Ruled and Quadric Surfaces Satisfying Δ^IIN = ΛN

Abstract

In the 3-dimensional Euclidean space $E^3$, a quadric surface is either ruled or of one of the following two kinds $z^2=a{s}^2+b{t}^2+c,abc\ne 0$ or $z=\frac{a}{2}{s}^2+\frac{b}{2}{t}^2,a>0,b>0$. In the present paper, we investigate these three kinds of surfaces whose Gauss map $\mathit{N}$ satisfies the property ${\Delta }^{II}\mathit{N}=\Lambda \mathit{N}$, where $\Lambda$ is a square symmetric matrix of order 3, and ${\Delta }^{II}$ denotes the Laplace operator of the second fundamental form $II$ of the surface. We prove that spheres with the nonzero symmetric matrix $\Lambda$, and helicoids with $\Lambda$ as the corresponding zero matrix, are the only classes of surfaces satisfying the above given property.

1. Introduction

As is known, the theory of the Gauss map is one of the interesting subjects for many researchers in Euclidean space; it has been investigated from several different areas by many different geometers. In 1983, B.Y. Chen introduced the concept of Euclidean immersions of a finite type [1]. A submanifold $M^n$ is said to be of finite type if each component of the position vector $\mathit{x}$ of $M^n$ can be written as a finite sum of the eigenfunctions of its Laplacian $\Delta$, where $\Delta$ is the Laplace operator of $M^n$ associated with the induced metric.

On the other hand, let $\mathit{x}$ be the position vector of a surface S in the Euclidean 3-space $E^3$ equipped with the induced metric. If we consider that $(x_1,x_2,x_3)$ are the component functions of $\mathit{x}$, then it is well known that

$${\Delta }^I\mathit{x}=({\Delta }^I{x}_1,{\Delta }^I{x}_2,{\Delta }^I{x}_3).$$

(1)

In this context, Chen and Piccini in [2] introduced the theory of the submanifolds of a finite type Gauss map in the same way. In a special case for $E^3$, one can ask:

Problem 1. 

Classify all surfaces in $E^3$ with finite type Gauss map.

Results concerning this problem can be found in ([3,4,5,6]).

Similarly, another type of study arose and became of great interest to researchers, namely, classifying all surfaces in $E^3$ whose Gauss map $\mathit{N}$ satisfies an equation of the form

$${\Delta }^I\mathit{N}=\Lambda \mathit{N}$$

(2)

In 2003, authors in [7] defined the concept of surfaces of finite type with respect to the 2nd or 3rd fundamental forms. So, an extension of (2) is studying surfaces in $E^3$ whose Gauss map $\mathit{N}$ satisfies an equation of the form

$${\Delta }^J\mathit{N}=\Lambda \mathit{N},J=II,III.$$

(3)

In [8], Kim and others studied the class of surfaces of revolutions, and they showed that the catenoid and sphere are the only surfaces of revolution satisfying (3) with respect to the fundamental form $II$.

2. Fundamentals

Let $\mathit{r}:=\mathit{r}(u^1,u^2)$ be a regular parametric representation of a surface S in $E^3$. We denote this using

$$I=g_{km}d{u}^{k}d{u}^m,II=b_{km}d{u}^{k}d{u}^m,III=e_{km}d{u}^{k}d{u}^m,k,m=1,2,$$

the components of the fundamental forms of S, respectively. For two sufficiently differentiable functions f and h on S, the first and second differential parameters of Beltrami regarding the fundamental form J are defined by [9].

$${\nabla }^J(f,h):=a^{km}{f}_{/k}{h}_{/m},$$

(4)

$${\Delta }^{J}h:=-a^{km}{\nabla }_k^J{h}_{/m}=-\frac{1}{\sqrt{\left|a\right|}}\frac{\partial h}{\partial u^k}\left(\sqrt{\left|a\right|}{a}^{km}\frac{\partial h}{\partial u^m}\right),$$

(5)

where $h_{/k}:=\frac{\partial h}{\partial u^k}$, $\left(a^{km}\right)$ denotes the components of the inverse tensor of $\left(g_{km}\right),\left(b_{km}\right)$ and $\left(e_{km}\right)$ for $J=I,II$, and $III$, respectively, $a=det\left(a_{km}\right)$, and ${\nabla }_k^J$ is the covariant derivative in the $u^k$ direction, corresponding to the fundamental form J.

The Gauss curvature K and the mean curvature H of S are given by

$$K=\frac{1}{R_1{R}_2}=\frac{b}{g}=\frac{e}{b},2H=\frac{1}{R_1}+\frac{1}{R_2}=g^{ik}{b}_{ik}=b^{ik}{e}_{ik},$$

(6)

where $g=det\left(g_{ik}\right),b=det\left(b_{ik}\right)$, $e=det\left(e_{ik}\right)$ and $R_1,R_2$ are the principal radii of curvature. For simplicity, this can be expressed as:

$$R=\frac{2H}{K}=e^{ik}{b}_{ik}=b^{ik}{g}_{ik},Q=4H^2-2K=g^{ik}{e}_{ik}.$$

(7)

The Weingarten equations are

$${\mathit{r}}_k=-g_{kj}{b}^{jr}{\mathit{N}}_{/r}=-b_{kj}{e}^{jr}{\mathit{N}}_{/r}$$

$${\mathit{N}}_k=-e_{kj}{b}^{jr}{\mathit{r}}_{/r}=-b_{kj}{g}^{jr}{\mathit{r}}_{/r}$$

(8)

We mention now the following well-known relations for later use

$${\Delta }^I\mathit{r}=-2H\mathit{N},$$

(9)

$${\Delta }^I\mathit{N}=2gra{d}^{I}H+Q\mathit{N},$$

(10)

$${\Delta }^{II}\mathit{r}=\frac{1}{2}Kgra{d}^{III}\frac{1}{K}-2\mathit{N},$$

(11)

$${\Delta }^{II}\mathit{N}=\frac{1}{2K}gra{d}^{I}K+2H\mathit{N},$$

(12)

$${\Delta }^{III}\mathit{r}=gra{d}^{III}R-R\mathit{N},$$

(13)

$${\Delta }^{III}\mathit{N}=2\mathit{N}.$$

(14)

We denote by $T_P\left(S\right)$ the tangent plane to S at a point $P\in S$. The Weingarten map is the symmetric linear transformation $A:T_P\left(S\right)\to T_P\left(S\right)$, which is defined as follows:

For $\mathit{z}=z^i{\mathit{r}}_{/i}$, then $A\left(\mathit{z}\right)=v^i{\mathit{r}}_{/i},v^i:=b_{jk}{g}^{ki}{z}_j$. In the special case that $g_{12}=b_{12}=0$, we find that

$$A\left(\mathit{z}\right)=\frac{1}{R_1}{z}^1{\mathit{r}}_{/1}+\frac{1}{R_2}{z}^2{\mathit{r}}_{/2}.$$

Firstly, we prove the following:

Theorem 1. 

For a sufficient differentiable function $f(u^1,u^2)$ defined on S, the following relations hold true:

1. 

${\nabla }^{II}(f,\mathit{N})+gra{d}^{I}f=0$,

2. 

${\nabla }^{II}(f,\mathit{r})+gra{d}^{III}f=0$,

3. 

${\nabla }^{III}(f,\mathit{r})+\frac{1}{K}gra{d}^{I}f=R{\nabla }^{II}(f,\mathit{r})$,

4. 

$A\left(gra{d}^{I}f\right)+{\nabla }^I(f,\mathit{N})=0$.

Proof. 

We prove (3) and (4). For proof of (1) and (2), see [7].

From the well-known relation

$$K{e}^{ik}-2H{b}^{ik}+g^{ik}=0,$$

then

$$e^{ik}=R{b}^{ik}-\frac{1}{K}{g}^{ik}.$$

From (4), and using the last equation, we find

$${\nabla }^{III}(f,\mathit{r})=e^{ik}{f}_{/k}{\mathit{r}}_i=(R{b}^{ik}-\frac{1}{K}{g}^{ik})f_{/k}{\mathit{r}}_i=R{\nabla }^{II}(f,\mathit{r})-\frac{1}{K}gra{d}^{I}f.$$

Hence

$${\nabla }^{III}(f,\mathit{r})+\frac{1}{K}gra{d}^{I}f=R{\nabla }^{II}(f,\mathit{r}).$$

Finally, for proving (4), without loss of generality, we may assume that $g_{12}=b_{12}=0$, then

$$\begin{array}{ccc}\hfill A\left(gra{d}^{I}f\right)& =& A\left(g^{ik}{f}_{/k}{\mathit{r}}_i\right)=\frac{b_{11}}{g_{11}}{g}^{11}{f}_{/1}{\mathit{r}}_{/1}+\frac{b_{22}}{g_{22}}{g}^{22}{f}_{/2}{\mathit{r}}_{/2}\hfill \\ & =& \frac{b_{11}}{{\left(g_{11}\right)}^2}{f}_{/1}{\mathit{r}}_{/1}+\frac{b_{22}}{{\left(g_{22}\right)}^2}{f}_{/2}{\mathit{r}}_{/2}.\hfill \end{array}$$

On the other hand,

$${\nabla }^I(f,\mathit{N})=g^{ik}{f}_{/k}{\mathit{N}}_i=g^{11}{f}_{/1}{\mathit{N}}_{/1}+g^{22}{f}_{/2}{\mathit{N}}_{/2}.$$

On account of $g_{12}=b_{12}=0$, the Weingarten Equations (8) become ${\mathit{N}}_{/1}=\frac{b_{11}}{g_{11}}{\mathit{r}}_{/1}$ and ${\mathit{N}}_{/2}=\frac{b_{22}}{g_{22}}{\mathit{r}}_{/2}$; therefore, the last equation becomes

$${\nabla }^I(f,\mathit{N})=-\frac{b_{11}}{{\left(g_{11}\right)}^2}{f}_{/1}{\mathit{r}}_{/1}-\frac{b_{22}}{{\left(g_{22}\right)}^2}{f}_{/2}{\mathit{r}}_{/2}=-A\left(gra{d}^{I}f\right).$$

□

We denote by $w=-<\mathit{r},\mathit{N}>$ the support function of S, where $<,>$ is the Euclidean inner product. We now prove the following relations:

Theorem 2. 

For the support function w of S, the following relations hold true:

1. 

${\Delta }^{I}w=Qw-2H-<gra{d}^{I}H,\mathit{r}>$,

2. 

${\Delta }^{II}w=2Hw-\frac{1}{2K}<gra{d}^{I}K,\mathit{r}>-2$,

3. 

${\Delta }^{III}w=2w-R$.

Proof. 

(1) Using (5), we find

$$\begin{array}{ccc}\hfill {\Delta }^{I}w& =& -g^{ik}{\nabla }_k^I{w}_{/i}=g^{ik}{\nabla }_k^I<{\mathit{r}}_{/i},\mathit{N}>+g^{ik}{\nabla }_k^I<\mathit{r},{\mathit{N}}_{/i}>\hfill \\ & =& <g^{ik}{\nabla }_k^I{\mathit{r}}_{/i},\mathit{N}>+<\mathit{r},g^{ik}{\nabla }_k^I{\mathit{N}}_{/i}>+2g^{ik}<{\mathit{r}}_{/i},{\mathit{N}}_{/k}>\hfill \\ & =& -<{\Delta }^I\mathit{r},\mathit{N}>-<\mathit{r},{\Delta }^I\mathit{N}>+2g^{ik}<{\mathit{r}}_{/i},{\mathit{N}}_{/k}>.\hfill \end{array}$$

From (6), (8), (9) and (10), the last equation becomes

$$\begin{array}{ccc}\hfill {\Delta }^{I}w& =& <2H\mathit{N},\mathit{N}>-<\mathit{r},2gra{d}^{I}H+Q\mathit{N}>+2g^{ik}<{\mathit{r}}_{/i},-e_{kj}{b}^{jr}{\mathit{r}}_{/r}>\hfill \\ & =& Qw-2H-<gra{d}^{I}H,\mathit{r}>.\hfill \end{array}$$

(2) From (5),we have

$$\begin{array}{ccc}\hfill {\Delta }^{II}w& =& -b^{ik}{\nabla }_k^{II}{w}_{/i}=b^{ik}{\nabla }_k^{II}<{\mathit{r}}_{/i},\mathit{N}>+b^{ik}{\nabla }_k^{II}<\mathit{r},{\mathit{N}}_{/i}>\hfill \\ & =& <b^{ik}{\nabla }_k^{II}{\mathit{r}}_{/i},\mathit{N}>+<\mathit{r},b^{ik}{\nabla }_k^{II}{\mathit{N}}_{/i}>+2b^{ik}<{\mathit{r}}_{/i},{\mathit{N}}_{/k}>\hfill \\ & =& -<{\Delta }^{II}\mathit{r},\mathit{N}>-<\mathit{r},{\Delta }^{II}\mathit{N}>+2b^{ik}<{\mathit{r}}_{/i},{\mathit{N}}_{/k}>.\hfill \end{array}$$

From (8), (11) and (12), the last equation becomes

$$\begin{array}{ccc}\hfill {\Delta }^{II}w& =& -<\mathit{r},\frac{1}{2K}gra{d}^{I}K+2H\mathit{N}>-<\frac{1}{2}Kgra{d}^{III}\frac{1}{K}-2\mathit{N},\mathit{N}>\hfill \\ & & +2b^{ik}<{\mathit{r}}_{/i},{\mathit{N}}_{/k}>\hfill \\ & =& 2Hw-<\mathit{r},\frac{1}{2K}gra{d}^{I}K>+2+2b^{ik}<{\mathit{r}}_{/i},-b_{kj}{g}^{jr}{\mathit{r}}_{/r}>\hfill \\ & =& 2Hw-<\mathit{r},\frac{1}{2K}gra{d}^{I}K>-2.\hfill \end{array}$$

(3) From (5), we have

$$\begin{array}{ccc}\hfill {\Delta }^{III}w& =& -e^{ik}{\nabla }_k^{III}{w}_{/i}=e^{ik}{\nabla }_k^{III}<{\mathit{r}}_{/i},\mathit{N}>+e^{ik}{\nabla }_k^{III}<\mathit{r},{\mathit{N}}_{/i}>\hfill \\ & =& <s^{ik}{\nabla }_k^{III}{\mathit{r}}_{/i},\mathit{N}>+<\mathit{r},e^{ik}{\nabla }_k^{III}{\mathit{N}}_{/i}>+2e^{ik}<{\mathit{r}}_{/i},{\mathit{N}}_{/k}>\hfill \\ & =& -<{\Delta }^{III}\mathit{r},\mathit{N}>-<\mathit{r},{\Delta }^{III}\mathit{N}>+2e^{ik}<{\mathit{r}}_{/i},{\mathit{N}}_{/k}>.\hfill \end{array}$$

From (7), (8), (13) and (14), the last equation becomes

$$\begin{array}{ccc}\hfill {\Delta }^{III}w& =& -<gra{d}^{III}R-R\mathit{N},\mathit{N}>-<\mathit{r},2\mathit{N}>+2e^{ik}<{\mathit{r}}_{/i},{\mathit{N}}_{/k}>\hfill \\ & =& R+2w+2e^{ik}<{\mathit{r}}_{/i},-b_{kj}{g}^{jr}{\mathit{r}}_{/r}>=2w-R.\hfill \end{array}$$

□

Let S now be a minimal surface. Then, we find $R=\frac{2H}{K}=0$. Thus, from the last equation, we find ${\Delta }^{III}w=2w$. So, we have:

Corollary 1. 

Let S be a minimal surface. Then, the support function w is of an eigenfunction of ${\Delta }^{III}$ with the corresponding eigenvalue $\lambda =2$.

Interesting research means that one can also follow the idea in [10] by defining the first and second Beltrami operators on the finite product of spaces as a generalization of the Euclidean 3-space. In this paper, we will continue this type of research by investigating new classes of surfaces, namely, ruled and quadric surfaces in $E^3$, which are of coordinate finite $II$-type; i.e., their Gauss map $\mathit{N}$ satisfies relation (3) with $J=II$. Our main results are the following:

Theorem 3. 

Helicoids are the only ruled surfaces in the Euclidean space $E^3$ satisfying (3).

Theorem 4. 

Spheres are the only quadric surfaces in the Euclidean space $E^3$ satisfying (3).

3. Proof of Our Main Results

As it is known, when S is a quadric in $E^3$, then it is either ruled or has one of the following forms

$$z^2=c+a{s}^2+b{t}^2,abc\ne 0$$

(15)

or

$$z=\frac{1}{2}a{s}^2+\frac{1}{2}b{t}^2,a>0,b>0.$$

(16)

In the following paragraph, we study the ruled surfaces and prove the first theorem. Next, we investigate the quadrics of the first kind (15), and we prove that the relation (3) cannot be satisfied unless $a=b=-1$, that is, S is a sphere. Lastly, we prove that the relation (3) cannot be satisfied for a quadric surface of the second form (16).

3.1. Ruled Surfaces

Let S be the ruled surface of a non-vanishing Gaussian curvature. Then, the parametrization for S is given by

$$S:\mathit{x}(u,v)=\mathit{\beta }\left(u\right)+v\mathit{\alpha }\left(u\right),u\in J,vs.\in \mathbb{R},$$

such that

$$⟨{\mathit{\beta }}^{\prime },\mathit{\alpha }⟩=0,⟨\mathit{\alpha },\mathit{\alpha }⟩=1,⟨{\mathit{\alpha }}^{\prime },{\mathit{\alpha }}^{\prime }⟩=1,$$

(17)

where ${}^{\prime }:=\frac{d}{du}$ and $⟨,⟩$ is the Euclidean inner product. The curve $\mathit{\beta }=\mathit{\beta }\left(u\right)$ is called the base curve and $\mathit{\alpha }=\mathit{\alpha }\left(u\right)$ a director vector field.

The components of the first fundamental form of S are

$$E=⟨{\mathit{x}}_{\mathit{u}},{\mathit{x}}_{\mathit{u}}⟩=q,F=⟨{\mathit{x}}_{\mathit{u}},{\mathit{x}}_{\mathit{v}}⟩=0,G=⟨{\mathit{x}}_{\mathit{v}},{\mathit{x}}_{\mathit{v}}⟩=1,$$

where

$${\mathit{x}}_{\mathit{u}}:=\frac{\partial \mathit{x}}{\partial u},{\mathit{x}}_{\mathit{v}}:=\frac{\partial \mathit{x}}{\partial v}$$

and

$$q:=⟨{\mathit{\beta }}^{\prime },{\mathit{\beta }}^{\prime }⟩+2⟨{\mathit{\alpha }}^{\prime },{\mathit{\beta }}^{\prime }⟩v+v^2$$

The Gauss map $\mathit{N}$ of S is

$$\mathit{N}(u,v)=\frac{1}{\sqrt{q}}\left({\mathit{\beta }}^{\prime }\times \mathit{\alpha }+v{\mathit{\alpha }}^{\prime }\times \mathit{\alpha }\right).$$

We express that

$$\mathsf{\Phi }:={\mathit{\beta }}^{\prime }\times \mathit{\alpha },\mathsf{\Psi }:={\mathit{\alpha }}^{\prime }\times \mathit{\alpha };$$

then, the vector $\mathit{N}$ becomes

$$\mathit{N}(u,v)=\frac{1}{\sqrt{q}}\left(\mathsf{\Phi }+v\mathsf{\Psi }\right).$$

The components of the second fundamental form are

$$L=⟨{\mathit{x}}_{\mathit{uu}},\mathit{N}⟩=\frac{p}{\sqrt{q}},M=⟨{\mathit{x}}_{\mathit{uv}},\mathit{N}⟩=\frac{2W}{\sqrt{q}},N=⟨{\mathit{x}}_{\mathit{vv}},\mathit{N}⟩=0,$$

where

$$p:=\left({\mathit{\beta }}^{\prime },\mathit{\alpha },{\mathit{\beta }}^{″}\right)+\left[\left({\mathit{\beta }}^{\prime },\mathit{\alpha },{\mathit{\alpha }}^{″}\right)-\left(\mathit{\alpha },{\mathit{\alpha }}^{\prime },{\mathit{\beta }}^{″}\right)\right]v-\left(\mathit{\alpha },{\mathit{\alpha }}^{\prime },{\mathit{\alpha }}^{″}\right)v^2$$

and

$$W:=\left(\mathit{\alpha },{\mathit{\alpha }}^{\prime },{\mathit{\beta }}^{\prime }\right).$$

For convenience, we write:

$$\begin{array}{ccc}\hfill k& :& =⟨{\mathit{\beta }}^{\prime },{\mathit{\beta }}^{\prime }⟩,l:=⟨{\mathit{\beta }}^{\prime },{\mathit{\alpha }}^{\prime }⟩,\hfill \\ \hfill m& :& =\left({\mathit{\alpha }}^{\prime },\mathit{\alpha },{\mathit{\alpha }}^{″}\right),n:=\left({\mathit{\beta }}^{\prime },\mathit{\alpha },{\mathit{\alpha }}^{″}\right)-\left(\mathit{\alpha },{\mathit{\alpha }}^{\prime },{\mathit{\beta }}^{″}\right),\hfill \\ \hfill r& :& =\left({\mathit{\beta }}^{\prime },\mathit{\alpha },{\mathit{\beta }}^{″}\right),\hfill \end{array}$$

and so we find

$$q=v^2+2lv+k,p=m{v}^2+nv+r.$$

The Gaussian curvature K of S is given by

$$K=-\frac{W^2}{q^2}.$$

We note that $W\ne 0$, otherwise the Gaussian curvature vanishes. The Laplace operator ${\Delta }^{II}$ is given as follows [11]

$$\begin{array}{c}\hfill {\Delta }^{II}=\frac{2\sqrt{q}}{W}\frac{{\partial }^2}{\partial u\partial v}-\frac{p\sqrt{q}}{W^2}\frac{{\partial }^2}{\partial v^2}-\frac{\sqrt{q}{p}_v}{W^2}\frac{\partial }{\partial v},\end{array}$$

(18)

where

$$p_v:=\frac{\partial p}{\partial v}.$$

.

Inserting $\mathit{N}$ on (18) gives

$$\begin{array}{ccc}\hfill {\Delta }^{II}\mathit{N}& =& \frac{1}{q^2}\left[\right(\frac{tpq{q}_{vv}}{2W^2}-\frac{3vp{q}_v^2}{4W^2}+\frac{vq{q}_v{p}_v}{W^2}-\frac{q^2{p}_v}{W^2}+\frac{qp{q}_v}{W^2}\hfill \\ & & +\frac{3v{q}_v{q}_u}{2W}-\frac{vq{q}_{uv}}{W}-\frac{q{q}_u}{W})\mathsf{\Psi }+\left(\frac{2q^2}{W}-\frac{vq{q}_v}{W}\right){\mathsf{\Psi }}^{\prime }-\frac{q{q}_v}{W}{\mathsf{\Phi }}^{\prime }\hfill \\ & & +\left(\frac{pq{q}_{vv}}{2W^2}-\frac{3p{q}_v^2}{4W^2}+\frac{q{q}_v{p}_v}{2W^2}+\frac{3q_u{q}_v}{2W}-\frac{q{q}_{uv}}{W}\right)\mathsf{\Phi }].\hfill \end{array}$$

(19)

Here again, we have

$$p_u:=\frac{\partial p}{\partial u},q_v:=\frac{\partial q}{\partial v},q_u:=\frac{\partial q}{\partial u}$$

and, as we mentioned before, the prime stands for the derivative with respect to u, that is

$${\mathsf{\Phi }}^{\prime }=\frac{d\mathsf{\Phi }}{du},{\mathsf{\Psi }}^{\prime }=\frac{d\mathsf{\Psi }}{du}.$$

Equation (19) can be expressed as follows:

$$\begin{array}{ccc}\hfill {\Delta }^{II}\mathit{N}& =& \frac{1}{q^2}[\frac{1}{W^2}{f}_1\left(v\right)\mathsf{\Psi }\left(\mathit{u}\right)+\frac{1}{W}{f}_2\left(v\right){\mathsf{\Psi }}^{\prime }\left(\mathit{u}\right)\hfill \\ & & +\frac{1}{W^2}{f}_3\left(v\right)\mathsf{\Phi }\left(\mathit{u}\right)+\frac{1}{W}{f}_4\left(v\right){\mathsf{\Phi }}^{\prime }\left(\mathit{u}\right)].\hfill \end{array}$$

(20)

We consider $f_j\left(v\right),j=1,2,3,4$ as polynomials in v with coefficients as functions in u, where we have deg$\left(f_j\right)\le 5$. We find:

$$\begin{array}{ccc}\hfill f_1\left(v\right)& =& 2m{t}^5+\left(4lm+n\right)v^4\hfill \\ & & +\left(2ln-3l^{2}m+3kl+2l^{\prime }W\right)v^3\hfill \\ & & +\left(2k^{\prime }W-4mlk-3n{l}^2+3nk-2l{l}^{\prime }W\right)v^2\hfill \\ & & +\left(3r(k-l^2)-2k(ln+km)+k^{\prime }lW-4k{l}^{\prime }W\right)v\hfill \\ & & -k(kn+k^{\prime }W),\hfill \end{array}$$

$$\begin{array}{c}\hfill f_2\left(v\right)=2l{v}^3+\left(2k+4n{l}^2\right)v^2+6klv+2k^2,\end{array}$$

$$\begin{array}{ccc}\hfill f_3\left(v\right)& =& \left(2ml-n\right)v^3+\left(m{l}^2-ln+4l^{\prime }-2r+3mk\right)v^2\hfill \\ & & +\left(2mlk-n{l}^2-4lr+2nk+3k^{\prime }A+2l{l}^{\prime }\right)v\hfill \\ & & +k(r+ln-2l^{\prime })+3l(k^{\prime }-lr),\hfill \end{array}$$

$$\begin{array}{c}\hfill f_4\left(v\right)=2v^3+6l{v}^2+\left(4l^2+2k\right)v+2kl.\end{array}$$

We denote by $(N_1,N_2,N_3),({\mathsf{\Phi }}_1,{\mathsf{\Phi }}_2,{\mathsf{\Phi }}_3)$ and $({\mathsf{\Psi }}_1,{\mathsf{\Psi }}_2,{\mathsf{\Psi }}_3)$ the coordinate functions of $\mathit{N},\mathsf{\Phi }$ and $\mathsf{\Psi }$, respectively. From (20), we obtain

$$\begin{array}{ccc}\hfill {\Delta }^{II}{N}_j& =& \frac{1}{q^2}[\frac{1}{W^2}{f}_1\left(v\right){\mathsf{\Psi }}_j\left(u\right)+\frac{1}{W}{f}_2\left(v\right){\mathsf{\Psi }}_j^{\prime }\left(u\right)\hfill \\ \hfill & & +\frac{1}{W^2}{f}_3\left(v\right){\mathsf{\Phi }}_j\left(u\right)+\frac{1}{W}{f}_4\left(v\right){\mathsf{\Phi }}_j^{\prime }\left(u\right)],\hfill \\ \hfill j& =& 1,2,3.\hfill \end{array}$$

(21)

Let $\Lambda =\left[{\lambda }_{ij}\right],i,j=1,2,3$. From (21) and (3), we find

$$\begin{array}{ccc}& & \frac{1}{W^2}{f}_1\left(v\right){\mathsf{\Psi }}_i\left(u\right)+\frac{1}{W}{f}_2\left(v\right){\mathsf{\Psi }}_i^{\prime }\left(u\right)+\frac{1}{W^2}{f}_3\left(v\right){\mathsf{\Phi }}_i\left(u\right)+\frac{1}{W}{f}_4\left(v\right){\mathsf{\Phi }}_i^{\prime }\left(u\right)\hfill \\ & =& {\lambda }_{i1}{q}^{\frac{3}{2}}\left({\mathsf{\Phi }}_i+v{\mathsf{\Psi }}_i\right)+{\lambda }_{i2}{q}^{\frac{3}{2}}\left({\mathsf{\Phi }}_i+v{\mathsf{\Psi }}_i\right)+{\lambda }_{i3}{q}^{\frac{3}{2}}\left({\mathsf{\Phi }}_i+v{\mathsf{\Psi }}_i\right),\hfill \\ \hfill i& =& 1,2,3.\hfill \end{array}$$

Consequently,

$$\begin{array}{ccc}& & -3m^2{\alpha }_i{v}^5+[\left(m^{\prime }W-m{W}^{\prime }-4mn-7l{m}^2\right){\alpha }_i+3mW{\alpha }_i^{\prime }]v^4\hfill \\ & & +[mW{\beta }_i^{\prime }-W^2{\alpha }_i^{″}+(2nW+7lmW){\alpha }_i^{\prime }+(n^{\prime }W-n{W}^{\prime }\hfill \\ & & +2l{m}^{\prime }W-2lm{W}^{\prime }-l^{\prime }mW-W^2-10lmn-2mr-n^2\hfill \\ & & -4k{m}^2){\alpha }_i]v^3+\left[\right(k{m}^{\prime }W-km{W}^{\prime }-\frac{1}{2}{k}^{\prime }mW+2l{n}^{\prime }W\hfill \\ & & -2ln{W}^{\prime }-l^{\prime }nA-r{A}^{\prime }+r^{\prime }W-3l{W}^2-3l{n}^2-6lmr\hfill \\ & & -6kmn){\alpha }_i+3lmW{\beta }_i^{\prime }-2l{W}^2{\alpha }_i^{″}-W^2{\beta }_i^{″}+(l^{\prime }W+5ln\hfill \\ & & +4km+r)W{\alpha }_i^{\prime }]v^2+[(\frac{1}{2}{k}^{\prime }W+3kn+3lr)W{\alpha }_i^{\prime }\hfill \\ & & +(k{n}^{\prime }W-kn{W}^{\prime }-\frac{1}{2}{k}^{\prime }nW+2l{r}^{\prime }W-2lr{W}^{\prime }-l^{\prime }rW\hfill \\ & & -k{A}^2-2l^2{A}^2-2k{n}^2+r^2-2lnr-4kmr){\alpha }_i\hfill \\ & & -2l{W}^2{\beta }_i^{″}-k{A}^2{\alpha }_i^{″}+\left(ln-r+2km+l^{\prime }W\right)W{\beta }_i^{\prime }]v\hfill \\ & & -W^4\left({\lambda }_{i1}{\alpha }_1+{\lambda }_{i2}{\alpha }_2+{\lambda }_{i3}{\alpha }_3\right)v+(k{r}^{\prime }W-kr{W}^{\prime }\hfill \\ & & -\frac{1}{2}{k}^{\prime }rA+l{r}^2-kl{W}^2-2knr){\alpha }_i-k{W}^2{\beta }_i^{″}\hfill \\ & & +2krW{\alpha }_i^{\prime }+(\frac{1}{2}{k}^{\prime }W-lr+kn)W{\beta }_i^{\prime }\hfill \\ & & -W^4\left({\lambda }_{i1}{\beta }_1+{\lambda }_{i2}{\beta }_2+{\lambda }_{i3}{\beta }_3\right)=0.\hfill \end{array}$$

(22)

It is easily verified that (22) are polynomials in v with functions in u as coefficients for $i=1,2,3$. This means that the coefficients of the powers of v in (22) must be zeros, and so we have the following equations:

$$3m^2{\alpha }_i=0,$$

(23)

$$\left(m^{\prime }W-m{W}^{\prime }-7\lambda m^2-4mn\right){\alpha }_i+3mW{\alpha }_i^{\prime }=0,$$

$$\begin{array}{ccc}& & mW{\beta }_i^{\prime }-W^2{\alpha }_i^{″}+(2nW+7lmW){\alpha }_i^{\prime }\hfill \\ & & +(n^{\prime }W-n{W}^{\prime }+2l{m}^{\prime }W-2lm{W}^{\prime }-l^{\prime }mW\hfill \\ & & -W^2-10lmn-2mr-n^2-4k{m}^2){\alpha }_i=0,\hfill \end{array}$$

(24)

$$\begin{array}{ccc}& & (k{m}^{\prime }W-km{W}^{\prime }-\frac{1}{2}{k}^{\prime }mW+2l{n}^{\prime }W-2ln{W}^{\prime }-l^{\prime }nW\hfill \\ & & -r{W}^{\prime }+r^{\prime }W-3l{W}^2-3l{n}^2-6lmr-6kmn){\alpha }_i+3lmW{\beta }_i^{\prime }\hfill \\ & & -2l{W}^2{\alpha }_i^{″}-W^2{\beta }_i^{″}+(l^{\prime }W+5ln+4km+r)W{\alpha }_i^{\prime }=0,\hfill \end{array}$$

(25)

$$\begin{array}{ccc}& & (k{n}^{\prime }W-kn{W}^{\prime }-\frac{1}{2}{k}^{\prime }nW+2l{r}^{\prime }W-2lr{W}^{\prime }-l^{\prime }rW-k{W}^2\hfill \\ & & -2l^2{W}^2-2k{n}^2+r^2-2lnr-4kmr){\alpha }_i-2l{W}^2{\beta }_i^{″}-k{W}^2{\alpha }_i^{″}\hfill \\ & & +(\frac{1}{2}{k}^{\prime }W+2kn+4lr)W{\alpha }_i^{\prime }+\left(ln-r+2km+l^{\prime }W\right)W{\beta }_i^{\prime }\hfill \\ & =& W^4({\lambda }_{i1}{\alpha }_1+{\lambda }_{i2}{\alpha }_2+{\lambda }_{i3}{\alpha }_3),\hfill \end{array}$$

(26)

$$\begin{array}{ccc}& & (k{r}^{\prime }W-kr{W}^{\prime }-\frac{1}{2}{k}^{\prime }rW+l{r}^2-kl{W}^2-2knr){\alpha }_i+2krW{\alpha }_i^{\prime }\hfill \\ & & +(\frac{1}{2}{k}^{\prime }W-lr+kn)W{\beta }_i^{\prime }-k{W}^2{\beta }_i^{″}\hfill \\ & =& W^4\left({\lambda }_{i1}{\beta }_1+{\lambda }_{i2}{\beta }_2+{\lambda }_{i3}{\beta }_3\right).\hfill \end{array}$$

(27)

From (23), one finds

$$m=\left({\mathit{\alpha }}^{\prime },\mathit{\alpha },{\mathit{\alpha }}^{″}\right)=0,$$

(28)

which means that ${\mathit{\alpha }}^{\prime },\mathit{\alpha },{\mathit{\alpha }}^{″}$ are linearly dependent vectors, so there exist two functions, ${\beta }_1={\beta }_1\left(u\right)$ and ${\beta }_2={\beta }_2\left(u\right)$, such that

$${\mathit{\alpha }}^{″}={\beta }_1\mathit{\alpha }+{\beta }_2{\mathit{\alpha }}^{\prime }.$$

(29)

On differentiating $⟨{\mathit{\alpha }}^{\prime },{\mathit{\alpha }}^{\prime }⟩=1,$ we obtain $⟨{\mathit{\alpha }}^{\prime },{\mathit{\alpha }}^{″}⟩=0.$ So, from (29), we have

$${\mathit{\alpha }}^{″}={\beta }_1\mathit{\alpha }.$$

(30)

Differentiating $⟨\mathit{\alpha },\mathit{\alpha }⟩=1$ twice, we obtain

$$⟨{\mathit{\alpha }}^{\prime },{\mathit{\alpha }}^{\prime }⟩+⟨\mathit{\alpha },{\mathit{\alpha }}^{″}⟩=0.$$

However, $⟨{\mathit{\alpha }}^{\prime },{\mathit{\alpha }}^{\prime }⟩=1,$ so by virtue of (30), we have ${\beta }_1\left(u\right)=-1.$ Hence, (30) becomes ${\mathit{\alpha }}^{″}=-\mathit{\alpha }$ or

$${\alpha }_j^{″}=-{\alpha }_j,j=1,2,3.$$

(31)

Using (28) and (31), Equation (24) reduces to

$$2nW{\alpha }_j^{\prime }+(n^{\prime }W-n{W}^{\prime }-n^2){\alpha }_j=0,j=1,2,3$$

or, in vector notation:

$$2nW{\mathit{\alpha }}^{\prime }+(n^{\prime }W-n{W}^{\prime }-n^2)\mathit{\alpha }=\mathbf{0}.$$

(32)

Differentiating $⟨\mathit{\alpha },\mathit{\alpha }⟩=1$, we obtain that the vectors $\mathit{\alpha },{\mathit{\alpha }}^{\prime }$ are linearly independent. Taking into account (32), we find that $nW=0.$$W\ne 0$, since, as mentioned before, the Gauss curvature vanishes; subsequently, we have $n=0$. Therefore, Equation (25) reduces to

$$-W^2{\mathit{\beta }}_j^{″}+(l^{\prime }W+r)W{\mathit{\alpha }}_j^{\prime }+(r^{\prime }W-r{W}^{\prime }-l{W}^2){\mathit{\alpha }}_j=0,j=1,2,3$$

or, in vector notation

$$-W^2{\mathit{\beta }}^{″}+(l^{\prime }W+r)W{\mathit{\alpha }}^{\prime }+(r^{\prime }W-r{W}^{\prime }-l{W}^2)\mathit{\alpha }=\mathbf{0}.$$

(33)

Applying the inner product of both sides of (33) with ${\mathit{\alpha }}^{\prime }$ and taking into account (17), we find that

$$-W^2⟨{\mathit{\beta }}^{″},{\mathit{\alpha }}^{\prime }⟩+rW+l^{\prime }{W}^2=0.$$

(34)

Taking the derivative of $l=⟨{\mathit{\beta }}^{\prime },{\mathit{\alpha }}^{\prime }⟩$, in view of (31) and (17), we find

$$l^{\prime }=⟨{\mathit{\beta }}^{″},{\mathit{\alpha }}^{\prime }⟩+⟨{\mathit{\beta }}^{\prime },{\mathit{\alpha }}^{″}⟩=⟨{\mathit{\alpha }}^{\prime },{\mathit{\beta }}^{″}⟩-⟨\mathit{\alpha },{\mathit{\beta }}^{\prime }⟩=⟨{\mathit{\alpha }}^{\prime },{\mathit{\beta }}^{″}⟩.$$

(35)

Hence, (34) becomes $rW=0,$ and so $r=0.$ Therefore, the vectors ${\mathit{\beta }}^{\prime },\mathit{\alpha },{\mathit{\beta }}^{″}$ are linearly dependent, that is, there exist two functions ${\beta }_3={\beta }_3\left(u\right)$ and ${\beta }_4={\beta }_4\left(u\right)$, such that

$${\mathit{\beta }}^{″}={\beta }_3\mathit{\alpha }+{\beta }_4{\mathit{\beta }}^{\prime }.$$

(36)

Applying the inner product of both sides of (36) with $\mathit{\alpha }$ and taking into account (17), we find that ${\beta }_3=⟨{\mathit{\beta }}^{″},\mathit{\alpha }⟩.$

Taking the derivative of $⟨{\mathit{\beta }}^{\prime },\mathit{\alpha }⟩=0,$ we find that $⟨{\mathit{\beta }}^{″},\mathit{\alpha }⟩+⟨{\mathit{\beta }}^{\prime },{\mathit{\alpha }}^{\prime }⟩=0,$ that is

$$⟨\mathit{\alpha },{\mathit{\beta }}^{″}⟩+l=0,$$

and so ${\beta }_3=-l$.

Applying the inner product of both sides of Equation (36) again with ${\mathit{\alpha }}^{\prime }$, and taking into account (17), we find that

$$⟨{\mathit{\beta }}^{″},{\mathit{\alpha }}^{\prime }⟩={\beta }_{4}l.$$

Using (35), we find $l^{\prime }={\beta }_{4}l.$ Thus, ${\beta }_4=\frac{l^{\prime }}{l}.$ Therefore,

$${\mathit{\beta }}^{″}=-l\mathit{\alpha }+\frac{l^{\prime }}{l}{\mathit{\beta }}^{\prime }.$$

(37)

We note that $l\ne 0$, because otherwise, if $l=0$ and taking into consideration $r=0$, Equation (33) would yield $W=0$; this has been excluded.

Now, because of $l\ne 0$ and $r=0$, from (33), (37) we obtain that

$$-\frac{l^{\prime }}{l}{W}^2{\mathit{\beta }}^{\prime }+l^{\prime }{W}^2{\mathit{\alpha }}^{\prime }=\mathbf{0}$$

which yields that $l^{\prime }({\mathit{\beta }}^{\prime }-l{\mathit{\alpha }}^{\prime })=\mathbf{0}.$

It is easily verified that $l^{\prime }=0$, because otherwise, if $l^{\prime }\ne 0,$ then ${\mathit{\beta }}^{\prime }=l{\mathit{\alpha }}^{\prime }.$ Therefore ${\mathit{\beta }}^{\prime },{\mathit{\alpha }}^{\prime }$ are linearly dependent, and so $W=0$, which is a contradiction. Thus, from (37), we have

$${\mathit{\beta }}^{″}=-l\mathit{\alpha }.$$

(38)

Additionally, differentiating k and using (38), we obtain that k is constant. Thus, Equations (26) and (27) become

$$\begin{array}{ccc}\hfill {\lambda }_{i1}{\alpha }_1+{\lambda }_{i2}{\alpha }_2+{\lambda }_{i3}{\alpha }_3& =& 0,\hfill \\ \hfill {\lambda }_{i1}{\beta }_1+{\lambda }_{i2}{\beta }_2+{\lambda }_{i3}{\beta }_3& =& 0,i=1,2,3,\hfill \end{array}$$

which means that $\left[{\lambda }_{ij}\right]$ is the zero matrix.

Since we chose the parameter u to be the arc length of the spherical curve $\mathit{\alpha }\left(u\right)$, and taking into consideration ${\mathit{\alpha }}^{″}=-\mathit{\alpha }$, then, without loss of generality we may assume that the parametrization of $\mathit{\alpha }\left(u\right)$ is

$$\mathit{\alpha }\left(u\right)=(cosu,sinu,0)$$

Integrating (38) twice yields

$$\mathit{\beta }\left(u\right)=(c_{1}u+c_2+lcosu,c_{3}u+c_4+lsinu,c_{5}u+c_6),$$

for some integrating constants $c_j,j=1,2,\dots ,6$.

Since $k=⟨{\mathit{\beta }}^{\prime },{\mathit{\beta }}^{\prime }⟩$ = const., one can easily conclude that $c_1=c_3=0.$ Hence $\mathit{\beta }\left(u\right)$ can be written

$$\mathit{\beta }\left(u\right)=(c_2+lcosu,c_4+lsinu,c_{5}u+c_6).$$

Hence, we find that

$$S:\mathit{x}(u,v)=(c_2+(l+v)cosu,c_4+(l+v)sinu,c_{5}u+c_6)$$

which represents a helicoid.

3.2. Quadrics of the First Kind

A parametric representation of this kind of quadric surface is

$$\mathit{r}(s,t)=\left(s,t,\sqrt{c+a{s}^2+b{t}^2}\right).$$

For simplecity, we put $\sigma :=c+a{s}^2+b{t}^2$. We have

$${\mathit{r}}_{\mathit{s}}=\left(1,0,\frac{as}{\sqrt{\sigma }}\right),$$

and

$${\mathit{r}}_{\mathit{t}}=\left(0,1,\frac{bt}{\sqrt{\sigma }}\right),$$

The components of the first fundamental form $I=Ed{s}^2+2Fdsdt+Gd{t}^2$ are the following:

$$E=<{\mathit{r}}_{\mathit{s}},{\mathit{r}}_{\mathit{s}}>=1+\frac{{\left(as\right)}^2}{\sigma },$$

$$F=<{\mathit{r}}_{\mathit{s}},{\mathit{r}}_{\mathit{t}}>=\frac{abst}{\sigma },$$

$$G=<{\mathit{r}}_{\mathit{t}},{\mathit{r}}_{\mathit{t}}>=1+\frac{{\left(bt\right)}^2}{\sigma },$$

and the Gauss map of $\mathit{r}$ is

$$\mathit{N}=\left(-\frac{as}{\sqrt{\Omega }},-\frac{bt}{\sqrt{\Omega }},\frac{\sqrt{\sigma }}{\sqrt{\Omega }}\right),$$

where $\Omega =a(a+1)s^2+b(b+1)t^2+c$. The components of the second fundamental form $II=Ld{s}^2+2Mdsdt+Nd{t}^2$ are the following

$$L=<{\mathit{r}}_{\mathit{ss}},\mathit{N}>=\frac{a\left(b{t}^2+c\right)}{\sigma \sqrt{\Omega }},$$

$$M=<{\mathit{r}}_{\mathit{st}},\mathit{N}>=-\frac{abst}{\sigma \sqrt{\Omega }},$$

$$N=<{\mathit{x}}_{\mathit{tt}},\mathit{N}>=\frac{b\left(a{s}^2+c\right)}{\sigma \sqrt{\Omega }}.$$

The natural frame $\left\{{\mathit{N}}_s,{\mathit{N}}_t\right\}$ of $\mathit{r}$ is defined by

$${\mathit{N}}_{\mathit{s}}=\left(-\frac{a[b(b+1)t^2+c]}{{\Omega }^{\frac{3}{2}}},\frac{ab(a+1)st}{{\Omega }^{\frac{3}{2}}},\frac{as[b(b+1)t^2-ac]}{{\Omega }^{\frac{3}{2}}\sqrt{\sigma }}\right),$$

and

$${\mathit{N}}_{\mathit{t}}=\left(\frac{ab(b+1)st}{{\Omega }^{\frac{3}{2}}},-\frac{b[a(a+1)s^2+c]}{{\Omega }^{\frac{3}{2}}},\frac{bt[a(a+1)s^2-bc]}{{\Omega }^{\frac{3}{2}}\sqrt{\sigma }}\right).$$

The Laplacian ${\Delta }^{II}$ of S can be found as follows:

$$\begin{array}{ccc}\hfill {▵}^{II}& =& -\frac{\sqrt{\Omega }}{c}[\frac{a{s}^2+c}{a}\frac{{\partial }^2}{\partial s^2}+2st\frac{{\partial }^2}{\partial s\partial t}+\frac{b{t}^2+c}{b}\frac{{\partial }^2}{\partial t^2}\hfill \\ & & +2s\frac{\partial }{\partial s}+2t\frac{\partial }{\partial t}].\hfill \end{array}$$

(39)

Applying (39) for the component functions $N_1=-\frac{as}{\sqrt{\Omega }}$ and $N_2=-\frac{bt}{\sqrt{\Omega }}$ of $\mathit{N}$, one finds

$$\begin{array}{ccc}\hfill {\Delta }^{II}{N}_1& =& \frac{3as}{c{\Omega }^2}\left[a(a+1)s^2+bc\right]\left[b(b+1)t^2\right]\hfill \\ & & +\frac{as}{{\Omega }^2}\left[-3(a+1)(s^2+c)-(b-1)\Omega \right],\hfill \end{array}$$

$$\begin{array}{ccc}\hfill {\Delta }^{II}{N}_2& =& \frac{3bt}{c{\Omega }^2}\left[b(b+1)t^2+ac\right]\left[a(a+1)s^2\right]\hfill \\ & & +\frac{bt}{{\Omega }^2}\left[-3(b+1)(t^2+c)-(a-1)\Omega \right].\hfill \end{array}$$

Let $\Lambda =\left[{\lambda }_{ij}\right],i,j=1,2,3$. On account of (3) and (39), we find

$$\begin{array}{ccc}\hfill & & \frac{3as}{c{\Omega }^2}\left[a(a+1)s^2+bc\right]\left[b(b+1)t^2\right]+\frac{as}{{\sigma }^2}\left[-3(a+1)(s^2+c)-(b-1)\Omega \right]\hfill \\ & =& {\lambda }_{11}\left(-\frac{as}{\sqrt{\Omega }}\right)+{\lambda }_{12}\left(-\frac{bt}{\sqrt{\Omega }}\right)+{\lambda }_{13}\left(\frac{\sqrt{\sigma }}{\sqrt{\Omega }}\right),\hfill \end{array}$$

(40)

$$\begin{array}{ccc}\hfill & & \frac{3bt}{{\Omega }^2}\left[b(b+1)t^2+ac\right]\left[a(a+1)s^2\right]+\frac{bt}{{\Omega }^2}\left[-3(b+1)(t^2+c)-(a-1)\Omega \right]\hfill \\ & =& {\lambda }_{21}\left(-\frac{as}{\sqrt{\Omega }}\right)+{\lambda }_{22}\left(-\frac{bt}{\sqrt{\Omega }}\right)+{\lambda }_{23}\left(\frac{\sqrt{\sigma }}{\sqrt{\Omega }}\right).\hfill \end{array}$$

(41)

$$\begin{array}{c}\hfill {\Delta }^{II}{N}_3={\Delta }^{II}\frac{\sqrt{\sigma }}{\sqrt{\Omega }}={\lambda }_{31}\left(-\frac{as}{\sqrt{\Omega }}\right)+{\lambda }_{32}\left(-\frac{bv}{\sqrt{\Omega }}\right)+{\lambda }_{33}\left(\frac{\sqrt{\sigma }}{\sqrt{\Omega }}\right)\end{array}$$

(42)

Putting $s=0$ in (40), it follows that

$$-{\lambda }_{12}bt+{\lambda }_{13}\sqrt{1+b{t}^2}=0.$$

(43)

On differentiation (43), with respect to t, we observe

$$-{\lambda }_{12}\sqrt{1+b{t}^2}+{\lambda }_{13}t=0.$$

(44)

Considering (43) and (44) as a system in ${\lambda }_{12}$ and ${\lambda }_{13}$, and since the determinant

$$\left|\begin{array}{cc}-bt& \sqrt{1+b{t}^2}\\ -\sqrt{1+b{t}^2}& t\end{array}\right|\ne 0.$$

We must have ${\lambda }_{12}={\lambda }_{13}=0$. Hence, (40) reduces to

$$\begin{array}{ccc}\hfill & & \frac{3}{c{\Omega }^2}\left[a(a+1)s^2+bc\right]\left[b(b+1)t^2\right]\hfill \\ & & +\frac{1}{{\Omega }^2}\left[-3(a+1)(s^2+c)-(b-1)\Omega \right]=-\frac{1}{\sqrt{\Omega }}{\lambda }_{11}.\hfill \end{array}$$

(45)

Putting $t=0$ in (45), we obtain that

$$\begin{array}{c}\hfill -{\lambda }_{11}{[c+a(a+1)s^2]}^{\frac{3}{2}}=-3(a+1)(s^2+c)-(b-1)[a(a+1)s^2+c].\end{array}$$

(46)

In the same way, one can see that ${\lambda }_{21}={\lambda }_{23}=0$. Then, (41) turns into

$$\begin{array}{ccc}\hfill & & \frac{3}{c{\Omega }^2}\left[b(b+1)t^2+ac\right]\left[a(a+1)s^2\right]\hfill \\ & & +\frac{1}{{\Omega }^2}\left[-3(a+1)(s^2+c)-(b-1)\Omega \right]=-\frac{1}{\sqrt{\Omega }}{\lambda }_{22}.\hfill \end{array}$$

(47)

Putting $s=0$ in (47), we obtain that

$$\begin{array}{c}\hfill -{\lambda }_{22}{[c+b(b+1)t^2]}^{\frac{3}{2}}=-3(b+1)(t^2+c)-(a-1)[b(b+1)t^2+c].\end{array}$$

(48)

From (46) and (48), we conclude that relations (46) and (48) are polynomials in s and t, respectively. Since $a\ne 0,b\ne 0$ and $c\ne 0$, it can be easily verified that a must equal $-1$ and b must equal $-1$. Hence, S is a sphere. We put $a=-1$ and $b=-1$, in (46) and (48), respectively, so one can find that ${\lambda }_{11}={\lambda }_{22}=-\frac{2}{\sqrt{c}}$.

On the other hand, relation (39) reduces to

$$\begin{array}{c}\hfill {\Delta }^{II}=-\frac{1}{\sqrt{c}}\left[(s^2-c)\frac{{\partial }^2}{\partial s^2}+2st\frac{{\partial }^2}{\partial s\partial t}+(t^2-c)\frac{{\partial }^2}{\partial t^2}+2s\frac{\partial }{\partial s}+2t\frac{\partial }{\partial t}\right]\end{array}$$

from which we get

$${\Delta }^{II}{N}_3=-\frac{2\sqrt{c-s^2-t^2}}{c}$$

Therefore, relation (42) becomes

$$-\frac{2\sqrt{c-s^2-t^2}}{c}=-{\lambda }_{31}\frac{s}{\sqrt{c}}-{\lambda }_{32}\frac{t}{\sqrt{c}}+{\lambda }_{33}\frac{\sqrt{c-s^2-t^2}}{\sqrt{c}}.$$

It can be easily seen that ${\lambda }_{31}={\lambda }_{32}=0$ and ${\lambda }_{33}=-\frac{2}{\sqrt{c}}$. Hence, we conclude that spheres are the only quadric surfaces of that kind (15) whose Gauss map satisfies (3). The corresponding matrix is

$$\Lambda =\left[\begin{array}{ccc}-\frac{2}{\sqrt{c}}& 0& 0\\ 0& -\frac{2}{\sqrt{c}}& 0\\ 0& 0& -\frac{2}{\sqrt{c}}\end{array}\right].$$

3.3. Quadrics of the Second Kind

A parametric representation of this kind of quadric surfaces is

$$\mathit{r}(s,t)=\left(s,t,\frac{a}{2}{s}^2+\frac{b}{2}{t}^2\right).$$

Then the components of the first and second fundamental forms are the following

$$E=1+{\left(as\right)}^2,F=abst,G=1+{\left(bt\right)}^2,$$

$$L=\frac{a}{\sqrt{g}},M=0,N=\frac{b}{\sqrt{g}},$$

where $g:=EG-F^2=1+{\left(as\right)}^2+{\left(bt\right)}^2.$

The Laplacian ${\Delta }^{II}$ of S is expressed as follows

$$\begin{array}{c}\hfill {\Delta }^{II}=-\sqrt{g}\left(\frac{1}{a}\frac{{\partial }^2}{\partial s^2}+\frac{1}{b}\frac{{\partial }^2}{\partial t^2}\right).\end{array}$$

(49)

The Gauss map of S is

$$\mathit{N}=\left(-\frac{as}{\sqrt{g}},-\frac{bt}{\sqrt{g}},\frac{1}{\sqrt{g}}\right).$$

We denote by $(N_1,N_2,N_3)$ the components of $\mathit{N}$, and by $\Lambda =\left[{\lambda }_{ij}\right],i,j=1,2,3$. From (3), we find that

$${\Delta }^{II}{N}_1={\Delta }^{II}\left(-\frac{as}{\sqrt{g}}\right)={\lambda }_{11}\left(-\frac{as}{\sqrt{g}}\right)+{\lambda }_{12}\left(-\frac{bt}{\sqrt{g}}\right)+{\lambda }_{13}\left(\frac{1}{\sqrt{g}}\right),$$

$${\Delta }^{II}{N}_2={\Delta }^I\left(-\frac{bt}{\sqrt{g}}\right)={\lambda }_{21}\left(-\frac{as}{\sqrt{g}}\right)+{\lambda }_{22}\left(-\frac{bt}{\sqrt{g}}\right)+{\lambda }_{23}\left(\frac{1}{\sqrt{g}}\right),$$

$${\Delta }^{II}{N}_3={\Delta }^I\left(-\frac{1}{\sqrt{g}}\right)={\lambda }_{31}\left(-\frac{as}{\sqrt{g}}\right)+{\lambda }_{32}\left(-\frac{bt}{\sqrt{g}}\right)+{\lambda }_{33}\left(\frac{1}{\sqrt{g}}\right).$$

By applying the operator ${\Delta }^{II}$ to the component functions $N_1$ and $N_2$ of $\mathit{N}$, we find by means of (49)

$$\begin{array}{ccc}\hfill {\Delta }^{II}\left(-\frac{as}{\sqrt{g}}\right)& =& -\frac{as}{g^2}\left[3a(1+b^2{t}^2)+b(g-3b^2{t}^2)\right]\hfill \\ & =& {\lambda }_{11}\left(-\frac{as}{\sqrt{g}}\right)+{\lambda }_{12}\left(-\frac{bt}{\sqrt{g}}\right)+{\lambda }_{13}\left(\frac{1}{\sqrt{g}}\right),\hfill \end{array}$$

(50)

and

$$\begin{array}{ccc}\hfill {\Delta }^{II}\left(-\frac{bt}{\sqrt{g}}\right)& =& -\frac{bt}{g^2}\left[3b(1+a^2{s}^2)+a(g-3s^2{u}^2)\right]\hfill \\ & =& {\lambda }_{21}\left(-\frac{as}{\sqrt{g}}\right)+{\lambda }_{22}\left(-\frac{bt}{\sqrt{g}}\right)+{\lambda }_{23}\left(\frac{1}{\sqrt{g}}\right).\hfill \end{array}$$

(51)

If we apply $s=0$ to (50), then the left side of Equation (50) vanishes. Therefore, we are left to

$${\lambda }_{12}bt-{\lambda }_{13}=0,$$

which immplies that ${\lambda }_{12}={\lambda }_{13}=0$. So, Equation (50) becomes

$$\begin{array}{c}\hfill {\left[3a(1+b^2{t}^2)+b(g-3b^2{t}^2)\right]}^2={\lambda }_{11}{g}^3.\end{array}$$

(52)

Similarly, if we put $t=0$ in (51), then the left side of (51) vanishes. In the same way, Equation (51) becomes

$$\begin{array}{c}\hfill {\left[3b(1+a^2{s}^2)+a(g-3a^2{s}^2)\right]}^2={\lambda }_{22}{g}^3.\end{array}$$

(53)

Equations (52) and (53) are nontrivial polynomials in t and s, respectively, with constant coefficients. These two polynomials can never be zero, unless $a=b=0$. This is clearly impossible, since $a,b>0$.

4. Conclusions

Firstly, we defined formulae for the first and second Laplace operators regarding the first, second, and third fundamental forms of a surface. As a result, many relations regarding the first and second Laplace operators were investigated and proved. Then, we defined an important class of surfaces, namely quadrics, in the Euclidean 3-space. Finally, we classified this class of surfaces using its Gauss map N, which satisfies the relation ${\Delta }^{II}\mathit{N}=\Lambda \mathit{N}$ for a real square symmetric matrix $\Lambda$ of order 3. We distinguished three types according to whether these surfaces are determined, with each type investigated in a subsection of Section 3. An interesting study can be drawn if this type of study can be applied to other classes of surfaces, such as spiral surfaces, the compact and non-compact cyclides of Dupin, or tubular surfaces, which have not been investigated yet.