Tubes of finite Chen-type

TL;DR

This paper investigates surfaces in three-dimensional space that have finite type with respect to the third fundamental form, specifically analyzing tubes and demonstrating they are of infinite III-type.

Contribution

The paper introduces the family of tubes in E3 and proves they are of infinite III-type, expanding understanding of surface classifications.

Findings

Tubes in E3 are of infinite III-type.

Surfaces of finite III-type are characterized in the study.

The paper provides new insights into the classification of surfaces based on fundamental forms.

Abstract

In this paper, we consider surfaces in the 3-dimensional Euclidean space E3 which are of finite III-type, that is, they are of finite type, in the sense of B.-Y. Chen, corresponding to the third fundamental form. We present an important family of surfaces, namely, tubes in E3 .We show that tubes are of infinite III-type.

Equations

$${\Delta }^I\mathbf{x}=-n\mathbf{H}.$$

$$\mathbf{x}={\mathbf{x}}_0+\sum _{i=1}^k{\mathbf{x}}_i,{\Delta }^J{\mathbf{x}}_i={\lambda }_i{\mathbf{x}}_i,i=1,...,k,$$

$${\sigma }_1$$

$${\sigma }_2$$

$${\sigma }_3$$

$${\sigma }_k$$

$$({\Delta }^J{)}^k\mathbf{x}+{\sigma }_1({\Delta }^J{)}^{k-1}\mathbf{x}+...+{\sigma }_k(\mathbf{x}-\mathbf{c})=0.$$

$$d\mathbf{x}={\omega }_1{\mathbf{\epsilon }}_1+{\omega }_2{\mathbf{\epsilon }}_2,d\mathbf{n}={\omega }_{31}{\mathbf{\epsilon }}_1+{\omega }_{32}{\mathbf{\epsilon }}_2,$$

$$d{\mathbf{\epsilon }}_1={\omega }_{12}{\mathbf{\epsilon }}_2-{\omega }_{31}{\mathbf{\epsilon }}_3,d{\mathbf{\epsilon }}_2=-{\omega }_{12}{\mathbf{\epsilon }}_1-{\omega }_{32}{\mathbf{\epsilon }}_3,$$

$${\omega }_{31}=-a{\omega }_1-b{\omega }_2,{\omega }_{32}=-b{\omega }_1-c{\omega }_2,{\omega }_{12}=q_1{\omega }_1+q_2{\omega }_2.$$

$${\omega }_1=-\frac{1}{a}{\omega }_{31},{\omega }_2=-\frac{1}{c}{\omega }_{32}.$$

$$K=ac,H=\frac{a+c}{2}.$$

$${\nabla }_1\mathbf{x}$$

$${\nabla }_1{\mathbf{\epsilon }}_1$$

$${\nabla }_1{\mathbf{\epsilon }}_2$$

$${\nabla }_1\mathbf{n}$$

$${\tilde{\nabla }}_{1}f=-\frac{1}{a}{\nabla }_{1}f,{\tilde{\nabla }}_{2}f=-\frac{1}{c}{\nabla }_{2}f.$$

$${\tilde{\nabla }}_1\mathbf{x}=-\frac{1}{a}{\mathbf{\epsilon }}_1,{\tilde{\nabla }}_2\mathbf{x}=-\frac{1}{c}{\mathbf{\epsilon }}_2,$$

$${\tilde{\nabla }}_1{\mathbf{\epsilon }}_1=p_1{\mathbf{\epsilon }}_2-\mathbf{n},{\tilde{\nabla }}_2{\mathbf{\epsilon }}_1=p_2{\mathbf{\epsilon }}_2,$$

$${\tilde{\nabla }}_1{\mathbf{\epsilon }}_2=-p_1{\mathbf{\epsilon }}_1,{\tilde{\nabla }}_2{\mathbf{\epsilon }}_2=-p_2{\mathbf{\epsilon }}_1-\mathbf{n},$$

$${\tilde{\nabla }}_1\mathbf{n}={\mathbf{\epsilon }}_1,{\tilde{\nabla }}_2\mathbf{n}={\mathbf{\epsilon }}_2,$$

$$\widetilde{\nabla}_{1}\frac{1}{c}=p_{2}\Big{(}\frac{1}{a}-\frac{1}{c}\Big{)},\ \ \ \widetilde{\nabla}_{2}\frac{1}{a}=p_{1}\Big{(}\frac{1}{a}-\frac{1}{c}\Big{)}.$$

$${\Delta }^{III}f=-{\tilde{\nabla }}_1{\tilde{\nabla }}_{1}f-{\tilde{\nabla }}_2{\tilde{\nabla }}_{2}f-p_2{\tilde{\nabla }}_{1}f+p_1{\tilde{\nabla }}_{2}f.$$

$${\Delta }^{III}\mathbf{x}=-{\tilde{\nabla }}_1{\tilde{\nabla }}_1\mathbf{x}-{\tilde{\nabla }}_2{\tilde{\nabla }}_2\mathbf{x}-p_2{\tilde{\nabla }}_1\mathbf{x}+p_1{\tilde{\nabla }}_2\mathbf{x}.$$

$$\Delta^{III}\boldsymbol{x}=\widetilde{\nabla}_{1}\Big{(}\frac{1}{a}\boldsymbol{\varepsilon_{1}}\Big{)}+\widetilde{\nabla}_{2}\Big{(}\frac{1}{c}\boldsymbol{\varepsilon_{2}}\Big{)}+\frac{1}{a}p_{2}\boldsymbol{\varepsilon_{1}}-\frac{1}{c}p_{1}\boldsymbol{\varepsilon_{2}}.$$

$$\Delta^{III}\boldsymbol{x}=\Big{(}\widetilde{\nabla}_{1}\frac{1}{a}\Big{)}\boldsymbol{\varepsilon_{1}}+\Big{(}\frac{1}{a}-\frac{1}{c}\Big{)}p_{2}\boldsymbol{\varepsilon_{1}}+\Big{(}\widetilde{\nabla}_{2}\frac{1}{c}\Big{)}\boldsymbol{\varepsilon_{2}}+\Big{(}\frac{1}{a}-\frac{1}{c}\Big{)}p_{1}\boldsymbol{\varepsilon_{2}}-\Big{(}\frac{1}{a}+\frac{1}{c}\Big{)}\boldsymbol{n}.$$

$$\Delta^{III}\boldsymbol{x}=\Big{(}\widetilde{\nabla}_{1}\big{(}\frac{1}{a}+\frac{1}{c}\big{)}\Big{)}\boldsymbol{\varepsilon_{1}}+\Big{(}\widetilde{\nabla}_{2}\big{(}\frac{1}{a}+\frac{1}{c}\big{)}\Big{)}\boldsymbol{\varepsilon_{2}}-\big{(}\frac{1}{a}+\frac{1}{c}\big{)}\boldsymbol{n}$$

$$\Delta^{III}\boldsymbol{x}=grad^{III}\Big{(}\frac{2H}{K}\Big{)}-\Big{(}\frac{2H}{K}\Big{)}\boldsymbol{n}.$$

$${\Delta }^{III}\mathbf{x}=A\mathbf{x},$$

$$T_r(\mathbf{\alpha })=\{ex{p}_{\mathbf{\alpha }(t)}\mathbf{u}\mid \mathbf{u}\in N_{\mathbf{\alpha }},\parallel \mathbf{u}\parallel =r,t\in (a,b)\}.$$

$$ϝ:\mathbf{x}(t,\phi )=\mathbf{\alpha }+r\cos \phi \mathbf{h}+r\sin \phi \mathbf{b}.$$

$$I$$

$$II$$

$$K=-\frac{\kappa \cos \phi }{r\delta }.$$

Full text

1. Introduction

Let $M^{n}$ be a (connected) submanifold in the m-dimensional Euclidean space $E^{m}$. Let $\boldsymbol{x},\boldsymbol{H}$ be the position vector field and the mean curvature field of $M^{n}$ respectively. Denote by $\Delta^{I}$ the second Beltrami-Laplace operator corresponding to the first fundamental form $I$ of $M^{n}$. Then, it is well known that [3]

[TABLE]

From this formula one can see that $M^{n}$ is a minimal submanifold if and only if all coordinate functions, restricted to $M^{n}$, are eigenfunctions of $\Delta^{I}$ with eigenvalue $\lambda=0$. Moreover in [12] T. Takahashi showed that the submanifold $M^{n}$ for which $\Delta^{I}\boldsymbol{x}=\lambda\boldsymbol{x}$, i.e., for which all coordinate functions are eigenfunctions of $\Delta^{I}$ with the same eigenvalue $\lambda\in\mathbb{Re}$, are precisely either the minimal submanifold with eigenvalue $\lambda=0$ or the minimal submanifold of hyperspheres $S^{m-1}$ with eigenvalue $\lambda>0$. Although the class of finite type submanifolds in an arbitrary dimensional Euclidean spaces is very large, very little is known about surfaces of finite type in the Euclidean 3-space $E^{3}$. Actually, so far, the only known surfaces of finite type corresponding to the first fundamental form in the Euclidean 3-space are the minimal surfaces, the circular cylinders and the spheres. So in [4] B.-Y. Chen mentions the following problem

Problem 1**.**

Determine all surfaces of finite Chen $I$-type in $E^{3}$.

In order to provide an answer to the above problem, important families of surfaces were studied by different authors by proving that finite type ruled surfaces, finite type quadrics, finite type tubes, finite type cyclides of Dupin and finite type spiral surfaces are surfaces of the only known examples in $E^{3}$. However, for another classical families of surfaces, such as surfaces of revolution, translation surfaces as well as helicoidal surfaces, the classification of its finite type surfaces is not known yet. For a more details, the reader can refer to [5].

Later in [8] O. Garay generalized T. Takahashi’s condition studied surfaces in $E^{3}$ for which all coordinate functions $\left(x_{1},x_{2},x_{3}\right)$ of $\boldsymbol{x}$ satisfy $\Delta^{I}\boldsymbol{x}_{i}=\lambda_{i}x_{i},i=1,2,3$, not necessarily with the same eigenvalue. Another generalization was studied in [6] for which surfaces in $E^{3}$ satisfy the condition $\Delta^{I}\boldsymbol{x}=A\boldsymbol{x}+B$ $({\ddagger})$ where $A\in\mathbb{Re}^{3\times 3};B\in\mathbb{Re}^{3\times 1}$. It was shown that a surface $S$ in $E^{3}$ satisfies $({\ddagger})$ if and only if it is an open part of a minimal surface, a sphere, or a circular cylinder. Surfaces satisfying $({\ddagger})$ are said to be of coordinate finite type.

In the thematic circle of the surfaces of finite type in the Euclidean space $E^{3}$, S. Stamatakis and H. Al-Zoubi in [10] restored attention to this theme by introducing the notion of surfaces of finite type corresponding to the second or the third fundamental forms of $S$ in the following way:

A surface $S$ is said to be of finite type corresponding to the fundamental form $J$, or briefly of finite $J$-type, $J=II,III$, if the position vector $\boldsymbol{x}$ of $S$ can be written as a finite sum of nonconstant eigenvectors of the operator $\Delta^{J}$, that is if

[TABLE]

where $\boldsymbol{x}_{0}$ is a fixed vector and $\boldsymbol{x}_{1},...,\boldsymbol{x}_{k}$ are nonconstant maps such that $\Delta^{J}\boldsymbol{\ x}_{i}=\lambda_{i}\boldsymbol{x}_{i},i=1,...,k$. If, in particular, all eigenvalues $\lambda_{1},\lambda_{2},...,\lambda_{k}$ are mutually distinct, then $S$ is said to be of $J$-type $k$, otherwise $S$ is said to be of infinite type. When $\lambda_{i}=0$ for some i = 1,…, k, then $S$ is said to be of null $J$-type $k$.

In general when $S$ is of finite type $k$, it follows from (1.1) that there exist a monic polynomial, say $R(x)\neq 0,$ such that $R(\Delta^{J})(\boldsymbol{x}-\boldsymbol{c})=\mathbf{0}.$ Suppose that $R(x)=x^{k}+\sigma_{1}x^{k-1}+...+\sigma_{k-1}x+\sigma_{k},$ then coefficients $\sigma_{i}$ are given by

[TABLE]

Therefore the position vector $\boldsymbol{x}$ satisfies the following equation, (see [3])

[TABLE]

In this paper we will pay attention to surfaces of finite $III$-type. Firstly, we will establish a formula for $\Delta^{III}\boldsymbol{x}$ by using Cartan’s method of the moving frame. Further, we continue our study by proving finite type surfaces for an important class of surfaces, namely, tubes in $E^{3}$.

2. Preliminaries

Let $S$ be a (connected) surface in the Euclidean 3-space $E^{3}$, whose Gaussian curvature $K$ never vanishes. Let $\wp=\{\boldsymbol{\varepsilon_{1}}(u,v),\boldsymbol{\varepsilon_{2}}(u,v),\boldsymbol{\varepsilon_{3}}(u,v)\}$ is a moving frame of $S$, $\boldsymbol{\varepsilon_{3}}=\boldsymbol{n}$ is the Gauss map of $S$ and $\det(\boldsymbol{\varepsilon_{1}},\boldsymbol{\varepsilon_{2}},\boldsymbol{\varepsilon_{3}})=1$. Then it is well known that there are linear differential forms $\omega_{1},\omega_{2},\omega_{31},\omega_{32}$ and $\omega_{12}$, such that [7]

[TABLE]

[TABLE]

and functions $a,b,c,q_{1},q_{2}$ of $u,v$ such that

[TABLE]

We can choose the moving frame of $S$, such that the vectors $\boldsymbol{\varepsilon_{1}},\boldsymbol{\varepsilon_{2}}$ are the principle directions of $S$. Then $a$, $c$ are the principle curvatures of $S$ and $b=0$, so the differential forms $\omega_{1}$ and $\omega_{2}$ become

[TABLE]

The Gauss and mean curvature are respectively

[TABLE]

Let $\nabla_{1}f,\nabla_{2}f$ be the derivatives of Pfaff of a function $f(u,v)\in C^{1}$ along the curves $\omega_{2}=0,\omega_{1}=0$ respectively. Then we have the following well known relations [2]

[TABLE]

We denote by $\widetilde{\nabla}_{1}f$ and $\widetilde{\nabla}_{2}f$ the derivatives of Pfaff of $f$ along the curves $\omega_{32}=0,\omega_{31}=0$ respectively. One finds

[TABLE]

It follows that

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where $p_{1}=-\frac{1}{a}q_{1},p_{2}=-\frac{1}{c}q_{2}$ are the geodesic curvatures of the spherical curves $\omega_{32}=0$ and $\omega_{31}=0$ respectively. The Mainardi-Codazzi equations have the following forms

[TABLE]

Let $f$ be a sufficient differentiable function on $S$. Then the second differential parameter of Beltrami corresponding to the fundamental form $III$ of $S$ is defined by

[TABLE]

Applying (2.5) to the position vector $\boldsymbol{x}$, gives

[TABLE]

From (2.1) we obtain

[TABLE]

Using (2.2) and (2.3), equation (2.6) becomes

[TABLE]

Taking into account the Mainardi-Codazzi equations (2.4), so equation (2.7) reduces to

[TABLE]

or equivalently, (see [10])

[TABLE]

Remark 1**.**

S. Stamatakis and H. Al-Zoubi proved in [10] relation (2.8) by using tensors calculus.

From (2.8) the following results were proved in [10].

Theorem 1**.**

A surface $S$ in $E^{3}$ is of 0-type 1 corresponding to the third fundamental form if and only if $S$ is minimal.

Theorem 2**.**

A surface $S$ in $E^{3}$ is of $III$-type 1 if and only if $S$ is part of a sphere.

Corollary 1**.**

*The Gauss map of every surface $S$ in $E^{3}$ is of $III$-type

1. The corresponding eigenvalue is $\lambda=2$.*

Up to now, the only known surfaces of finite $III$-type in $E^{3}$ are parts of spheres, the minimal surfaces and the parallel of the minimal surfaces which are of null $III$-type 2. So the following question seems to be interesting:

Problem 2**.**

Other than the surfaces mentioned above, which surfaces in $E^{3}$ are of finite $III$-type?

Another generalization of the above problem is to study surfaces in $E^{3}$ with the position vector $\boldsymbol{x}$ satisfying

[TABLE]

where $A\in\mathbb{Re}^{3\times 3}$.

From this point of view, we also pose the following problem

Problem 3**.**

Classify all surfaces in $E^{3}$ with the position vector $\boldsymbol{x}$ satisfying relation (2.9).

Concerning this problem, in [11] S. Stamatakis and H. Al-Zoubi studied the class of surfaces of revolution and they proved that: A surface of revolution $S$ satisfies (2.9) if and only if $S$ is a catenoid or part of a sphere. Recently, the same authors in [1] studied the class of ruled surfaces and the class of quadric surfaces. In particular, they proved that helicoids and spheres are the only ruled and quadric surfaces satisfying (2.9) respectively.

This paper provides the first attempt at the study of finite type families of surfaces in $E^{3}$ corresponding to the third fundamental form. Our main result is the following

Theorem 3**.**

All tubes in $E^{3}$ are of infinite type.

Our discussion is local, which means that we show in fact that any open part of a tube is of infinite Chen type.

3. Tubes in $E^{3}$

Let $C:\boldsymbol{\alpha}=\boldsymbol{\alpha}(\emph{t})$, $\mathit{t}\epsilon(a,b)$ be a regular unit speed curve of finite length which is topologically imbedded in $E^{3}$. The total space $N_{\boldsymbol{\alpha}}$ of the normal bundle of $\boldsymbol{\alpha}((a,b))$ in $E^{3}$ is naturally diffeomorphic to the direct product $(a,b)\times E^{2}$ via the translation along $\boldsymbol{\alpha}$ with respect to the induced normal connection. For a sufficiently small $r>0$ the tube of radius $r$ about the curve $\boldsymbol{\alpha}$ is the set:

[TABLE]

Assume that ${\mathbf{t},\mathbf{h},\mathbf{b}}$ is the Frenet frame and $\kappa$ the curvature of the unit speed curve $\boldsymbol{\alpha}=\boldsymbol{\alpha}(\emph{t})$. For a small real number $r$ satisfies $0<r<min\frac{1}{|\kappa|}$, the tube $T_{r}(\boldsymbol{\alpha})$ is a smooth surface in $E^{3}$, [9]. Then a parametric representation of the tube $T_{r}(\boldsymbol{\alpha})$ is given by

[TABLE]

It is easily verified that the first and the second fundamental forms of $\digamma$ are given by

[TABLE]

where $\delta:=(1-r\kappa\cos\varphi)$ and $\tau$ is the torsian of the curve $\boldsymbol{\alpha}$. The Gauss curvature of $\digamma$ is given by

[TABLE]

Notice that $\kappa\neq 0$ since the Gauss curvature vanishes. The Beltrami operator corresponding to the third fundamental form of $\digamma$ can be expressed as follows

[TABLE]

where $\beta:=\kappa\acute{}\cos\varphi+\kappa\tau\sin\varphi$ and $\acute{}:=\frac{d}{dt}$.

Before we start of proving our main result, we mention and prove the following special case of tubular surfaces for later use.

3.1. Anchor rings

A tube in the Euclidean 3-space is called an anchor ring if the curve $C$ is a plane circle (or is an open portion of a plane circle). In this case, the torsian $\tau$ of $\alpha$ vanishes identically and the curvature $\kappa$ of $\alpha$ is a nonzero constant. Then the position vector $\boldsymbol{x}$ of the anchor ring can be expressed as

[TABLE]

$a>r,\ \ a\epsilon\mathbb{R}.$

The first fundamental form is

[TABLE]

while the second is

[TABLE]

Hence, the Beltrami operator is given by

[TABLE]

Let $x_{1}$ be the first coordinate function of $\boldsymbol{x}$. By virtue of (3.5) one can find

[TABLE]

Moreover, by a direct computation, we obtain

[TABLE]

[TABLE]

It can be seen that $\Delta^{III}(\cos t\cos\varphi)=2\cos t\cos\varphi$, and for each integer $k>0$, it is easy to see that

[TABLE]

Thus, by induction, one finds

[TABLE]

where $d_{j,m}$ are constants, $j=1,2,...,m-1$, and

[TABLE]

Notice that $d_{m-1,m}\neq 0$, for each integer $m\geq 1$. Now, if $\digamma$ is of finite type, then there exist real numbers, $c_{1},c_{2},...,c_{m}$ such that

[TABLE]

Since $x_{1}=(a+r\cos t)\cos\varphi$ is the first coordinate of $\boldsymbol{x}$, (3.11), one gets

[TABLE]

From (3.6-3.8), (3.10) and (3.12) we obtain that

[TABLE]

which can be rewritten as

[TABLE]

where $F(u)$ is a polynomial in $u=\cos t$ of degree $2m-2$.

This is impossible for any $m\geq 1$ since $d_{m-1,m}\neq 0$. Consequently, we have the following

Corollary 2**.**

Every anchor ring in the Euclidean 3-space is of infinite type.

4. Proof of the main theorem

Applying relation (3) on the position vector $\boldsymbol{x}$ of (3.1) gives

[TABLE]

which can be rewritten as

[TABLE]

where $\mathbf{P_{1}}(u,v)$ is a vector valued polynomial in $u,v$ of degree 3 with functions in $t$ as coefficients. Moreover, by a long computation, we obtain

[TABLE]

where $\mathbf{P_{2}}(u,v)$ is a vector valued polynomial in $u,v$ of degree 7 with functions in $t$ as coefficients.

We need the following lemma which can be proved directly by using (3).

Lemma 1**.**

For any natural numbers m and n we have

[TABLE]

where $P$ is a polynomial in $u,v$ of degree $n$ + 4 with functions in $t$ as coefficients.

Using lemma 1 and relation (3) one finds

[TABLE]

where

[TABLE]

It can be seen that $d_{\lambda}\neq 0$, for each natural number $\lambda$. Moreover, we have

[TABLE]

Let $\digamma$ be of finite type. Then there exist real numbers, $c_{1},c_{2},...,c_{\lambda}$ such that

[TABLE]

Using (4.1-4.4), one has

[TABLE]

where $Q_{i},i=1,2,3$, are polynomials in $u,v$ with functions in $t$ as coefficients.

Now, if $\beta\neq 0.$ From (4.6) we find

[TABLE]

This is impossible, since $Q_{1}$ is polynomial in $\cos\varphi$ and $\sin\varphi$. Assume now $\beta=0$. Then $\kappa\acute{}=0$ and $\kappa\tau=0$ so $\kappa=const.\neq 0$ and $\tau=0$. Therefore the curve $C$ is a circle, and so $\digamma$ is anchor ring. Hence, $\digamma$ is of infinite type according to Corollary (2). This completes our proof.