The Chen type of Hasimoto surfaces in the Euclidean 3-space

1.   Introduction

In the theory of curves in Riemannian manifolds, one of the important and interesting problems is the characterizations of a regular curve. The authors in ^[1] examined the curvatures of the Hasimoto surface according to Bishop's frame and gave some characterization of the parameter curves for these surfaces. The geometric properties of Hasimoto surfaces are investigated by ^[2]. In ^[3], Schief and Rogers studied the binormal motion of curves with constant curvatures. In ^[4], the authors studied the intrinsic geometry of the nonlinear Schrödinger (NLS) equation in $\mathbb{E}^{3}$.

Hasimoto surfaces, also known as Frenet frames surfaces, are a class of surfaces in Euclidean 3-space that arise from solutions to the Schrödinger equation in quantum mechanics. Specifically, they are associated with the motion of a charged particle in a magnetic field.

The construction of Hasimoto surfaces involves the use of the Frenet-Serret frame, which is a set of orthonormal vectors that describe the local geometry of a curve or surface. In the case of Hasimoto surfaces, the Frenet-Serret frame is used to describe the motion of a curve in Euclidean 3-space under the influence of a magnetic field. This leads to the construction of a surface in 3-space that has certain interesting properties, such as having a constant mean curvature.

In ^[5], authors studied Hasimoto surfaces in Minkowski 3-space. On the other hand, M. Elzawy in ^[6] investigated Hasimoto surfaces in Galilean space $G_{3}$.

In this work, we briefly give the geometric properties of Hasimoto surfaces in the Euclidean 3-space. Especially, we obtain the curvatures of Hasimoto surface according to Bishop's frame. Then, we investigate the second Laplace operator for the first, second, and third fundamental forms of Hasimoto surfaces.

Further, one can follow the idea in ^[7] by defining the first and second Beltrami operators using the definition of the fractional vector operators. It is also interesting, as future work, to do interdisciplinary research, apply a mix or a blend of the techniques followed in ^[8,9,10,11] and combine them with the methods of this paper to obtain more new properties of Hasimoto surfaces.

2.   Preliminaries

Let $\phi:\; I\rightarrow \mathcal{M}^{2}$ be a regular unit speed curve on the orientable surface $\mathcal{M}^{2}$. Let $\left\{ \boldsymbol{T}, \boldsymbol{N}, \boldsymbol{B}\right\}$ be the orthonormal moving Frenet frame along the curve $\phi$ in $\mathcal{M}^{2}$ such that $\boldsymbol{T} = \phi^{\prime }$ is the unit vector field tangent to $\phi,$ $\boldsymbol{N}$ is the unit vector field in the direction $\boldsymbol{T}^{\prime }$ normal to $\phi$ (principal normal) and $\boldsymbol{B} = \boldsymbol{T}\wedge \boldsymbol{N}$ (binormal vector). Then we have the following Frenet equations

Functions $k$ and $\tau$ are the curvature and the torsion of $\phi$.

Introduce a new frame, called Darboux frame $\left\{ \boldsymbol{T}, \; \boldsymbol{\eta}, \; \boldsymbol{g}\right\}$ with

where $\boldsymbol{g} = \boldsymbol{\eta} \wedge \boldsymbol{T}$ and $\beta$ is the angle between the vector fields $\boldsymbol{N}$ and $\boldsymbol{\eta}$.

The derivative formulas of (2.2) can be given as follows:

where $k_{g}$ is the geodesic curvature, $k_{\eta}$ is the normal curvature, $t_{r}$ is the geodesic torsion of the curve $\phi$ and $\boldsymbol{T}_{s} = \frac{d \boldsymbol{T}}{d\, s}$. From now on we will use the prime $'$ to denote the derivative with respect to the parameter $s$.

Here Darboux curvatures are defined by

Theorem 1. ^[12] Suppose $r = r(s, t)$ is an NLS surface such that $r = r(s, t)$ is a unit speed curve with a normal vector field for all $t$. Then the following is satisfied:

where $\alpha$, $\lambda$ and $\gamma$ are smooth functions given by

where $\delta = k_{g_{t}}k_{\eta}-k_{\eta_{t}}k_{g}$ and $\boldsymbol{T}_{t} = \frac{d \boldsymbol{T}}{d\, t}$.

From (2.4) and (2.6) we obtain

Using compatibility conditions $\boldsymbol{T}_{s t} = \boldsymbol{T}_{t s}$, $\boldsymbol{\eta}_{s t} = \boldsymbol{\eta}_{t s}$ and $\boldsymbol{g}_{s t} = \boldsymbol{g}_{t s}$, we get

The mean curvature $H_{mean}$ and the Gaussian curvature $K_{G}$ are, respectively, defined by

and

The Laplace-Beltrami operator of a smooth function $\varphi :$ $\mathcal{M}^{2} \rightarrow \mathbb{R},$ with respect to the first fundamental form $I$ of the surface $\mathcal{M}^{2}$ is the operator $\Delta ^{I}$, defined in ^[13,14,15] as follows:

The second differential parameter of Beltrami of a function $\varphi :$ $\mathcal{M}^{2}\rightarrow \mathbb{R}, (s, t)\rightarrow \varphi (s, t)$ with respect to the second fundamental form $II$ of $\mathcal{M}^{2}$ is the operator $\Delta ^{II}$ which is defined by ^[13,14,15]

where $LN-M^{2}\neq 0$ since the surface has no parabolic points.

In classical literature, one writes the third fundamental form as

where

The second Beltrami differential operator with respect to the third fundamental form $III$ is defined by ^[13,14,15]

where $e = \det (e_{ij})$ and $e^{ij}$ denote the components of the inverse tensor of $e_{ij}.$

3.   Hasimoto surfaces

In this section, Hasimoto surfaces are investigated by using the Darboux frame and discussing the geometric properties of Hasimoto surfaces. We find the Gaussian $K_{G}$ and mean curvatures $H_{mean}$ of this surface.

Let $r = r(s, t)$ be the position vector of a curve $\phi$ moving on a surface $\mathcal{M}^{2}$ in $\mathbb{E}^{3}$ such that $r(s, t)$ is a unit speed curve for all $t$. If the surface $\mathcal{M}^{2}$ is a Hasimoto surface, then, the position vector $r$ satisfies the following condition

This is called the vortex filament or smoke ring equation.

Lemma 1. ^[16] The evolution equations for curvature and torsion are

The coefficients of the first fundamental form of the surface $r = r(s, t)$ are

The unit normal vector of the Hasimoto surface is given by

Lemma 2. The components $b_{ij}$ and $e_{ij}$ of the second and the third fundamental tensors in coordinates are the following

From (3.2) and (3.4) we have

where $k\neq 0$, since the surface has no parabolic points.

4.   $J$-Hasimoto surface in the Euclidean 3-space

We consider a surface $\mathcal{M}^{2}$ in $\mathbb{E}^{3}$ parametrized by

Definition 1. A surface in the three-dimensional Euclidean space is said to be $J$-harmonic if it satisfies the condition $\Delta^{J}r = 0$, where $\Delta^{J}$ denotes the Laplace operator with respect to the fundamental forms $I$–$III$.

4.1.   $I$-harmonic Hasimoto surfaces in $\mathbb{E}^{3}$

Theorem 2. ^[16] The Laplacian $\Delta^{I}$ of the Hasimoto surface $r = r(s, t)$ can be expressed as

where

$k_{g_{t}} = \frac{\partial\, k_{g}}{\partial\, t},$ $k_{\eta_{t}} = \frac{\partial\, k_{\eta}}{\partial\, t}$.

Remark 1.

Corollary 1. ^[16] Therefore, $r$ is $I$-harmonic if and only if $H_{mean} = 0$.

4.2.   $II$-harmonic Hasimoto surfaces in $\mathbb{E}^{3}$

In this section, we classify Hasimoto surface with non-degenerate second fundamental form in $\mathbb{E}^{3}$ satisfying the equation

By a straightforward computation, the Laplacian $\Delta ^{II}$ of the second fundamental form $II$ on $\mathcal{M}^{2}$ with the help of (2.8) and (3.4) turns out to be

where

Theorem 3. The formula of the Laplacian $\Delta^{II}$ takes the following form

where

and $|LN-M^{2}| = \varepsilon k k^{\prime \prime }\neq 0$, $\varepsilon = \pm 1$ since the surface has no parabolic points.

Proof. From (3.1), we have

Substituting the last equations into (4.4) gives (4.5).□

If $\Delta^{II}r(s, t) = 0$, then we get $k^{2} k^{\prime \prime 2} = 0$. It contradicts the non-degeneracy of the second fundamental form on $\mathcal{M}^{2}$.

Theorem 4. There do not exist Hasimoto surfaces in $\mathbb{E}^{3}$ which satisfy the condition $\Delta^{II}r(s, t) = 0$.

4.3.   $III$-harmonic Hasimoto surfaces in $\mathbb{E}^{3}$

In this section, we classify Hasimoto surface with non-degenerate third fundamental form in $\mathbb{E}^{3}$ satisfying the equation

Theorem 5. The formula of the Laplacian $\Delta^{III}$ takes the following form

where

Proof. By (2.9), the Laplacian operator $\Delta ^{III}$ of $r$ can be expressed as

Using (3.5), we have (4.7).□

Remark 2. We observe that

S. Stamatakis, H. Al-Zoubi proved in ^[14] the relation

From (4.8)–(4.10) we have

Theorem 6. Let $\mathcal{M}^{2}$ be a Hasimoto surface in $\mathbb{E}^{3}.$ Then $\mathcal{M}^{2}$ is $III$-harmonic if and only if $\mathcal{M}^{2}$ has zero mean curvature.

5.   Hasimoto surface in the Euclidean 3-space satisfying $\Delta^{J}\boldsymbol{N} =0$

In this section, we consider the Gauss map $\boldsymbol{N}$ of the surface $\mathcal{M}^{2}$ with parametric representation (3.3).

5.1.   Hasimoto surface in the Euclidean 3-space satisfying $\Delta^{I}\boldsymbol{N}=0$

Let $r = r(s, t)$ be a Hasimoto surface. From (2.7), (3.3) and (3.2), we write the Laplacian operator of the Gauss map as ^[16]

where

Theorem 7. ^[16] Let $r = r(s, t)$ be a Hasimoto surface. There are no Hasimoto surfaces in $\mathbb{E}^{3}$, satisfying the condition $\Delta^{I} \boldsymbol{N} = 0$.

5.2.   Hasimoto surface in the Euclidean 3-space satisfying $\Delta^{II}\boldsymbol{N}=0$

Using (3.3) and (4.4), we obtain

Theorem 8.

where

Proof. Using (4.4) and

where

we have (5.2).□

Suppose that the Hasimoto surface has a $II$- harmonic Gauss map. Then, the vector $\Delta^{II} \boldsymbol{N}$ given from (5.2) is zero. Thus, we have

Therefore

Case Ⅰ. Suppose $c > 0$. Then from $-\frac{k^{\prime \prime }}{k} = c$ we have $k^{2}+\tau^{2} < 0$, it is impossible.

Case Ⅱ. Suppose $c < 0$. Then $k^{\prime \prime } = \alpha^{2}k$, where $c = -\alpha^{2}$.

The solution of the last equation is

where $\delta_{1}$ and $\delta_{2}$ are smooth functions on open set of $\mathbb{R}$.

Hence, the equation $k^{\prime \prime } = k(k^{2}+\tau^{2})$ implies that

Theorem 9. Let $r = r(s, t)$ be a Hasimoto surface. $\Delta^{II}\boldsymbol{N} = 0$ if and only if

and

where $\varepsilon = \mp 1$ and $\delta_{1}$ and $\delta_{2}$ are smooth functions on open set of $\mathbb{R}$.

5.3.   Hasimoto surface in the Euclidean 3-space satisfying $\Delta^{III}\boldsymbol{N}=0$

In ^[14] S. Stamatakis, H. Al-Zoubi proved the relation

From (5.3), it can be seen that the Gauss map $\boldsymbol{N}$ of $\mathcal{M}^{2}$ in $\mathbb{E}^{3}$ is of finite $III$-type $1$, the corresponding eigenvalue is $2$. Then

Theorem 10. The Gauss map $\boldsymbol{N}$ of a Hasimoto surface $\mathcal{M}^{2}$ in $\mathbb{E}^{3}$ is of finite $III$-type $1$, the corresponding eigenvalue is $2$.

6.   Conclusions

In the beginning, a brief introduction and definition for the Hasimoto surfaces were given in the Euclidean 3-space. Then, we investigate Hasimoto surfaces by using the Darboux frame and discuss its geometric properties of it. Consequentially, we define a formula for the Laplace operator regarding the first, second, and third fundamental forms. Finally, we classify the Hasimoto surfaces satisfying the relations $\Delta ^{J}\boldsymbol{r} = 0$, and $\Delta ^{J}\boldsymbol{N} = 0$ for $J = I, II$ and $III$, where $\boldsymbol{N}$ is the Gauss map of $\mathcal{M}^{2}$ in $\mathbb{E}^{3}$. We distinguish three types according to whether these surfaces are determined, with each type investigated in a subsection of section 4 and for the Gauss map in Section 5. An interesting study can be drawn, if this type of study can be applied to the general definition of surfaces of finite Chen $k$-type.

Acknowledgments

Most of this work was done by Dr. Bendehiba Senoussi and I the correspondence author would like to thank him for the great work he did.

Conflict of interest

The authors declare that they have no conflict of interest.