# Legendre curves on 3-dimensional $C_{12}$-Manifolds

Gherici Beldjilali  Benaoumeur Bayour and Habib Bouzir  Gherici Beldjilali –
Laboratory of Quantum Physics and Mathematical Modeling (LPQ3M)
University of Mascara, Algeria<EMAIL_ADDRESS>Benaoumeur Bayour – Departement of Mathematics, University of Mascara,
Algeria<EMAIL_ADDRESS>Habib Bouzir – Laboratory of Quantum
Physics and Mathematical Modeling (LPQ3M)
University of Mascara, Algeria<EMAIL_ADDRESS>
###### Abstract

Legendre curves play a very important and special role in geometry and
topology of almost contact manifolds.There are certain results known for
Legendre curves in 3-dimensional normal almost contact manifolds. The aim of
this paper is to study Legendre curves of three-dimensional $C_{12}$-manifolds
which are non-normal almost contact manifolds and classifying all biharmonic
Legendre curves in these manifolds.

###### keywords:

Almost contact metric structure, $C_{12}$-manifolds, Legendre curves.

Primary 53D15, Secondary 53B25.

##  1 Introduction

In [8], D. Chinea and C. Gonzalez have defined 12 classes of almost contact
metric manifolds. In dimension 3, these manifolds are reduced to five classes:
$|C|$ class of cosymplectic manifolds, $C_{5}$ class of $\beta$-Kenmotsu
manifolds, $C_{6}$ class of $\alpha$-Sasakian manifolds, $C_{9}$-manifolds and
$C_{12}$-manifolds.
Only the last two classes can never be normal. For this reason, all work
concerning curves on almost contact metric manifolds focuses on the first
three classes. for example, Legendre curves on contact manifolds have been
studied by C. Baikoussis and D.E. Blair in the paper [1]. M. Belkhelfa et al.
[3] have investigated Legendre curves in Riemannian and Lorentzian manifolds.
Also, Legendre curves have been studied on Kenmotsu manifolds in [11], on
$\alpha$-Sasaki minifolds in [9], on quasi-Sasakian manifolds in [12], on
Trans-Sasaki manifolds in [10] and others. $C_{12}$-manifolds arose in a
natural way from the classification of almost contact metric structures by D.
Chinea and C. Gonzalez [8] in 1990. Recently, these manifolds was studied and
most of its properties were deduced in [6], especially [4]. As
$C_{12}$-manifolds differs from the above almost contact metric manifolds, at
least in the normality property, in the present paper, we will show that the
results related to Legendre curves in 3-dimentional $C_{12}$-manifolds are
fundamentally different from the results in other almost contact metric
manifolds. The present paper is organized as follows: After the introduction,
we recall some required preliminaries on almost contact geometry and Frenet
curves in general Riemannian geometry in Section 3. In Section 4, we give a
very brief review of 3-dimentional $C_{12}$-manifolds with concrete example.
The next section is focused on the study of Legendre curves on 3-dimentional
$C_{12}$-manifolds. In the last section, we demonstrate a nice property of
biharmonic Legendre curves in 3-dimentional $C_{12}$-manifolds.

##  2 Almost contact manifold

An odd-dimensional Riemannian manifold $(M^{2n+1},g)$ is said to be an almost
contact metric manifold if there exists on $M$ a $(1,1)$-tensor field
$\varphi$, a vector field $\xi$ (called the structure vector field) and a
$1$-form $\eta$ such that

$\left\\{\begin{array}[]{lll}\eta(\xi)=1,\\\ \varphi^{2}(X)=-X+\eta(X)\xi,\\\
g(\varphi X,\varphi Y)=g(X,Y)-\eta(X)\eta(Y),\end{array}\right.$ (2.1)

for any vector fields $X$, $Y$ on $M$. In particular, in an almost contact
metric manifold we also have

$\varphi\xi=0\qquad and\qquad\eta\circ\varphi=0.$

The fundamental 2-form $\phi$ is defined by

$\phi(X,Y)=g(X,\varphi Y).$

It is known that the almost contact structure $(\varphi,\xi,\eta)$ is said to
be normal if and only if

$N^{(1)}(X,Y)=N_{\varphi}(X,Y)+2d\eta(X,Y)\xi=0,$ (2.2)

for any $X$, $Y$ on $M$, where $N_{\varphi}$ denotes the Nijenhuis torsion of
$\varphi$, given by

$N_{\varphi}(X,Y)=\varphi^{2}[X,Y]+[\varphi X,\varphi Y]-\varphi[\varphi
X,Y]-\varphi[X,\varphi Y].$ (2.3)

For more background on almost contact metric manifolds, we recommend the
references [2, 5, 13].

##  3 Legendre curves

Let $(M,g)$ be a 3-dimensional Riemannian manifold with Levi-Civita connection
$\nabla$ and $\gamma:I\rightarrow M$ parameterized by the arc length. $\gamma$
is said to be a Frenet curve if there exists an orthonormal frame
$\\{E_{1}=\dot{\gamma},E_{2},E_{3}\\}$ along $\gamma$ such that

$\left\\{\begin{array}[]{lll}\nabla_{\dot{\gamma}}E_{1}=\kappa E_{2},\\\
\nabla_{\dot{\gamma}}E_{2}=-\kappa E_{1}+\tau E_{3},\\\
\nabla_{\dot{\gamma}}E_{3}=-\tau E_{2}.\end{array}\right.$ (3.1)

The curvature $\kappa$ is defined by the formula

$\kappa=|\nabla_{\dot{\gamma}}\dot{\gamma}|.$ (3.2)

The second unit vector field $E_{2}$ is thus obtained by

$\nabla_{\dot{\gamma}}\dot{\gamma}=\kappa E_{2}.$ (3.3)

Next, the torsion $\tau$ and the third unit vector field $E_{3}$ are defined
by the formulas

$\tau=|\nabla_{\dot{\gamma}}E_{2}+\kappa E_{1}|\qquad
and\qquad\nabla_{\dot{\gamma}}E_{2}+\kappa E_{1}=\tau E_{3}.$ (3.4)

A Frenet curve $\gamma:I\rightarrow M$ in an almost contact metric manifold is
said to be a Legendre curve [1], if it is an integral curve of the contact
distribution $\mathcal{D}=ker\;\eta$. Formally, it is also said that a Frenet
curve $\gamma$ in an almost contact metric manifold is a Legendre curve if and
only if $\eta(\dot{\gamma})=0$ and $g(\dot{\gamma},\dot{\gamma})=1$.

##  4 Three dimensional $C_{12}$-manifold

In the classification of D. Chinea and C. Gonzalez [8] of almost contact
metric manifolds there is a class $C_{12}$-manifolds which can be integrable
but never normal. In this classification, $C_{12}$-manifolds are defined by

$(\nabla_{X}\phi)(Y,Z)=\eta(X)\eta(Z)(\nabla_{\xi}\eta)\varphi
Y-\eta(X)\eta(Y)(\nabla_{\xi}\eta)\varphi Z.$ (4.1)

In [4] and [6], The $(2n+1)$-dimensional $C_{12}$-manifolds is characterized
by:

$(\nabla_{X}\varphi)Y=\eta(X)\big{(}\omega(\varphi
Y)\xi+\eta(Y)\varphi\psi\big{)},$ (4.2)

for any $X$ and $Y$ vector fields on $M$, where
$\omega=-\big{(}\nabla_{\xi}\xi\big{)}^{\flat}=-\nabla_{\xi}\eta$ and $\psi$
is the vector field given by

$\omega(X)=g(X,\psi)=-g(X,\nabla_{\xi}\xi),$

for all $X$ vector field on $M$. Moreover, in [4] the $(2n+1)$-dimensional
$C_{12}$-manifolds is also characterized by

${\rm d}\eta=\omega\wedge\eta\qquad{\rm d}\phi=0\qquad and\qquad
N_{\varphi}=0.$ (4.3)

Here, we emphasize that the almost $C_{12}$-manifolds is defined by the
following:

######  Defintion 4.1.

Let $(M^{2n+1},\varphi,\xi,\eta,g)$ be an almost contact manifold. $M$ is
called almost $C_{12}$-manifold if there exists a one-form $\omega$ which
satisfies

${\rm d}\eta=\omega\wedge\eta\qquad and\qquad{\rm d}\phi=0.$

In addition, if $N_{\varphi}=0$ we say that $M$ is a $C_{12}$-manifold.

In [4], the authors studied the 3-dimensional unit $C_{12}$-manifold i.e. the
case where $\omega$ is closed and $\psi$ is a unit vector field. We will deal
here with the general case, i.e. $\psi$ is not necessarily unitary. For that,
taking $V={\rm e}^{-\rho}\psi$ where ${\rm e}^{\rho}=|\psi|$, we get
immediately that $\\{\xi,V,\varphi V\\}$ is an orthonormal frame. We refer to
this basis as Fundamental basis. Using this frame, one can get the following:

######  Proposition 4.2.

For any $C_{12}$-manifold, for all vector field $X$ on $M$ we have

  * 1)

$\nabla_{X}\xi=-{\rm e}^{\rho}\eta(X)V$

  * 2)

$\nabla_{\xi}V={\rm e}^{\rho}\xi$

  * 3)

$\nabla_{V}V=\varphi V(\rho)\varphi V$

  * 4)

$\nabla_{\xi}\varphi V=0$

  * 5)

$\nabla_{V}\varphi V=-\varphi V(\rho)V.$

###### Proof.

For the first, using (4.2) for $Y=\xi$ we get

$\displaystyle(\nabla_{X}\varphi)\xi$ $\displaystyle=$
$\displaystyle\eta(X)\varphi\psi$ $\displaystyle=$ $\displaystyle{\rm
e}^{\rho}\eta(X)\varphi V,$

knowing that $(\nabla_{X}\varphi)Y=\nabla_{X}\varphi Y-\varphi\nabla_{X}Y$ and
applying $\varphi$ we obtain

$\displaystyle\nabla_{X}\xi$ $\displaystyle=$ $\displaystyle{\rm
e}^{\rho}\eta(X)\varphi^{2}V$ $\displaystyle=$ $\displaystyle-{\rm
e}^{\rho}\eta(X)V.$

For the second, we have

$\displaystyle 2{\rm d}\omega(\xi,X)=0\Leftrightarrow g(\nabla_{\xi}\psi,X)$
$\displaystyle=$ $\displaystyle g(\nabla_{X}\psi,\xi)$ $\displaystyle=$
$\displaystyle-g(\psi,\nabla_{X}\xi)$ $\displaystyle=$ $\displaystyle{\rm
e}^{2\rho}\eta(X),$

which gives $\nabla_{\xi}\psi={\rm e}^{2\rho}\xi$ and then

$\displaystyle\nabla_{\xi}V$ $\displaystyle=$ $\displaystyle\nabla_{\xi}({\rm
e}^{-\rho}\psi)$ $\displaystyle=$ $\displaystyle-\xi(\rho)V+{\rm
e}^{\rho}\xi.$

On the other hand, we have

$\displaystyle\xi(\rho)$ $\displaystyle=$ $\displaystyle\frac{1}{2}{\rm
e}^{-2\rho}\xi({\rm e}^{2\rho})$ $\displaystyle=$
$\displaystyle\frac{1}{2}{\rm e}^{-2\rho}\xi\big{(}g(\psi,\psi)\big{)}$
$\displaystyle=$ $\displaystyle{\rm e}^{-2\rho}g(\nabla_{\xi}\psi,\psi)=0,$

then,

$\nabla_{\xi}V={\rm e}^{\rho}\xi.$

For $\nabla_{V}V$, we have

$\displaystyle 2{\rm d}\omega(\psi,X)=0\Leftrightarrow g(\nabla_{\psi}\psi,X)$
$\displaystyle=$ $\displaystyle g(\nabla_{X}\psi,\psi)$ $\displaystyle=$
$\displaystyle\frac{1}{2}Xg(\psi,\psi)$ $\displaystyle=$ $\displaystyle{\rm
e}^{2\rho}g({\rm grad}\rho,X),$

i.e. $\nabla_{\psi}\psi={\rm e}^{2\rho}{\rm grad}\rho$ which gives
$\nabla_{V}V={\rm grad}\rho-V(\rho)V.$
Also, we have

$\displaystyle{\rm grad}\rho$ $\displaystyle=$
$\displaystyle\xi(\rho)\xi+V(\rho)V+\varphi V(\rho)\varphi V$ $\displaystyle=$
$\displaystyle V(\rho)V+\varphi V(\rho)\varphi V,$

then,

$\nabla_{V}V=\varphi V(\rho)\varphi V.$

For the rest, just use the formula $\nabla_{X}\varphi
Y=(\nabla_{X}\varphi)Y+\varphi\nabla_{X}Y$ noting that
$(\nabla_{V}\varphi)X=(\nabla_{\varphi V}\varphi)X=0$. ∎

It remains to count $\nabla_{\varphi V}V$ and $\nabla_{\varphi V}\varphi V$.
For that, we have the following lemma

######  Lemma 4.3.

For any $3$-dimensional $C_{12}$-manifold, we have

  * 1)

$\nabla_{\varphi V}V=(-{\rm e}^{\rho}+{\rm div}V)\varphi V,$

  * 2)

$\nabla_{\varphi V}\varphi V=({\rm e}^{\rho}-{\rm div}V)V.$

###### Proof.

Since $\\{\xi,V,\varphi V\\}$ is an orthonormal frame then,

$\nabla_{\varphi V}V=a\;\xi+b\;V+c\;\varphi V,$

Using Proposition 4.2, we have

$a=g(\nabla_{\varphi V}V,\xi)=-g(V,\nabla_{\varphi V}\xi)=0$

and $b=g(\nabla_{\varphi V}V,V)=0$. for get $c$ we have

$\displaystyle{\rm div}V$ $\displaystyle=$ $\displaystyle
g(\nabla_{\xi}V,\xi)+g(\nabla_{\varphi V}\psi,\varphi V)$ $\displaystyle=$
$\displaystyle{\rm
e}^{\rho}+g(\nabla_{\varphi\psi}\psi,\varphi\psi)\Leftrightarrow
g(\nabla_{\varphi V}V,\varphi V)=-{\rm e}^{\rho}+{\rm div}V,$

then,

$\nabla_{\varphi V}V=(-{\rm e}^{\rho}+{\rm div}V)\varphi V.$

Applying $\varphi$ with (4.2), we obtain

$\nabla_{\varphi V}\varphi V=({\rm e}^{\rho}-{\rm div}V)V.$

∎

According to the Proposition 4.2 and Lemma 4.3, the 3-dimensional
$C_{12}$-manifold is completely controllable. That is:

######  Corollary 4.4.

For any $C_{12}$-manifold, we have

$\begin{array}[]{lll}\nabla_{\xi}\xi=-{\rm e}^{\rho}V,&\nabla_{\xi}V={\rm
e}^{\rho}\xi,&\nabla_{\xi}\varphi V=0,\\\ \nabla_{V}\xi=0,&\nabla_{V}V=\varphi
V(\rho)\varphi V,&\nabla_{V}\varphi V=-\varphi V(\rho)V,\\\ \nabla_{\varphi
V}\xi=0,&\nabla_{\varphi V}V=(-{\rm e}^{\rho}+{\rm div}V)\varphi
V,&\nabla_{\varphi V}\varphi V=({\rm e}^{\rho}-{\rm div}V)V.\end{array}$

######  Example 4.5.

We denote the Cartesian coordinates in a $3$-dimensional Euclidean space
$M=\mathbb{R}^{3}$ by $(x,y,z)$ and define a symmetric tensor field $g$ by

$\displaystyle g={\rm e}^{2y}\left(\begin{array}[]{ccc}1+\alpha^{2}&0&-1\\\
0&\alpha^{2}&0\\\ -1&0&1\end{array}\right),$

where $\alpha=\alpha(x,y)\geq 0$ every where is a function on
$\mathbb{R}^{3}$. Further, we define an almost contact metric
$(\varphi,\xi,\eta)$ on $\mathbb{R}^{3}$ by

$\varphi=\left(\begin{array}[]{ccc}0&-1&0\\\ 1&0&0\\\
0&-1&0\end{array}\right),\quad\xi={\rm e}^{-y}\left(\begin{array}[]{c}0\\\
0\\\ 1\end{array}\right),\quad\begin{array}[]{lll}\eta={\rm
e}^{y}(-1,0,1).\end{array}$

The fundamental 1-form $\eta$ and the 2-form $\phi$ have the forms,

$\eta={\rm e}^{y}(dz-dx)\qquad and\qquad\phi=-2\alpha^{2}{\rm e}^{2y}dx\wedge
dy,$

and hence

${\rm d}\eta={\rm e}^{y}\Big{(}dx\wedge dy+dy\wedge dz\Big{)}=dy\wedge\eta,$
${\rm d}\phi=0.$

By a direct computation the non trivial components of $N_{kj}^{(1)\;i}$ are
given by

$N_{12}^{(1)\;3}=1,\quad N_{23}^{(1)\;3}=1.$

i.e. $N^{(1)}\neq 0$. But, $\forall i,j,k\in\\{1,2,3\\}$

$\displaystyle(N_{\varphi})_{kj}^{i}=0,$

implying that $(\varphi,\xi,\eta)$ becomes integable non normal. We have
$\omega=dy$ i.e. ${\rm d}\omega=0$ and knowing that $\omega$ is the $g$-dual
of $\psi$ i.e. $\omega(X)=g(X,\psi)$, we have immediately that

$\psi=\frac{{\rm e}^{-2y}}{\alpha^{2}}\frac{\partial}{\partial y}.$ (4.5)

Thus, $(\varphi,\xi,\psi,\eta,\omega,g)$ is a 1-parameter family of
$C_{12}$-structure on $\mathbb{R}^{3}$.
Notice that

$|\psi|^{2}=\omega(\psi)=g(\psi,\psi)=\frac{{\rm e}^{-2y}}{\alpha^{2}}$

implies $V=\frac{{\rm e}^{-y}}{\alpha}\frac{\partial}{\partial y}$ is a unit
vector field, then

$\left\\{\xi={\rm e}^{-y}\frac{\partial}{\partial z},\quad V=\frac{{\rm
e}^{-y}}{\alpha}\frac{\partial}{\partial y},\quad\varphi V=\frac{{\rm
e}^{-y}}{\alpha}\big{(}\frac{\partial}{\partial x}+\frac{\partial}{\partial
z}\big{)}\right\\}$

form an orthonormal basis. To verify result in formula (4.2), the components
of the Levi-Civita connection corresponding to $g$ are given by:

$\begin{array}[]{lll}\nabla_{\xi}\xi=-\frac{{\rm
e}^{-y}}{\alpha}V,&\nabla_{\xi}V=\frac{{\rm
e}^{-y}}{\alpha}\xi,&\nabla_{\xi}\varphi V=0,\\\
\nabla_{V}\xi=0,&\nabla_{V}V=-\frac{{\rm e}^{-y}}{\alpha^{2}}\alpha_{1}\varphi
V,&\nabla_{V}\varphi V=-\varphi\nabla_{V}V,\\\ \nabla_{\varphi
V}\xi=0,&\nabla_{\varphi V}V=\frac{{\rm
e}^{-y}}{\alpha^{2}}(\alpha+\alpha_{2})\varphi V,&\nabla_{\varphi V}\varphi
V=\varphi\nabla_{\varphi V}V,\end{array}$

where $\alpha_{2}=\frac{\partial\alpha}{\partial y}$. Then, one can easily
check that for all $i,j\in\\{1,2,3\\}$

$\displaystyle(\nabla_{e_{i}}\varphi)e_{j}$ $\displaystyle=$
$\displaystyle\nabla_{e_{i}}\varphi e_{j}-\varphi\nabla_{e_{i}}e_{j}$
$\displaystyle=$ $\displaystyle\eta(e_{i})\big{(}\omega(\varphi
e_{j})\xi+\eta(e_{j})\varphi\psi\big{)}.$

Through the rest of this paper $(M,\varphi,\xi,\psi,\eta,\omega,g)$ always
denotes a 3-dimensional $C_{12}$-manifold and $\\{\xi,V,\varphi V\\}$ it’s
fundamental frame.

##  5 Legendre curves on 3-dimensional $C_{12}$-manifold

Let $\gamma(s)$ be a Legendre curve in a 3-dimensional $C_{12}$-manifold $M$.
Let us compute the equations of motion for the associated Frenet frame is
$\\{\dot{\gamma},\varphi\dot{\gamma},\xi\\}$. Differentiating
$\eta(\dot{\gamma})=0$ along $\gamma$ we get

$g(\nabla_{\dot{\gamma}}\dot{\gamma},\xi)=-g(\nabla_{\dot{\gamma}}\xi,\dot{\gamma}),$

with the help of Proposition 4.2, we get
$g(\nabla_{\dot{\gamma}}\dot{\gamma},\xi)=0$, then,

$\nabla_{\dot{\gamma}}\dot{\gamma}=\kappa\;\varphi\dot{\gamma},$ (5.1)

where
$\kappa=|g(\nabla_{\dot{\gamma}}\dot{\gamma},\varphi\dot{\gamma})\varphi\dot{\gamma}|$.
For $\nabla_{\dot{\gamma}}\varphi\dot{\gamma}$, using (4.2) we obtain

$\displaystyle\nabla_{\dot{\gamma}}\varphi\dot{\gamma}$ $\displaystyle=$
$\displaystyle(\nabla_{\dot{\gamma}}\varphi)\dot{\gamma}+\varphi\nabla_{\dot{\gamma}}\dot{\gamma}$
(5.2) $\displaystyle=$ $\displaystyle\kappa\varphi^{2}\dot{\gamma}$
$\displaystyle=$ $\displaystyle-\kappa\dot{\gamma}.$

Using Proposition 4.2, we get

$\nabla_{\dot{\gamma}}\xi=0.$ (5.3)

We compare the equations (5.1)-(5.3) and (3.1), we conclude the following:

######  Proposition 5.1.

In a 3-dimensional $C_{12}$-manifold, a Legendre curve is a plane Frenet
curve.

Conversely, suppose that $\gamma$ is a plane Frenet curve in a 3-dimensional
$C_{12}$-manifold. That is $\tau=0$ and the equations of motion (3.1) becomes

$\left\\{\begin{array}[]{lll}\nabla_{\dot{\gamma}}E_{1}=\kappa E_{2},\\\
\nabla_{\dot{\gamma}}E_{2}=-\kappa E_{1},\\\
\nabla_{\dot{\gamma}}E_{3}=0.\end{array}\right.$ (5.4)

Putting $\eta(\dot{\gamma})=\sigma$ then, since $\dot{\gamma}$ is different
from $\xi$ and both of them are unit vectors we observe that
$0\leq|g(\xi,\dot{\gamma})=\sigma|<1$ on $M$. We note that $\dot{\gamma}$ is
not collinear with $\xi$. So, it can be verified that the vector fields

$E_{1}=\dot{\gamma},\qquad
E_{2}=\frac{\varphi\dot{\gamma}}{\sqrt{1-\sigma^{2}}},\qquad
E_{3}=\frac{\xi-\sigma\dot{\gamma}}{\sqrt{1-\sigma^{2}}},$ (5.5)

form an orthonormal frame along $\gamma$ and consequently, we can write

$\psi=\omega(\dot{\gamma})E_{1}+\frac{\omega\varphi\dot{\gamma})}{\sqrt{1-\sigma^{2}}}E_{2}-\frac{\sigma\omega(\dot{\gamma})}{\sqrt{1-\sigma^{2}}}E_{3}.$
(5.6)

From (5.5) and (5.6), one can get

$\displaystyle\nabla_{\dot{\gamma}}E_{3}$ $\displaystyle=$
$\displaystyle\nabla_{\dot{\gamma}}\Big{(}\frac{\xi-\sigma\dot{\gamma}}{\sqrt{1-\sigma^{2}}}\Big{)}$
$\displaystyle=$
$\displaystyle\frac{-1}{\sqrt{1-\sigma^{2}}}\Big{(}\dot{\sigma}+\sigma\omega(\dot{\gamma})\Big{)}E_{1}-\frac{\sigma}{1-\sigma^{2}}\Big{(}\omega(\varphi\dot{\gamma})+\kappa\sqrt{1-\sigma^{2}}\Big{)}E_{2}$
$\displaystyle+\frac{\sigma}{1-\sigma^{2}}\Big{(}\dot{\sigma}+\sigma\omega(\dot{\gamma})\Big{)}E_{3}.$

Since $\nabla_{\dot{\gamma}}E_{3}=0$ then, we get

$\left\\{\begin{array}[]{ll}\dot{\sigma}+\sigma\omega(\dot{\gamma})=0\\\
\sigma\big{(}\omega(\varphi\dot{\gamma})+\kappa\sqrt{1-\sigma^{2}})=0.\end{array}\right.$
(5.7)

This establishes the following theorem:

######  Theorem 5.2.

Let $M$ be a 3-dimensional $C_{12}$-manifold and $\gamma:I\rightarrow M$ a
plane Frenet curve in $M$ (i.e. $\tau=0$), set $\sigma=\eta(\dot{\gamma})$. If
at a certain point $t_{0}\in I$, $\sigma(t_{0})=0$, then $\gamma$ is a
Legendre curve.

##  6 Biharmonic curves on 3-dimensional $C_{12}$-manifold

The purpose of this last section is to study the curves $\gamma:I\rightarrow
M$ which are biharmonic, i.e. which satisfy $\Delta^{2}\gamma=0$ where
$\Delta$ is the Laplacian of $I$ i.e. $\Delta=-\frac{d^{2}}{ds^{2}}$ with $s$
is the arclength parameter. in [7], Chen defined a biharmonic submanifold
$N\subset\mathbb{E}^{n}$ of the Euclidean space as its mean curvature vector
field $H$ satisfies $\Delta H=0$.
We note $H=\nabla_{\dot{\gamma}}\dot{\gamma}$, then

$\Delta H=-\nabla_{\dot{\gamma}}\nabla_{\dot{\gamma}}H=0.$

Let $\gamma$ be a Legendre curve 3-dimensional $C_{12}$-manifold parametrized
by arc length and its associated Frenet frame is
$\\{\dot{\gamma},\varphi\dot{\gamma},\xi\\}$. So, we have

$H=\nabla_{\dot{\gamma}}\dot{\gamma}=\kappa\varphi\dot{\gamma}.$

With the help of (4.2), we get

$\nabla_{\dot{\gamma}}H=\dot{\kappa}\varphi\dot{\gamma}-\kappa^{2}\dot{\gamma},$

then,

$\displaystyle\Delta H$ $\displaystyle=$
$\displaystyle-\nabla_{\dot{\gamma}}\big{(}\dot{\kappa}\varphi\dot{\gamma}-\kappa^{2}\dot{\gamma}\big{)}$
$\displaystyle=$ $\displaystyle
3\dot{\kappa}\kappa\dot{\gamma}-(\ddot{\kappa}-\kappa^{3})\varphi\dot{\gamma},$

therefore, $\Delta H=0$ if and only if

$\left\\{\begin{array}[]{ll}\dot{\kappa}\kappa=0\\\
\ddot{\kappa}-\kappa^{3}=0.\end{array}\right.$ (6.1)

Hence, $\kappa=0$ and we have the following result:

######  Theorem 6.1.

All biharmonic Legendre curves $\gamma$ in a 3-dimensional $C_{12}$-manifold
are straight lines, i.e. totally geodesic.

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