Quadric surfaces of finite Chen Ⅱ-type

1.   Introduction

The theory of sub-manifolds of finite type has led to significant insights and results in differential geometry, helping to identify and characterize sub-manifolds with special geometric properties.

In ^[1], Chen mentioned the concept of Euclidean immersions of finite type concerning the first fundamental form Ⅰ of a surface $\aleph$. According to Chen's theory, a surface $\aleph$ is said to be of finite type if its coordinate functions can be expressed as a finite sum of eigenfunctions of the Beltrami operator $\Delta ^{I}.$

For instance, Chen ^[2] posed the problem of classifying finite type sub-manifolds in 3-dimensional Euclidean space $E^{3}$. This initiated a comprehensive study of the spectral properties of the Laplacian on these sub-manifolds, leading to the classification of minimal surfaces, spheres, and circular cylinders as specific examples of finite-type sub-manifolds.

If we consider the surface $\aleph$ in $E^{3}$, its position vector

can be written as:

where $F_{i}(v^{1}, \ v^{2})$ are eigenfunctions of the Beltrami operator $\Delta ^{I}$, and $\mathit{\boldsymbol{e}}_{i}$ are constant vectors in $E^{3}$.

For a surface to be of finite Ⅱ-type, its shape operator (related to the second fundamental form) must also have a similar decomposition into a finite sum of eigenfunctions of the Beltrami operator.

To understand the implications of a surface $\aleph$ being of finite type $l$, we start by considering the relevant equation involving the second fundamental form, $\Delta ^{II},$ which is the Laplacian operator applied to the components of the second fundamental form.

When $\aleph$ is of finite type $l$, there exists a monic polynomial

such that

Here, $\boldsymbol{X}$ represents the position vector of a point on the surface, and $\boldsymbol{c}$ is a constant vector.

Suppose the polynomial

Then the coefficients $\gamma_{i}$ are determined by the specific relationship between the eigenfunctions of $\Delta ^{II},$ and the position vector components. These coefficients $\gamma_{i}$ are related to the eigenvalues $\mu_{i}$ of $\Delta ^{II}$ acting on the coordinate functions of $\boldsymbol{X}$. In detail, $\gamma_{i}$ are typically given in terms of symmetric polynomials of these eigenvalues. Specifically, they can be expressed as follows:

Therefore the position vector $\boldsymbol{X}$ satisfies the following equation (see ^[3]):

Finite-type immersions involve studying sub-manifolds whose coordinate functions are finite sums of eigenfunctions of the Laplace-Beltrami operator. This notion provides a way to classify sub-manifolds based on the spectral properties of the Laplacian acting on the coordinate functions.

These classifications help in understanding the geometric and topological properties of sub-manifolds. For instance, the result that spheres are the only quadric surfaces of finite Ⅱ-type in $E^{3}$ provides a clear distinction between spheres and other quadric surfaces like ellipsoids, hyperboloids, and paraboloids, based on their curvature properties.

A recent study in ^[4] authors investigated the Hasimoto surfaces according to their finite Chen type, while in ^[5,6] interesting researches were done by studying the class of translation surfaces according to it's finite Chen Ⅲ-type once in $E^{3}$, and on the other hand in Sol$_{3}$.

Takahashi in ^[7] mentioned that a surface $M^{2}$ whose position vector $\boldsymbol{X}$ satisfies

is either a minimal with $\mu = 0$ or $M^{2}$ lies in an ordinary sphere $S^{2}$ with a fixed nonzero eigenvalue.

Garay in his article ^[8] made a generalization of Takahashi's condition. In his study, he considered surfaces in $E^{3}$ satisfying

where $\left(X_{1}, X_{2}, X_{3}\right)$ are the coordinate functions of the position vector $\boldsymbol{X}$ and $\mu_{i}$, are different eigenvalues. Garay's work expands on the problem of identifying surfaces in $E^{3}$ that satisfy this eigenvalue condition. The coordinate functions of these surfaces are expressed as eigenfunctions of the Laplace-Beltrami operator associated with distinct eigenvalues, contributing to the understanding of surfaces of finite type in a more general context.

Another related general problem was presented in ^[9], which investigated surfaces in $E^{3}$ satisfying

where $K$ is a $3\times3$ matrix; $L$ is a $3\times1$ matrix. It was proven that minimal surfaces, spheres, and circular cylinders are the only surfaces in $E^{3}$ satisfying Eq (1.2). Surfaces meeting this criterion are said to be of coordinate finite type.

As an application, the alignment of molecules in relation to quadric surfaces has meaningful applications in understanding molecular orientations, interactions, and behaviors under external influences. Quadric surfaces, such as ellipsoids, hyperboloids, and paraboloids, serve as mathematical representations of properties like potential energy distributions, molecular shapes, and field effects. The shapes of molecules can often be described using quadric surfaces such as ellipsoids which is common for anisotropic molecules like liquid crystal rods or elongated organic molecules, or spheres which represent isotropic molecules such as noble gases or symmetric compounds like $CH4$. These quadric shapes help model how molecules orient or align in space (see ^[10,11]).

We consider a (connected) surface $\aleph$ in a Euclidean 3-space $E^{3}$ referred to any system of coordinates $v^{1}, v^{2},$ whose Gaussian curvature never vanishes. Let $B_{st}$ be the components of the second fundamental form

of $\aleph.$ For any two sufficiently differentiable functions $\varphi (v^{1}, v^{2})$ and $\psi(v^{1}, v^{2})$ on $\aleph$, the first Beltrami operator with respect to the second fundamental form of $\aleph$ is given by

where

and $\left(B^{st}\right)$ denotes the inverse tensor of $\left(B_{st}\right)$.

The second Beltrami operator regarding the second fundamental form of $\aleph$ is defined by ^[12]

where

In ^[13], authors proved that, for the position vector

of $\aleph$, the relation is

where $\mathit{\boldsymbol{G}}$ is the Gauss map, $K$ the Gauss curvature, and $H$ the mean curvature of $\aleph$.

The main result in this study presents the following as detailed below.

Theorem 1. Among all quadric surfaces in $E^{3}$, the only one that satisfies the finite Ⅱ-type condition is the sphere.

This result highlights a unique geometric property of spheres compared to other quadric surfaces such as ellipsoids, hyperboloids, and paraboloids.

2.   Quadric surfaces

Let $\aleph$ be a $\mathcal{C}^{r}$ quadric surface in $\mathbb{E}^{3}$ defined on a region $\mathcal{U}\subset \mathbb{R}^{2}$. Then, $\aleph$ is one of the following three kinds ^[14,15]:

$1^{st} Kind: \aleph$ is a ruled surface.

$2^{nd} Kind: \aleph$ is of the form

$3^{rd} Kind: \aleph$ is of the form

The class of ruled surfaces has been studied in ^[16], so we will complete our study by investigating the second and third kinds of surfaces mentioned above in terms of their finite Chen type.

2.1.   Quadrics of the second kind

A parametrization of a part of a quadric of this kind is ^[15]

For simplicity, we use

The metrics I, II of $\aleph$, are respectively,

where

The Laplacian $\triangle^{II}$ of $\aleph$ can be expressed as follows:

For a function $\varphi(\upsilon) \in \mathcal{C}^{\infty}(\mathcal{U})$, on account of Eq (2.2), we get

On use of Eq (2.2), it can be easily proved:

Lemma 1. The relation

holds true, where $F(\upsilon, \nu)$ is a polynomial of degree at most $m + 4$, and when $\nu = 0$, $deg(F(\upsilon, 0))$ is at most $m + 2$.

We denote by $(x_{1}, x_{2}, x_{3})$ the components of $\boldsymbol{x}(\upsilon, \nu)$. On account of Eq (2.3), we have

Applying Eq (2.2) for the relation (2.4), we find

where

Note that $f_{2}(\upsilon, \nu)$ is a polynomial of degree at most 5, and if we put $\nu = 0$, then $f_{2}(\upsilon, 0)$ is a polynomial in $\upsilon$ of degree at most 3.

From Lemma 1, we get

where $f_{3}(\upsilon, \nu)$ is a polynomial of degree at most 9, with

We will also prove:

Lemma 2. The relation

holds true, where

Proof. The proof goes by induction on $l$.

Base case: For $l = 1$, the formula comes true from Eq (2.4) applied to $\varphi = \upsilon$.

Inductive step: Assume that the lemma is true for $l-1$. So,

Proof for $l$. Taking into account $\nu = 0$, relation (2.3), and Lemma 1, we obtain

This completes the proof. □

For the second component $x_{2}$, we have

Also

and

Similarly, $g_{2}(\upsilon, \nu)$ is a polynomial of degree at most 5, and if we put $\upsilon = 0$, then $g_{2}(0, \nu)$ is a polynomial in $\nu$ of degree at most 3.

Lemma 3. The relation

holds true, where $G(\upsilon, \nu)$ is a polynomial of degree at most $m + 4$, and when $\upsilon = 0$, then $deg(G(0, \nu))$ is at most $m + 2$.

So, using the above lemma, one can find that

where $g_{3}(\upsilon, \nu)$ is a polynomial of degree at most 9, with

By induction, one can also obtain:

Lemma 4. The relation

is valid, and

Let now $\aleph$ be of finite Ⅱ-type $l$. Then, there exist real numbers $c_{i}, i = 1, \dotsc, l$ such that

Applying Eq (2.10) to the coordinate functions $x_{1} = \upsilon$ and $x_{2} = \nu$ of the position vector (2.1) of $\aleph$, we obtain

From Lemma 2, relations (2.4)–(2.6), and (2.11), it follows that

which can be written as

Inserting $\nu = 0$ in (2.13), we obtain a nontrivial polynomial in $\upsilon$ with constant coefficients. Therefore, the above equation can be rewritten as

with

Since $\alpha\neq 0$, the relation (2.14) implies that $\alpha$ must be equal $-1$.

Following the same procedure for the second component $x_{2}$, by using relations (2.7)–(2.9), (2.12), and Lemma 4, we get

where $Q(\upsilon, \nu)$ is a polynomial of degree at most $4l$. Putting $\upsilon = 0$, then Eq (2.15) is a nontrivial polynomial in $\nu$ with constant coefficients. However $\beta \neq 0$, so from (2.15) $\beta$ equals $-1$. Therefore, $\aleph$ is a sphere.

Let

Then,

Thus, relation (2.2) reduces to

So, relations (2.4) and (2.7) become

For the third coordinate

after simple calculation, we conclude

Thus, we find that

That is, spheres are the only quadric surfaces of the kind (2) of finite Ⅱ-Chen type.

2.2.   Quadrics of the third kind

A parametrization of a part of a quadric of this kind is

The matrix of the components of the first fundamental form of $\aleph$ is the following:

Denote

The matrix of the components of the second fundamental form Ⅱ is given as follows:

Thus, $\triangle ^{II}$ of $\aleph$ becomes

By applying the operator $\triangle^{II}$ to the components

we get

For the third coordinate

we find

Applying Eq (2.17) for the relation (2.18), we find

where

with

On one hand, using (2.17), one can find:

Lemma 5. For $n > 0$, we find

and, on the other hand, we have:

Lemma 6.

From Lemma 6, we obtain that

where $deg(G(\upsilon, \nu))$ is at most $t+2$.

Taking into account (2.20), and (2.22), we get

where $G_{2}(\upsilon, \nu)$ is a polynomial in $\upsilon, \nu$ of degree at most 4. Similarly, we get

where $G_{3}(\upsilon, \nu)$ is a polynomial in $\upsilon, \nu$ of degree at most 6. In general, we have

Lemma 7. The relation

holds true for $l > 2$, with

On account of relation (1.1) applied to the component $x_{3}$, we have

From (2.18), (2.19), (2.23), (2.24), and Lemma 7, we get

or

where

Relation (2.25) must hold true for all $(\upsilon, \nu)$. This is clearly impossible since the first term of (2.25), which is the constant term of (2.25), must equal 0, something that cannot be satisfied since $\alpha, \beta > 0$.

3.   Conclusions

This research article was divided into three sections, where after the introduction, the needed definitions and relations regarding this interesting field of study were given. Then, a formula for the Laplace operator corresponding to the second fundamental form Ⅱ was proved once for the position vector and another for the Gauss map of a surface. Finally, we classify the quadric surfaces of finite Chen type regarding the second fundamental form. An interesting study can be drawn if this type of study can be applied to other classes of surfaces that have not been investigated yet, such as spiral surfaces, or tubular surfaces.

Acknowledgements

The author would like to thank the referees for their useful remarks.

Conflict of interest

The author declares that he has no conflict of interest.