( \mathfrak{g}_{ij}\right) =\left( f_{2}\right) ^{2}\left( 1+\left( f_{1}^{\prime }\right) ^{2}\right) .$ The orthonormal basis of the normal space of $\left( M^{2}\mathit{,}c_{1}\otimes c_{2}\right) $ is% \begin{equation*} N_{1}=\left( 0,0,0,1\right) \text{ and }N_{2}=\frac{1}{\sqrt{1+\left( f_{1}^{\prime }\right) ^{2}}}\left( f_{1}^{\prime },-1,0,0\right) . \end{equation*} Thereby, the nonzero components of the second fundamental form are% \begin{equation} \begin{split} h_{11}^{1}&= f_{2}\left( g^{\prime }v-g\right) \sqrt{1+\left( f_{1}^{\prime }\right) ^{2}}\left( f_{2}^{\prime \prime }-f_{2}^{\prime }\frac{% f_{1}^{\prime }f_{1}^{\prime \prime }}{1+\left( f_{1}^{\prime }\right) ^{2}}% \right) ,\\ \text{ }h_{22}^{1}&=-\left( f_{2}\right) ^{2}\sqrt{1+\left( f_{1}^{\prime }\right) ^{2}}g^{\prime \prime }, \\ h_{11}^{2}&=-\frac{% f_{1}^{\prime \prime }}{\sqrt{1+\left( f_{1}^{\prime }\right) ^{2}}}. \end{split} \label{4.3} \end{equation} The following results classify spherical product surfaces in $\mathbb{I}^{4}$ with vanishing curvature. \begin{theorem} Let $\left( M^{2}\mathit{,}c_{1}\otimes c_{2}\right) $ be an isotropic flat spherical product surface in $ \mathbb{I}^{4}.$ Then either it is a non-isotropic plane or one of the following holds: \begin{itemize} \item[(i)] $c_{1}$ is a planar curve in $\mathbb{I}^{3}$ lying in the non-isotropic plane $z=const.;$ \item[(ii)] $c_{1}$ is a line in $\mathbb{I}^{3};$ \item[(iii)] $c_{1}$ is a curve in $\mathbb{I}^{3}$ of the form \begin{equation*} c_{1}\left( u\right) =\left( u,f_{1}\left( u\right) ,\lambda \int \sqrt{% 1+\left( f_{1}^{\prime }\right) ^{2}}du+\xi \right) ,\text{ }\lambda ,\xi \in %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ,\text{ }\lambda \neq 0; \end{equation*} \item[(iv)] $c_{2}$ is a line in $\mathbb{I}^{2}.$ \end{itemize} \label{theorem3} \end{theorem} \begin{proof} Let us assume that the spherical product surface $\left( M^{2}\mathit{,}c_{1}\otimes c_{2}\right) $ is isotropic flat. Then, from $% \left( \ref{2.1}\right) $ and $\left( \ref{4.3}\right) ,$ we have \begin{equation} f_{2}^{3}g^{\prime \prime }\left( g^{\prime }v-g\right) \left( f_{2}^{\prime \prime }-f_{2}^{\prime }\frac{f_{1}^{\prime }f_{1}^{\prime \prime }}{% 1+\left( f_{1}^{\prime }\right) ^{2}}\right) \left( 1+\left( f_{1}^{\prime }\right) ^{2}\right) =0. \label{4.4} \end{equation}% It immediately implies that either $g$ is a linear function (which implies the statement (iv) of the theorem) or \begin{equation} f_{2}^{\prime \prime }-f_{2}^{\prime }\frac{f_{1}^{\prime }f_{1}^{\prime \prime }}{1+\left( f_{1}^{\prime }\right) ^{2}}=0. \label{4.5} \end{equation}% For equation $\left( \ref{4.5}\right) $ we have three cases: \bigskip \textbf{Case 1.} $f_{2}$ is constant. In this case, the generating curve $% c_{1}$ is a planar curve in $\mathbb{I}^{3}$ lying in the non-isotropic plane $% z=const.$ This gives statement (i) of the theorem. \bigskip \textbf{Case 2. }$f_{1}$ and $f_{2}$ are linear. Thus $c_{1}$ is a line in $% \mathbb{I}^{3},$ which implies statement (ii). \bigskip \textbf{Case 3. }$f_{1}$ and $f_{2}$ are non-linear. After solving $\left( \ref{4.5}\right) ,$ we derive% \begin{equation*} f_{2}=\lambda \int \sqrt{1+\left( f_{1}^{\prime }\right) ^{2}}du+\xi ,\text{ }\lambda \neq 0,\xi \in\mathbb{R}, \end{equation*} which completes the proof. \end{proof} \begin{theorem} There does not exist an isotropic minimal spherical product surface in $\mathbb{I}^{4}$, except totally geodesic ones. \label{theorem4} \end{theorem} \begin{proof}Suppose that $\left( M^{2}\mathit{,}c_{1}\otimes c_{2}\right) $ is an isotropic minimal spherical product surface in $\mathbb{% I}^{4}.$ By taking $r=2$ and $r=1$ in $\left( \ref{2.2}\right)$, respectively, and considering $\left( \ref{4.3}\right) ,$ we get% \begin{equation} \frac{\left( f_{2}\right) ^{2}f_{1}^{\prime \prime }}{\sqrt{1+\left( f_{1}^{\prime }\right) ^{2}}}=0 \label{4.6} \end{equation}% and \begin{equation} f_{2}\left( g^{\prime }v-g\right) \left( f_{2}^{\prime \prime }-f_{2}^{\prime }% \frac{f_{1}^{\prime }f_{1}^{\prime \prime }}{1+\left( f_{1}^{\prime }\right) ^{2}}\right) -g^{\prime \prime }\left( 1+\left( f_{1}^{\prime }\right) ^{2}+\left( f_{2}^{\prime }v\right) ^{2}\right) =0. \label{4.7} \end{equation}% It follows from $\left( \ref{4.6}\right) $ that $f_{1}$ is a linear function. By considering it into $\left( \ref{4.7}\right) ,$ we derive% \begin{equation} \left( f_{2}f_{2}^{\prime \prime }\right) \left( g^{\prime }v-g\right) -g^{\prime \prime }\left( 1+a^{2}+\left( f_{2}^{\prime }v\right) ^{2}\right) =0,\text{ }a\in \mathbb{R}. \label{4.8} \end{equation}% For $\left( \ref{4.8}\right) ,$ we have to distinguish two cases: \bigskip \textbf{Case 1.} $g$ is linear. We have again two cases: \bigskip \textbf{Case 1.1.} $g\left(v\right)=av$ is a solution for $\left( \ref{4.8}\right)$. It yields from $\left( \ref{4.3}\right) $ that $\left( M^{2}\mathit{,}% c_{1}\otimes c_{2}\right) $ is totally geodesic. \bigskip \textbf{Case 1.2.} $g\left( v\right) =av+b,$ $a,b\neq 0.$ $\left( \ref{4.8}\right) $ gives that $f_{2}$ is linear and it follows from $\left( \ref{4.3}\right) $ that $% \left( M^{2},c_{1}\otimes c_{2}\right) $ is again totally geodesic. \bigskip \textbf{Case 2.} $g$ is non-linear. There exist two cases depending on the function $f_{2}$: \bigskip \textbf{Case 2.1. }$f_{2}$ is linear, $f_{2}\left( u\right) =cu+d.$ By $\left( \ref{4.8}\right) $ we derive% \begin{equation*} g^{\prime \prime }\left( 1+a^{2}+c^{2}v^{2}\right) =0, \end{equation*}% which is not possible. \bigskip \textbf{Case 2.2. }$f_{2}$ is non-linear. Equation $\left( 4.8\right) $ can then be rewritten as% \begin{equation} \frac{g^{\prime }v-g}{g^{\prime \prime }}-\frac{1+a^{2}}{f_{2}f_{2}^{\prime \prime }}-\frac{\left( f_{2}^{\prime }\right) ^{2}}{f_{2}f_{2}^{\prime \prime }}v^{2}=0 \label{4.9} \end{equation}% After taking partial derivative of $\left( 4.9\right) $ with respect to $u$, we deduce% \begin{equation*} \left( \frac{1+a^{2}}{f_{2}f_{2}^{\prime \prime }}\right) ^{\prime }+\left( \frac{\left( f_{2}^{\prime }\right) ^{2}}{f_{2}f_{2}^{\prime \prime }}% \right) ^{\prime }v^{2}=0, \end{equation*}% which yields a contradiction since $f_{2}$ is a non-linear function and $v$ is an independent variable. Therefore the proof is completed. \end{proof} %%%% Acknowledgment %%%%%%%% \section*{Acknowledgement} The authors would like to thank the referees for their valuable comments which helped to improve the manuscript. This paper was written during the visit of the second author to Firat University, Turkey, in April-May 2015. 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