# Some characterizations of Rectifying and osculating curves on a smooth
immersed surface

Absos Ali Shaikh∗ and Pinaki Ranjan Ghosh
Department of Mathematics,
University of Burdwan, Golapbag,
Burdwan-713104,
West Bengal, India<EMAIL_ADDRESS><EMAIL_ADDRESS>
Department of Mathematics,
University of Burdwan, Golapbag,
Burdwan-713104,
West Bengal, India<EMAIL_ADDRESS>
###### Abstract.

The present paper deals with some characterizations of rectifying and
osculating curves on a smooth surface with respect to the reference frame
$\\{\vec{T},\ \vec{N},\ \vec{T}\times\vec{N}\\}$. We have computed the
components of position vectors of rectifying and osculating curves along
$\vec{T},\ \vec{N},\ \vec{T}\times\vec{N}$ and then investigated their
invariancy under isometry of surfaces, and it is shown that they are invariant
iff either the normal curvature of the curve is invariant or the position
vector of the curve is in the direction of the tangent vector to the curve.

††footnotetext:  ∗ Corresponding author.
$\mathbf{2010}$ Mathematics Subject Classification: 53C22, 53A04, 53A05.
Key words and phrases: Rectifying curves, osculating curves, isometry of
surfaces, normal curvature, first fundamental form, second fundamental form.

##  1\. Introduction

When we discuss about a smooth curves in the Euclidean space $\mathbb{R}^{3}$,
one of the beautiful tool is Serret-Frenet frame, consists of three mutually
orthogonal unit vectors $\\{\vec{t},\ \vec{n},\ \vec{b}\\}$ such that
$\\{\vec{t},\ \vec{n}\\}$, $\\{\vec{n},\ \vec{b}\\}$ and $\\{\vec{b},\
\vec{t}\\}$ are respectively spanned three mutually orthogonal planes named as
osculating, normal and rectifying plane in the Euclidean space
$\mathbb{R}^{3}$. Now curves with position vectors lie in the above defined
three planes are respectively called osculating, normal and rectifying curves.
But whenever we consider to study a curve on a smooth surface immersed in
$\mathbb{R}^{3}$, at every point of the curve another three orthogonal unit
vectors $\\{\vec{T},\ \vec{N},\ \vec{T}\times\vec{N}\\}$ comes naturally,
where $\vec{T}$ is any tangent vector to the surface at that point and
$\vec{N}$ being the normal to the surface.

Bang-Yen Chen ([3]) introduced and characterized the rectifying curves in
$\mathbb{R}^{3}$ and also studied ([4]) the relationship among rectifying
curves, centrodes and extremal curves. For more interesting properties of
osculating, normal and rectifying curves we refer the reader to see [5], [6]
and [8].

In $2011$ Camci et. al. ([7]) first studied curves on a surface immersed in
$\mathbb{R}^{3}$ whose position vectors lie in the plane spanned by
$\\{\vec{T},\ \vec{N}\\}$, $\\{\vec{N},\ \vec{T}\times\vec{N}\\}$,
$\\{\vec{T},\ \vec{T}\times\vec{N}\\}$ and characterized such curves. The
present authors ([9], [10], [12]) studied rectifying, osculating and normal
curves on a smooth immersed surface in $\mathbb{R}^{3}$ and obtained their
characterizations under isometry of surfaces. The present authors ([11]) also
studied curves on a surface whose position vectors lie in its tangent plane.

Motivating by the above studies, the objective of this paper is to study
rectifying and osculating curves on a smooth immersed surface with respect to
the orthogonal frame $\\{\vec{T},\ \vec{N},\ \vec{T}\times\vec{N}\\}$. In
section 2 we discuss some basic definitions and elementary results. Section 3
and 4 are respectively deal with the characterization of rectifying and
osculating curves on a smooth surface with respect to the reference frame
$\\{\vec{T},\ \vec{N},\ \vec{T}\times\vec{N}\\}$. We obtain a sufficient
condition for which component of rectifying curves along $\vec{T}$ and
$\vec{T}\times\vec{N}$ respectively to be invariant under rectifying curve
preserving isometry of surfaces (see Theorem 3.1., Theorem 3.2.). It is also
shown that the component of the position vector of an osculating curve along
$\vec{T}\times\ \vec{N}$ is invariant under osculating curve preserving
isometry (see Theorem $4.2.$).

##  2\. preliminaries

Let $\gamma(s)$ be an arc length parametrized curve on a surface patch
$\phi(u,v)$ of a smooth surface $S$. The tangent, normal and binormal vector
to the curve $\gamma(s)$ at any point $\gamma(s)$ are respectively given by

$\displaystyle\vec{t}(s)$ $\displaystyle=$
$\displaystyle\phi_{u}u^{\prime}+\phi_{v}v^{\prime},$
$\displaystyle\vec{n}(s)$ $\displaystyle=$
$\displaystyle\frac{1}{\kappa(s)}(\phi_{u}u^{\prime\prime}+\phi_{v}v^{\prime\prime}+\phi_{uu}u^{\prime
2}+2\phi_{uv}u^{\prime}v^{\prime}+\phi_{vv}v^{\prime 2}),$
$\displaystyle\vec{b}(s)$ $\displaystyle=$
$\displaystyle\frac{1}{\kappa(s)}\Big{[}\\{v^{\prime\prime}u^{\prime}-u^{\prime\prime}v^{\prime}\\}\vec{N}+u^{\prime
3}(\phi_{u}\times\phi_{uu})+2u^{\prime 2}v^{\prime}(\phi_{u}\times\phi_{uv})$
$\displaystyle+u^{\prime}v^{\prime 2}(\phi_{u}\times\phi_{vv})+u^{\prime
2}v^{\prime}(\phi_{v}\times\phi_{uu})+2u^{\prime}v^{\prime
2}(\phi_{v}\times\phi_{uv})+v^{\prime 3}(\phi_{v}\times\phi_{vv})\Big{]}.$

We note that $\gamma(s)$ is a rectifying (respectively, osculating) curve if
$\gamma(s)=\lambda\vec{t}+\mu\vec{b}$ (respectively,
$\alpha(s)=\lambda\vec{t}+\mu\vec{n}$), where $\lambda$, $\mu$ are smooth
functions.

######  Definition 2.1.

Two surfaces $S$ and $\bar{S}$ are isometric if there exists a diffeomorphism
$f:S\rightarrow\bar{S}$ such that the length of any curve preserves under $f$.

######  Definition 2.2.

Let $\gamma(s)$ be a curve parametrized by arc length on a smooth surface $S$
in $\mathbb{R}^{3}$. Then $\gamma^{\prime\prime}(s)$ lies in the plane
perpendicular to $\gamma^{\prime}(s)$, i.e., lies in the plane spanned by the
vectors $\vec{N}$ and $\gamma^{\prime}(s)\times\vec{N}$. So we have two non-
zero components of $\gamma^{\prime\prime}(s)$ along $\vec{N}$ and
$\gamma^{\prime}(s)\times\vec{N}$. Now

$\displaystyle\kappa_{g}(s)=\gamma^{\prime\prime}(s)\cdot(\gamma^{\prime}(s)\times\vec{N}),$
$\displaystyle\kappa_{n}(s)=\gamma^{\prime\prime}(s)\cdot\vec{N},$

where $\kappa_{g}(s)$ and $\kappa_{n}(s)$ are respectively known as geodesic
and normal curvature of $\gamma(s)$. Since
$\gamma^{\prime\prime}(s)=\kappa(s)\vec{n}(s)$, hence
$\kappa_{n}(s)=Lu^{\prime 2}+2Mu^{\prime}v^{\prime}+Nv^{\prime 2}$, where $L$,
$M$ and $N$ are coefficients of second fundamental form of the surface. For
more information of isometry, fundamental forms, normal curvature we refer the
reader to see [1] and [2].

##  3\. rectifying curves on a surface

It is well-known that at every point of any smooth curve in $\mathbb{R}^{3}$
there exists a Serret-Frenet frame. Now if the curve lies on a surface $S$
then there exists another frame of reference
$\\{\vec{T},\vec{N},\vec{T}\times\vec{N}\\}$ at each point of the curve
related to the surface. Throughout this section and the next section we will
deduce the components of position vectors of rectifying and osculating curves
along $\vec{T},\ \vec{N},\ \vec{T}\times\vec{N}$ and then investigate their
invariancy under surface isometry. The equation of a rectifying curve on $S$
is given by

(3.1) $\displaystyle\gamma(s)$ $\displaystyle=$
$\displaystyle\lambda(s)(\phi_{u}u^{\prime}+\phi_{v}v^{\prime})+\frac{\mu(s)}{\kappa(s)}\Big{[}u^{\prime
3}(\phi_{u}\times\phi_{uu})+2u^{\prime
2}v^{\prime}(\phi_{u}\times\phi_{uv})+u^{\prime}v^{\prime
2}(\phi_{u}\times\phi_{vv})$ $\displaystyle+u^{\prime
2}v^{\prime}(\phi_{v}\times\phi_{uu})+2u^{\prime}v^{\prime
2}(\phi_{v}\times\phi_{uv})+v^{\prime
3}(\phi_{v}\times\phi_{vv})+(v^{\prime\prime}u^{\prime}-v^{\prime}2u^{\prime\prime})\vec{N}\Big{]},$

for some functions $\lambda(s)$ and $\mu(s)$.

######  Theorem 3.1.

Let $f:S\rightarrow\bar{S}$ be an isometry. If $\gamma$ and $\bar{\gamma}$ are
rectifying curves on $S$ and $\bar{S}$ respectively, then for the component of
$\gamma(s)$ along any tangent vector $\vec{T}=a\phi_{u}+b\phi_{v},\ a,\
b\in\mathbb{R}$, to the surface $S$ at $\gamma(s)$, the following holds:

(3.2)
$\displaystyle\bar{\gamma}\cdot\vec{\bar{T}}-\gamma\cdot\vec{T}=\frac{\mu(s)}{\kappa(s)}(av^{\prime}+bu^{\prime})\Big{(}\bar{\kappa}_{n}(s)-\kappa_{n}(s)\Big{)}.$

###### Proof.

From $(\ref{r1})$, we see that

(3.3)
$\gamma\cdot\phi_{u}=\lambda(Eu^{\prime}+Fv^{\prime})+\frac{\mu(s)}{\kappa(s)}\Big{[}Lu^{\prime
2}v^{\prime}+2Mv^{\prime 2}u^{\prime}+Nv^{\prime 3}\Big{]}.$

Similarly for the rectifying curve $\bar{\gamma}(s)$ on $\bar{S}$, we have

$\bar{\gamma}\cdot\bar{\phi}_{u}=\lambda(\bar{E}u^{\prime}+\bar{F}v^{\prime})+\frac{\mu(s)}{\kappa(s)}\Big{[}\bar{L}u^{\prime
2}v^{\prime}+2\bar{M}v^{\prime 2}u^{\prime}+\bar{N}v^{\prime 3}\Big{]}.$

Since $f$ is isometry, hence $\bar{E}=E$, $\bar{F}=F$ and $\bar{G}=G$.
Therefore

(3.4)
$\bar{\gamma}\cdot\bar{\phi}_{u}=\lambda(Eu^{\prime}+Fv^{\prime})+\frac{\mu(s)}{\kappa(s)}\Big{[}\bar{L}u^{\prime
2}v^{\prime}+2\bar{M}v^{\prime 2}u^{\prime}+\bar{N}v^{\prime 3}\Big{]}.$

Taking the difference of $(\ref{rt2})$ and $(\ref{rt3})$, we get

(3.5)
$\bar{\gamma}\cdot\bar{\phi}_{u}-\gamma\cdot\phi_{u}=\frac{v^{\prime}\mu(s)}{\kappa(s)}\Big{(}\bar{\kappa}_{n}(s)-\kappa_{n}(s)\Big{)}.$

Similarly the following relation holds:

(3.6)
$\bar{\gamma}\cdot\bar{\phi}_{v}-\gamma\cdot\phi_{v}=\frac{u^{\prime}\mu(s)}{\kappa(s)}\Big{(}\bar{\kappa}_{n}(s)-\kappa_{n}(s)\Big{)}.$

Now with the help of $(\ref{rt4})$ and $(\ref{rt5})$ we get

$\displaystyle\bar{\gamma}\cdot\vec{\bar{T}}-\gamma\cdot T$ $\displaystyle=$
$\displaystyle\bar{\gamma}\cdot(a\bar{\phi}_{u}+b\bar{\phi}_{v})-\gamma\cdot(a\phi_{u}+b\phi_{v}),$
$\displaystyle=$ $\displaystyle
a(\bar{\gamma}\cdot\bar{\phi}_{u}-\gamma\cdot\phi_{u})+b(\bar{\gamma}\cdot\bar{\phi}_{v}-\gamma\cdot\phi_{v}),$
$\displaystyle=$ $\displaystyle
a\frac{v^{\prime}\mu(s)}{\kappa(s)}\Big{(}\bar{\kappa}_{n}(s)-\kappa_{n}(s)\Big{)}+b\frac{u^{\prime}\mu(s)}{\kappa(s)}\Big{(}\bar{\kappa}_{n}(s)-\kappa_{n}(s)\Big{)},$
$\displaystyle=$
$\displaystyle\frac{\mu(s)}{\kappa(s)}(av^{\prime}+bu^{\prime})\Big{(}\bar{\kappa}_{n}(s)-\kappa_{n}(s)\Big{)}.$

This proves our claim. ∎

######  Theorem 3.2.

Let $f:S\rightarrow\bar{S}$ be an isometry. If $\gamma$ and $\bar{\gamma}$ are
rectifying curves on $S$ and $\bar{S}$ respectively, then for the component of
$\gamma(s)$ along $\vec{T}\times\vec{N}$, the following holds:

(3.7)
$\bar{\gamma}\cdot(\vec{\bar{T}}\times\vec{\bar{N}})-\gamma\cdot(\vec{T}\times\vec{N})=\frac{\mu(s)}{\kappa(s)}\Big{(}\bar{\kappa}_{n}(s)-\kappa_{n}(s)\Big{)}\Big{\\{}vaF-
uaEb+vbG-ubF\Big{\\}}.$

###### Proof.

The component of $\gamma(s)$ along $\vec{T}\times\vec{N}$ is given by

$\displaystyle\gamma\cdot(\vec{T}\times\vec{N})$ $\displaystyle=$
$\displaystyle\gamma\cdot\\{(a\varphi_{u}+b\varphi_{v})\times\vec{N}\\},$
$\displaystyle=$
$\displaystyle\gamma\cdot\\{Fa\phi_{u}-Ea\phi_{v}+Gb\phi_{u}-Fb\phi_{v}\\},$
$\displaystyle=$
$\displaystyle(Fa+Gb)\gamma\cdot\phi_{u}-(Ea+Fb)\gamma\cdot\phi_{v}.$

Hence

$\bar{\gamma}\cdot(\vec{\bar{T}}\times\vec{\bar{N}})-\gamma\cdot(\vec{T}\times\vec{N})=(Fa+Gb)(\bar{\gamma}\cdot\bar{\phi}_{u}-\gamma\cdot\phi_{u})-(Ea+Fb)(\bar{\gamma}\cdot\bar{\phi}_{v}-\gamma\cdot\phi_{v}).$

Now using $(\ref{rt4})$ and $(\ref{rt5})$ we get

(3.8)
$\bar{\gamma}\cdot(\vec{\bar{T}}\times\vec{\bar{N}})-\gamma\cdot(\vec{T}\times\vec{N})=\frac{\mu(s)}{\kappa(s)}\Big{(}\bar{\kappa}_{n}(s)-\kappa_{n}(s)\Big{)}\Big{\\{}vaF-
uaE+vbG-ubF\Big{\\}}.$

This proves the result. ∎

######  Corollary 3.2.1.

Let $f:S\rightarrow\bar{S}$ be an isometry and $\gamma(s)$ be an rectifying
curve on $S$. Then the component of the rectifying curve $\gamma(s)$ along
$\vec{T}\times\vec{N}$ (respectively, $\vec{T}$) is invariant iff any one of
the following holds:

  * (i)

the position vector of $\gamma(s)$ is in the direction of tangent vector to
$\gamma$.

  * (ii)

The normal curvature is invariant.

###### Proof.

From $(\ref{rt6})$,
$\bar{\gamma}(s)\cdot(\vec{\bar{T}}\times\vec{\bar{N}})=\gamma(s)\cdot(\vec{T}\times\vec{N})$
(respectively, $\bar{\gamma}(s)\cdot\vec{\bar{T}}=\gamma(s)\cdot T$) iff

$\displaystyle\frac{\mu(s)}{\kappa(s)}\Big{(}\bar{\kappa}_{n}(s)-\kappa_{n}(s)\Big{)}\Big{\\{}vaF-
uaE+vbG-ubF\Big{\\}}=0$ $\displaystyle\Big{(}respectively,\
\frac{\mu(s)}{\kappa(s)}(av^{\prime}+bu^{\prime})\Big{(}\bar{\kappa}_{n}(s)-\kappa_{n}(s)\Big{)}=0\Big{)}$
$\displaystyle\text{i.e., iff }\mu(s)=0\text{ or
}\bar{\kappa}_{n}(s)=\kappa_{n}(s).$

If $\mu(s)=0$ then from the definition of rectifying curve, we see that
$\gamma(s)=\lambda(s)\vec{t}(s)$, i.e., the position vector of the rectifying
curve $\gamma(s)$ is in the direction of tangent vector to itself.
Otherwise if $\mu(s)\neq 0$, then $\bar{\kappa}_{n}(s)=\kappa_{n}(s)$, i.e.,
the normal curvature is invariant. ∎

######  Corollary 3.2.2.

Let $f:S\rightarrow\bar{S}$ be an isometry and $\gamma(s)$ be an rectifying
curve on $S$. If the component of the rectifying curve $\gamma(s)$ along
$\vec{T}\times\vec{N}$ (respectively, $\vec{T}$) is invariant and the position
vector of $\gamma(s)$ is not in the direction of the tangent vector to
$\gamma(s)$, then $\gamma(s)$ is asymptotic iff $\bar{\gamma}(s)$ is
asymptotic.

###### Proof.

From Corollary $3.2.1.$,
$\bar{\gamma}(s)\cdot(\vec{\bar{T}}\times\vec{\bar{N}})=\gamma(s)\cdot(\vec{T}\times\vec{N})$
(respectively $\bar{\gamma}(s)\cdot\vec{\bar{T}}=\gamma(s)\cdot T$) and the
position vector of $\gamma(s)$ is not in the direction of the tangent vector
to $\gamma(s)$ iff $\kappa_{n}(s)=\bar{\kappa}_{n}(s)$.
Therefore $\gamma(s)$ is asymptotic iff $\kappa_{n}(s)=0$, i.e., iff
$\bar{\kappa}_{n}(s)=0$, i.e., iff $\bar{\gamma}(s)$ is asymptotic. ∎

##  4\. osculating curves on a surface

The equation of an osculating curve on a surface patch $\phi(u,v)$ of $S$ is
given by

(4.1)
$\alpha(s)=\lambda_{1}(s)(\phi_{u}u^{\prime}+\phi_{v}v^{\prime})+\frac{\lambda_{2}(s)}{\kappa(s)}\Big{\\{}(\phi_{uu}u^{\prime
2}+2\phi_{uv}u^{\prime}v^{\prime}+\phi_{vv}v^{\prime
2})+(\phi_{u}u^{\prime\prime}+\phi_{v}v^{\prime\prime})\Big{\\}},$

for some functions $\lambda_{1}(s)$ and $\lambda_{2}(s)$.

######  Theorem 4.1.

Let $f:S\rightarrow\bar{S}$ be an isometry. If $\alpha$ and $\bar{\alpha}$ are
respectively osculating curves on $S$ and $\bar{S}$, then for the normal
component of $\alpha(s)$, we have

(4.2)
$\bar{\alpha}\cdot\vec{\bar{N}}-\alpha\cdot\vec{N}=\frac{\lambda_{2}(s)}{\kappa(s)}\Big{(}\bar{\kappa}_{n}(s)-\kappa_{n}(s)\Big{)}.$

###### Proof.

From $(\ref{o1})$, we see that

(4.3) $\displaystyle\alpha\cdot\vec{N}$ $\displaystyle=$
$\displaystyle\lambda_{1}(s)(\phi_{u}u^{\prime}+\phi_{v}v^{\prime})+\frac{\lambda_{2}(s)}{\kappa(s)}\Big{\\{}(\phi_{uu}u^{\prime
2}+2\phi_{uv}u^{\prime}v^{\prime}+\phi_{vv}v^{\prime 2})$
$\displaystyle+(\phi_{u}u^{\prime\prime}+\phi_{v}v^{\prime\prime})\Big{\\}}\cdot\vec{N},$
$\displaystyle=$
$\displaystyle\frac{\lambda_{2}(s)}{\kappa(s)}\Big{(}\phi_{uu}\cdot\vec{N}u^{\prime
2}+2\phi_{uv}\cdot\vec{N}u^{\prime}v^{\prime}+\phi_{vv}\cdot\vec{N}v^{\prime
2}\Big{)},$ $\displaystyle=$
$\displaystyle\frac{\lambda_{2}(s)}{\kappa(s)}\Big{(}Lu^{\prime
2}+2Mu^{\prime}v^{\prime}+Nv^{\prime 2}\Big{)}.$

Similarly for the isometry of $\alpha$ and $\vec{N}$, we have

(4.4)
$\bar{\alpha}\cdot\vec{\bar{N}}=\frac{\lambda_{2}(s)}{\kappa(s)}\Big{(}\bar{L}u^{\prime
2}+2\bar{M}u^{\prime}v^{\prime}+\bar{N}v^{\prime 2}\Big{)}.$

In view of $(\ref{o2})$ and $(\ref{o3})$, we obtain

$\displaystyle\bar{\alpha}\cdot\vec{\bar{N}}-\alpha\cdot\vec{N}$
$\displaystyle=$
$\displaystyle\frac{\lambda_{2}(s)}{\kappa(s)}\Big{(}\bar{L}u^{\prime
2}+2\bar{M}u^{\prime}v^{\prime}+\bar{N}v^{\prime
2}\Big{)}-\frac{\lambda_{2}(s)}{\kappa(s)}\Big{(}Lu^{\prime
2}+2Mu^{\prime}v^{\prime}+Nv^{\prime 2}\Big{)},$ $\displaystyle=$
$\displaystyle\frac{\lambda_{2}(s)}{\kappa(s)}\Big{(}\bar{\kappa}_{n}(s)-\kappa_{n}(s)\Big{)}.$

This proves our claim. ∎

######  Corollary 4.1.1.

Let $f:S\rightarrow\bar{S}$ be an isometry and $\alpha(s)$ be an osculating
curve on $S$. Then the normal component of the osculating curve $\alpha(s)$ is
invariant iff any one of the following holds:

  * (i)

the position vector of $\alpha(s)$ is in the direction of tangent vector to
$\alpha$.

  * (ii)

The normal curvature is invariant.

###### Proof.

From $(\ref{ot1})$, $\bar{\alpha}\cdot\vec{\bar{N}}=\alpha\cdot\vec{N}$ iff
$\frac{\lambda_{2}(s)}{\kappa(s)}\Big{(}\bar{\kappa}_{n}(s)-\kappa_{n}(s)\Big{)}=0$,

$\text{i.e., iff }\lambda_{2}(s)=0\text{ or
}\bar{\kappa}_{n}(s)=\kappa_{n}(s).$

If $\lambda_{2}(s)=0$ then from the definition of osculating curve, we have
$\alpha(s)=\lambda_{1}(s)\vec{t}(s)$, i.e., the position vector of the
osculating curve $\alpha(s)$ is in the direction of tangent vector to itself.
Otherwise if $\lambda_{2}(s)\neq 0$, then $\kappa_{n}(s)=\bar{\kappa}_{n}(s)$,
i.e., the normal curvature is invariant. ∎

######  Corollary 4.1.2.

Let $f:S\rightarrow\bar{S}$ be an isometry and $\alpha(s)$ be an osculating
curve on $S$. If the normal component of the osculating curve $\alpha(s)$ is
invariant and the position vector of $\alpha(s)$ is not in the direction of
the tangent vector to $\alpha$, then $\alpha(s)$ is asymptotic iff
$\bar{\alpha}(s)$ is asymptotic.

###### Proof.

From Corollary $4.1.1.$,
$\bar{\alpha}(s)\cdot\vec{\bar{N}}=\alpha(s)\cdot\vec{N}$ and the position
vector of $\alpha(s)$ is not in the direction of the tangent vector to
$\alpha(s)$ iff $\kappa_{n}(s)=\bar{\kappa}_{n}(s)$. Therefore $\alpha(s)$ is
asymptotic iff $\kappa_{n}(s)=0$, i.e., iff $\bar{\kappa}_{n}(s)=0$, i.e., iff
$\bar{\alpha}(s)$ is asymptotic. ∎

######  Theorem 4.2.

Let $f:S\rightarrow\bar{S}$ be an isometry. If $\alpha$ and $\bar{\alpha}$ are
respectively osculating curves on $S$ and $\bar{S}$, then the component of the
osculating curve $\alpha(s)$ along $\vec{T}\times\vec{N}$ is invariant, i.e.,
$\alpha\cdot(\vec{T}\times\vec{N})=\bar{\alpha}\cdot(\vec{\bar{T}}\times\vec{\bar{N}})$

###### Proof.

Since $f$ is isometry, for the surface patch $\phi(u,v)$ of the surface $S$,
$f\circ\phi(u,v)$ is also a surface patch for the surface $\bar{S}$. Let the
coefficients of first fundamental form of $\phi$ and $f\circ\phi=\bar{\phi}$
be $\\{E,F,G\\}$ and $\\{\bar{E},\bar{F},\bar{G}\\}$ respectively. Then

(4.5) $G=\bar{G},\ F=\bar{F}\text{ and }E=\bar{E},$

and

(4.6) $\bar{E}_{u}=E_{u},\ \bar{E}_{v}=E_{v},\ \bar{F}_{u}=F_{u},\
\bar{F}_{v}=F_{v},\ \bar{G}_{u}=G_{u},\text{ and }\bar{G}_{v}=G_{v}.$

Now

(4.7) $\begin{cases}\phi_{uu}\cdot\phi_{u}=\frac{1}{2}E_{u},\
\phi_{uv}\cdot\phi_{u}=\frac{1}{2}E_{v},\
\phi_{uu}\cdot\phi_{v}=F_{u}-\frac{1}{2}E_{v},\\\
\phi_{uv}\cdot\phi_{v}=\frac{1}{2}G_{u},\
\phi_{vv}\cdot\phi_{v}=\frac{1}{2}G_{v},\
\phi_{vv}\cdot\phi_{u}=F_{v}-\frac{1}{2}G_{u}.\end{cases}$

From $(\ref{o1})$ and $(\ref{fff3})$, we have

$\displaystyle\alpha\cdot(\vec{T}\times\vec{N})$ $\displaystyle=$
$\displaystyle\alpha\cdot\\{(a\phi_{u}+b\phi_{v})\times\vec{N}\\},$
$\displaystyle=$
$\displaystyle\alpha\cdot(Fa\phi_{u}-Ea\phi_{v}+Gb\phi_{u}-Fb\phi_{v}),$
$\displaystyle=$
$\displaystyle\Big{[}\lambda_{1}(s)(\phi_{u}u^{\prime}+\phi_{v}v^{\prime})+\frac{\lambda_{2}(s)}{\kappa(s)}\Big{\\{}(\phi_{uu}u^{\prime
2}+2\phi_{uv}u^{\prime}v^{\prime}+\phi_{vv}v^{\prime 2})$
$\displaystyle+(\phi_{u}u^{\prime\prime}+\phi_{v})v^{\prime\prime}\Big{\\}}\Big{]}\cdot(Fa\phi_{u}-Ea\phi_{v}+Gb\phi_{u}-Fb\phi_{v}),$
$\displaystyle=$
$\displaystyle\lambda_{1}(s)(av^{\prime}-bu^{\prime})(F^{2}-EG)+\frac{\lambda_{2}(s)}{\kappa(s)}\Big{[}u^{\prime
2}a\Big{\\{}\frac{1}{2}FE_{u}-E\Big{(}F_{u}-\frac{1}{2}E_{v}\Big{)}\Big{\\}}$
$\displaystyle+u^{\prime
2}b\Big{\\{}\frac{1}{2}GE_{u}-F\Big{(}F_{u}-\frac{1}{2}E_{v}\Big{)}\Big{\\}}+2u^{\prime}v^{\prime}a\Big{\\{}\frac{1}{2}FE_{u}-\frac{1}{2}EG_{u}\Big{\\}}2u^{\prime}v^{\prime}b\Big{\\{}\frac{1}{2}GE_{u}$
$\displaystyle-\frac{1}{2}FG_{u}\Big{\\}}+v^{\prime
2}a\Big{\\{}F\Big{(}F_{v}-\frac{1}{2}G_{u}\Big{)}-\frac{1}{2}EG_{v}\Big{\\}}-v^{\prime
2}b\Big{\\{}G\Big{(}F_{v}-\frac{1}{2}G_{u}\Big{)}-\frac{1}{2}FG_{v}\Big{\\}}\Big{]}.$

By the virtue of $(\ref{fff1})$, $(\ref{fff2})$ and $(\ref{fff3})$, the last
equation yields

$\displaystyle\alpha\cdot(\vec{T}\times\vec{N})$ $\displaystyle=$
$\displaystyle\lambda_{1}(s)(av^{\prime}-bu^{\prime})(\bar{F}^{2}-\bar{E}\bar{G})+\frac{\lambda_{2}(s)}{\kappa(s)}\Big{[}u^{\prime
2}a\Big{\\{}\frac{1}{2}\bar{F}\bar{E}_{u}-\bar{E}\Big{(}\bar{F}_{u}-\frac{1}{2}\bar{E}_{v}\Big{)}\Big{\\}}$
$\displaystyle+u^{\prime
2}b\Big{\\{}\frac{1}{2}\bar{G}\bar{E}_{u}-\bar{F}\Big{(}\bar{F}_{u}-\frac{1}{2}\bar{E}_{v}\Big{)}\Big{\\}}+2u^{\prime}v^{\prime}a\Big{\\{}\frac{1}{2}\bar{F}\bar{E}_{u}-\frac{1}{2}\bar{E}\bar{G}_{u}\Big{\\}}2u^{\prime}v^{\prime}b\Big{\\{}\frac{1}{2}\bar{G}\bar{E}_{u}$
$\displaystyle-\frac{1}{2}\bar{F}\bar{G}_{u}\Big{\\}}+v^{\prime
2}a\Big{\\{}\bar{F}\Big{(}\bar{F}_{v}-\frac{1}{2}\bar{G}_{u}\Big{)}-\frac{1}{2}\bar{E}\bar{G}_{v}\Big{\\}}-v^{\prime
2}b\Big{\\{}\bar{G}\Big{(}\bar{F}_{v}-\frac{1}{2}\bar{G}_{u}\Big{)}-\frac{1}{2}\bar{F}\bar{G}_{v}\\}\Big{]},$
$\displaystyle=$
$\displaystyle\bar{\alpha}\cdot(\vec{\bar{T}}\times\vec{\bar{N}}).$

This proves the result. ∎

##  5\. acknowledgment

The second author greatly acknowledges to The University Grants Commission,
Government of India for the award of Junior Research Fellow.

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# Submanifolds with harmonic mean curvature vector field in contact
$3$-manifolds 111Colloquium Mathematicum 100 (2004), no. 2, 163–179. Minor
misprints are corrected.

Jun-ichi Inoguchi

###### Abstract

Biharmonic or polyharmonic curves and surfaces in $3$-dimensional contact
manifolds are investigated.

AMS Mathematics Subject Classification: 2000 53C42 53D10

Keywords and Phrases: Biharmonicity, polyharmonicity, Sasaki manifolds

## Introduction

This paper concerns curves and surfaces in 3-dimensional contact manifolds,
whose mean curvature vector field is in the kernel of certain elliptic
differential operators.

First we study submanifolds whose mean curvature vector field is in the kernel
of Laplacian (submanifolds with harmonic mean curvature vector fields).

The study of such submanifolds is inspired by a conjecture of Bang-yen Chen
[14]:

Harmonicity of the mean curvature vector field implies harmonicity of the
immersion ?

The harmonicity equation $\Delta\mathbb{H}=0$ for the mean curvature vector
field $\mathbb{H}$ of an immersed submanifold
$\mathbf{x}:M^{m}\to\mathbf{E}^{n}$ in Euclidean $n$-space is equivalent to
the biharmonicity of the immersion: $\Delta\Delta\mathbf{x}=0$, since
$\Delta\mathbf{x}=-m\mathbb{H}$.

A submanifold $\mathbf{x}:M\to\mathbf{E}^{n}$ is said to be a biharmonic
submanifold if $\Delta\mathbb{H}=0$.

In 1985, Chen proved the nonexistence of proper biharmonic surfaces in
Euclidean 3-space. The conjecture by Chen is still open.

Some partial and positive answers have been obtained by several authors
[16]-[19], [25]-[27].

The biharmonicity equation is regarded as a special case of the following
condition:

$\Delta\mathbb{H}=\lambda\>\mathbb{H},\ \lambda\in\mathbf{R}.$

Namely the mean curvature vector field is an eigenfunction of the Laplacian.

The study of Euclidean submanifolds with $\Delta\mathbb{H}=\lambda\mathbb{H}$
was initiated by Chen in 1988 (See [14]).

It is known that submanifolds in $\mathbf{E}^{n}$ satisfying
$\Delta\mathbb{H}=\lambda\mathbb{H}$ are either biharmonic ($\lambda=0$), of
$1$-type or null $2$-type. In particular all surfaces in $\mathbf{E}^{3}$ with
$\Delta\mathbb{H}=\lambda\mathbb{H}$ are of constant mean curvature. Moreover
a surface in $\mathbf{E}^{3}$ satisfies $\Delta\mathbb{H}=\lambda\mathbb{H}$
if and only if it is minimal, an open portion of a totally umbilical sphere or
an open portion of a circular cylinder.

F. Defever [17] showed that hypersurfaces satisfying
$\Delta\mathbb{H}=\lambda\mathbb{H}$ are of constant mean curvature. Note that
Chen [12], [13] studied spacelike submanifolds with
$\Delta\mathbb{H}=\lambda\mathbb{H}$ in Minkowski space, hyperbolic space or
de Sitter space. M. Barros and O. J. Garay showed that Hopf cylinders in
$S^{3}$ with $\Delta\mathbb{H}=\lambda\mathbb{H}$ are Hopf cylinders over
circles in the $2$-sphere $S^{2}$. A. Ferrández, P. Lucas and M. A. Meroño
[24] studied such submanifolds in anti de Sitter $3$-space $H^{3}_{1}$.

In non-constant curvature ambient spaces, results on biharmonic submanifolds
are very few.

Recently, T. Sasahara [37]–[38] studied Legendre surfaces in the Sasakian
space form $\mathbf{R}^{5}(-3)$ satisfying
$\Delta\mathbb{H}=\lambda\mathbb{H}$. Moreover Sasahara introduced the notion
of “$\varphi$-position vector field” and “$\varphi$-mean curvature vector
field” for submanifolds in Sasakian space form $\mathbf{R}^{2n+1}(-3)$.
Sasahara investigated submanifolds in $\mathbf{R}^{2n+1}(-3)$ whose
$\varphi$-mean curvature vector field $\mathbb{H}_{\varphi}$ satisfies
$\Delta\mathbb{H}_{\varphi}=\lambda\mathbb{H}_{\varphi}$. In particular he
classified curves and surfaces in $\mathbf{R}^{3}(-3)$ with
$\Delta\mathbb{H}_{\varphi}=\lambda\mathbb{H}_{\varphi}$. Since both
$\mathbf{R}^{2n+1}(-3)$ and $S^{2n+1}$ are typical examples of Sasakian space
form, it seems to be interesting to study biharmonic submanifolds in general
Sasakian space forms.

Based on these observations, in the first part of this paper, we shall study
harmonicity of mean curvature vector fields of curves and surfaces in
3-dimensional Sasakian space forms. Several results for 3-dimensional sphere
$S^{3}$ due to Spanish research group (Barros, Garay Ferrández, Lucas and
Meroño) will be generalised to $3$-dimensional Sasakian space forms.

Next, in the second part, we shall study another “biharmonicity” suggested by
J. Eells and J. H. Sampson [23]. A smooth map $\phi:M\to N$ between Riemannian
manifolds is said to be a biharmonic map (or polyharmonic map of order $2$) if
its bitension field $\mathscr{T}_{2}(\phi)$ vanishes. In [9], “biharmonic”
curves and surfaces in $S^{3}$ are classified. We shall classify Legendre
curves and Hopf cylinders in $3$-dimensional Sasakian space forms, which are
biharmonic in this sense.

In particular we shall show the existence of non-minimal biharmonic Hopf
cylinders in Sasakian space forms of holomorphic sectional curvature greater
than $1$ (Berger spheres).

The author would like to thank Dr. Cezar Dumitru Oniciuc (University “AL. I.
Cuza ”) and Dr. Tooru Sasahara (Hokkaido University) for their useful
comments.

## Part I

##  1 Preliminaries

###  1.1 Contact manifolds

We begin by recalling fundamental ingredients of contact Riemannian geometry
from [7].

Let $M$ be a $(2n+1)$-manifold. A one form $\eta$ is called a contact form on
$M$ if $(d\eta)^{n}\wedge\eta\not=0$. A $(2n+1)$-manifold $M$ together with a
contact form is called a contact manifold. The contact distribution $D$ of
$(M,\eta)$ is defined by

$D=\left\\{X\in TM\ |\ \eta(X)=0\right\\}.$

On a contact manifold $(M,\eta)$, there exists a unique vector field $\xi$
such that

$\eta(\xi)=1,\ \ d\eta(\xi,\cdot)=0.$

This vector field $\xi$ is called the Reeb vector field or characteristic
vector field of $(M,\eta)$.

Moreover there exists an endomorphism field $\varphi$ and a Riemannian metric
$g$ on $M$ such that

(1) $\varphi^{2}=-I+\eta\otimes\xi,\ \eta(\xi)=1,$ (2) $g(\varphi X,\varphi
Y)=g(X,Y)-\eta(X)\eta(Y),\ \ g(\xi,\cdot)=\eta,$ (3) $d\eta(X,Y)=2g(X,\varphi
Y)$

for all vector fields $X,\ Y$ on $M$. On an almost contact manifold
$(M,\eta;\xi,\varphi)$, there exists a Riemannian metric $g$ satisfying (2).
Such a metric $g$ is called an compatible metric of $M$. A contact manifold
$(M,\eta)$ together with structure tensors $(\xi,\varphi,g)$ is called a
contact Riemannian manifold.

###### Proposition 1.1

Let $(M,\eta,\xi,\varphi,g)$ be a contact Riemannian manifold.
Then $M$ $\xi$ is a Killing vector field if and only if

(4) $\nabla_{X}\xi=-\varphi X,\ \ X\in\mathfrak{X}(M).$

Here $\nabla$ is the Levi-Civita connection of $(M,g)$.

###### Definition 1.1

A contact Riemannian manifold $(M,\eta,\xi,\varphi,g)$ is said to be a Sasaki
manifold if

(5) $(\nabla_{X}\varphi)Y=g(X,Y)\xi-\eta(Y)X,\ \ X,Y\in\mathfrak{X}(M).$

Note that on a Sasaki manifold, $\xi$ is a Killing vector field.

Let $(M,\eta;\xi,\varphi,g)$ be a contact Riemannian manifold. A tangent plane
at a point of $M$ is said to be a holomorphic plane if it is invariant under
$\varphi$. The sectional curvature of a holomorphic plane is called
holomorphic sectional curvature. If the sectional curvature function of $M$ is
constant on all holomorphic planes in $TM$, then $M$ is said to be of constant
holomorphic sectional curvature. Complete and connected Sasaki manifolds of
constant holomorphic sectional curvature are called Sasakian space forms. Let
us denote by $R$ the Riemannian curvature tensor field of the metric $g$ which
is defined by

$R(X,Y):=\nabla_{X}\nabla_{Y}-\nabla_{Y}\nabla_{X}-\nabla_{[X,Y]},\ \
X,Y\in\mathfrak{X}(M).$

When $(M,\eta;\xi,\varphi,g)$ is a Sasakian space form of constant holomorphic
sectional curvature $c$, then $R$ is described by the following formula:

$\displaystyle R(X,Y)Z$ $\displaystyle=$
$\displaystyle\frac{c+3}{4}\\{g(Y,Z)X-g(Z,X)Y\\}$
$\displaystyle+\frac{c-1}{4}\>\\{\eta(Z)\eta(X)Y-\eta(Y)\eta(Z)X$
$\displaystyle+g(Z,X)\eta(Y)\xi-g(Y,Z)\eta(X)\xi$ $\displaystyle-g(Y,\varphi
Z)\varphi X-g(Z,\varphi X)\varphi Y+2g(X,\varphi Y)\varphi Z\ \>\\}.$

Note that even if the holomorphic sectional curvature is negative, a Sasakian
space form is not negatively curved. In fact, the sectional curvature of plane
sections containing $\xi$ is $1$ on any Sasaki manifold.

It is known that every $3$-dimensional Sasakian space form is realised as a
Lie group together with a left invariant Sasaki structure. More precisely the
following is known (cf. [6]):

###### Proposition 1.2

Simply connected $3$-dimensional Sasakian space form of constant holomorphic
sectional curvature is isomorphic to

  1. (1)

special unitary group $\mathrm{SU}(2);$

  2. (2)

Heisenberg group $\mathbf{R}^{3}(-3);$

  3. (3)

the universal covering group of the special linear group
$\mathrm{SL}_{2}\mathbf{R}$

together with canonical left invariant Sasaki structure. In particular simply
connected Sasakian space form of constant holomorphic sectional curvature $1$
is the $\mathrm{SU}(2)$ with biinvariant metric of constant curvature $1$
(hence isometric to the unit $3$-sphere $S^{3}$).

###  1.2 Boothby-Wang fibration

Let $(M^{2n+1},\eta;\xi,\varphi,g)$ be a contact Riemannian manifold. Then $M$
is said to be regular if $\xi$ generates a one-parameter group $K$ of
isometries on $M$, such that the action of $K$ on $M$ is simply transitive.
Note that if $M$ is regular, then both $\varphi$ and $\eta$ are automatically
$K$-invariant, i.e, $\pounds_{\xi}\varphi=0$ and $\pounds_{\xi}\eta=0$. The
Killing vector field $\xi$ induces a regular one-dimensional Riemannian
foliation on $M$. We denote by ${\overline{M}}:=M/K$ the orbit space (the
space of all leaves) of a regular contact Riemannian manifold $M$ under the
$K$-action.

Let $\bar{X}_{\bar{p}}$ be a tangent vector of the orbit space $\overline{M}$
at ${\bar{p}}=\pi(p)$. Then there exists a tangent vector ${\bar{X}}^{*}_{p}$
of $M$ at $p$ which is orthogonal to $\xi$ such that
$\pi_{*p}{\bar{X}}^{*}_{p}=\bar{X}_{\bar{p}}$. The tangent vector
${\bar{X}}^{*}_{p}$ is called the horizontal lift of $\bar{X}_{\bar{p}}$ to
$M$ at $p$. The horizontal lift operation
$*:{\bar{X}}_{\bar{p}}\mapsto{\bar{X}}^{*}_{p}$ is naturally extended to
vector fields.

The contact structure on $M$ induces an almost Hermitian structure on the
orbit space ${\overline{M}}$:

(6) $J{\bar{X}}=\pi_{*}(\varphi{\bar{X}}^{*}),\
\bar{X}\in\mathfrak{X}(\bar{M}).$

Let us denote by $\bar{\nabla}$ the Levi-Civita connection of $\bar{M}$. Then,
by using the fundamental equations for Riemannian submersions due to O’Neill
[33], we have the following results.

###### Proposition 1.3

([32]) Let $M$ be a regular contact Riemannian manifold. Then for any
$\bar{X},\bar{Y}\in\mathfrak{X}(\bar{M}):$

(7)
$\nabla_{{\bar{X}}^{*}}\bar{Y}^{*}=(\bar{\nabla}_{\bar{X}}{\bar{Y}})^{*}-g(\bar{X}^{*},\varphi\bar{Y}^{*})\xi.$

###### Proposition 1.4

([32]) Sasakian space forms are regular Sasaki manifolds. The orbit space of a
Sasakian space form of constant holomorphic sectional curvature $c$ is a
complex space form of constant holomorphic sectional curvature $c+3$.

W. M. Boothby and H. C. Wang [8] proved that if $M$ is a compact regular
contact manifold, then the natural projection $\pi:M\rightarrow{\bar{M}}$
defines a principal circle bundle over a symplectic manifold ${\bar{M}}$ and
the symplectic form $\Omega$ of ${\overline{M}}$ determines an integral
cocycle. Furthermore the contact form $\eta$ gives a connection form of this
circle bundle and satisfies $\pi^{*}\Omega=d\eta$. The fibering
$\pi:M\rightarrow{\bar{M}}$ is called the Boothby-Wang fibering of a regular
compact contact manifold $M$. Based on this result, we call the fibering
$\pi:M\to\bar{M}$ of a regular contact Riemannian manifold $M$, the “Boothby-
Wang fibering” of $M$ even if $M$ is noncompact.

The unit sphere $S^{2n+1}$ is a typical example of regular compact Sasaki
manifold. For $S^{2n+1}$, the Boothby-Wang fibering coincides with the Hopf
fibering $S^{2n+1}\rightarrow\mathbb{C}P^{n}$.

In $3$-dimensional case, the Boothby-Wang fibering of Sasakian space forms
have the following matrix group models [6]:

$\displaystyle\pi:\mathrm{SU}(2)\to S^{2}(c)$ $\displaystyle=$
$\displaystyle\mathrm{SU}(2)/\mathrm{U}(1),$
$\displaystyle\pi:\mathbf{R}^{3}(-3)\to\mathbf{C}$ $\displaystyle=$
$\displaystyle\mathbf{R}^{3}(-3)/\mathbf{R},$
$\displaystyle\pi:\mathrm{SL}_{2}\mathbf{R}\to H^{2}(c)$ $\displaystyle=$
$\displaystyle\mathrm{SL}_{2}\mathbf{R}/\mathrm{SO}(2).$

Here $S^{2}(c)$ and $H^{2}(c)$ are sphere and hyperbolic space of curvature
$c$, respectively.

###  1.3 Hopf cylinders

Now we shall restrict our attention to $3$-dimensional regular contact
Riemannian manifold $M$.

Let ${\bar{\gamma}}$ be a curve parameterized by arc length in
${\overline{M}}$ with curvature ${\bar{\kappa}}$. Taking the inverse image
$S_{\bar{\gamma}}:=\pi^{-1}\\{{\bar{\gamma}}\\}$ of ${\bar{\gamma}}$ in
$M^{3}$.

Here we compute the fundamental quantities of $S_{\bar{\gamma}}$.

Let us denote by $\bar{P}=({\bar{\mathbf{p}}}_{1},{\bar{\mathbf{p}}}_{2})$ the
Frenet frame field of $\bar{\gamma}$. By using the complex structure $J$ of
${\overline{M}}^{2}$, ${\bar{\mathbf{p}}}_{2}$ is given by

${\bar{\mathbf{p}}}_{2}=J{\bar{\mathbf{p}}}_{1}$

Then the Frenet-Serret formula of $\bar{\gamma}$ is given by

$\bar{\nabla}_{\bar{\gamma}^{\prime}}P=P\left(\begin{array}[]{cc}0&-\bar{\kappa}\\\
\bar{\kappa}&0\end{array}\right).$

Here the function $\bar{\kappa}$ is the (signed) curvature of $\bar{\gamma}$.

Let $\mathbf{t}=(\bar{\mathbf{p}}_{1})^{*}$ the horizontal lift of
$\bar{\mathbf{p}}_{1}$ with respect to the Boothby-Wang fibering. Then
$(\mathbf{t},\xi)$ gives an orthonormal frame field of $S$. We choose a unit
normal vector field $\mathbf{n}$ by $\mathbf{n}=(\bar{\mathbf{p}}_{2})^{*}$.
Since ${\bar{\mathbf{p}}}_{2}$ is defined by
${\bar{\mathbf{p}}}_{2}=J{\bar{\mathbf{p}}}_{1}$,
$\mathbf{n}=\varphi\>\mathbf{t}$. In fact,

$(\bar{\mathbf{p}}_{2})^{*}=(J\bar{\mathbf{p}}_{1})^{*}=\varphi(\bar{\mathbf{p}}_{1})^{*}=\varphi\>\mathbf{t}.$

Let us denote by $\nabla^{S}$ the Levi-Civita connection of $S$. The second
fundamental form $I\\!I$ derived from $\mathbf{n}$ is defined by the Gauß
formula:

(8) $\nabla_{X}Y=\nabla^{S}_{X}Y+I\\!I(X,Y)\mathbf{n},\ \
X,Y\in\mathfrak{X}(S).$

By using (7),

$\nabla_{\mathbf{t}}\>\mathbf{t}=(\bar{\nabla}_{\bar{\mathbf{p}}_{1}}\bar{\mathbf{p}}_{1})^{*}-g(\mathbf{t},\varphi\>\mathbf{t})\xi=(\bar{\kappa}\circ\pi)\mathbf{n}.$

Hence $\nabla^{S}_{\mathbf{t}}{\mathbf{t}}=0$. Since $\xi$ is Killing, we have
$\nabla^{S}_{\mathbf{t}}\xi=\nabla^{S}_{\xi}\xi=0$. Thus $S_{\bar{\gamma}}$ is
flat. The second fundamental form $I\\!I$ is described as

$I\\!I(\mathbf{t},\mathbf{t})=\bar{\kappa}\circ\pi,\ \
I\\!I(\mathbf{t},\xi)=-1,\ \ I\\!I(\xi,\xi)=0.$

The mean curvature is $H=(\bar{\kappa}\circ\pi)/2$ and the mean curvature
vector field $\mathbb{H}$ is $\mathbb{H}=H\>\mathbf{n}$.

In case $M=S^{3}$, $S_{\bar{\gamma}}$ is called the Hopf cylinder. In
particular if ${\bar{\gamma}}$ is closed, then $S_{\bar{\gamma}}$ is a flat
torus in $S^{3}$ and called the Hopf torus over ${\bar{\gamma}}$ (H. B.
Lawson, cf. [31], [35]). The Hopf torus over a geodesic in $S^{2}(4)$
coincides with the Clifford minimal torus. We call the flat surface
$S_{\bar{\gamma}}$ in a regular contact Riemannian manifold $M$ a Hopf
cylinder over the curve $\bar{\gamma}$ in $\overline{M}$.

###  1.4 Curves in Riemannian $3$-manifolds

Let $(M,g)$ be a Riemannian manifold and $\gamma=\gamma(s):I\to M$ a curve
parametrised by the arclength parameter in $M$. We regard $\gamma$ as a
1-dimensional Riemannian manifold with respect to the metric induced by $g$.

We recall the following definition (cf. [2]).

###### Definition 1.2

If $\gamma(s)$ is a unit speed curve in a Riemannian $3$-manifold $(M^{3},g)$,
we say that $\gamma$ is a Frenet curve if there exists an orthonomal frame
field $P=(\mathbf{p}_{1},\mathbf{p}_{2},\mathbf{p}_{3})$ along $\gamma$ and
two nonnegative functions $\kappa$ and $\tau$ such that $P$ satisfies the
following Frenet-Serret formula:

$\nabla_{\gamma^{\prime}}P=P\left(\begin{array}[]{ccc}0&-\kappa&0\\\
\kappa&0&-\tau\\\ 0&\tau&0\end{array}\right),\ \
\mathbf{p}_{1}=\gamma^{\prime}(s).$

The functions $\kappa$ and $\tau$ are called the curvature and torsion of
$\gamma$ respectively.

Geodesics can be regarded as Frenet curves with $\kappa=0$. A curve with
constant curvature and zero torsion is called a (Riemannian) circle. A helix
is a curve whose curvature and torsion are constants. Riemannian circles are
regarded as degenerate helices. Helices, which are not circles, are frequently
called proper helices.

Note that, in general ambient space $(M^{3},g)$, geodesics may have non-
vanishing torsion. In fact, as we shall see later, Legendre geodesics in a
Sasakian 3-manifold have constant torsion $1$.

The Frenet-Serret formula of $\gamma$ implies that the mean curvature vector
field $\mathbb{H}$ of a Frenet curve $\gamma$ is given by

$\mathbb{H}=\nabla_{\gamma^{\prime}}\gamma^{\prime}=\kappa\mathbf{p}_{2}.$

Let us denote by $\Delta$ the Laplace operator acting on the space
$\Gamma(\gamma^{*}TM)$ of all smooth sections of the vector bundle:

$\gamma^{*}TM:=\bigcup_{s\in I}T_{\gamma(s)}M$

over $I$. Then $\Delta$ is given explicitly by

$\Delta=-\nabla_{\gamma^{\prime}}\nabla_{\gamma^{\prime}}.$

###### Lemma 1.1

The mean curvature vector field $\mathbb{H}$ of a Frenet curve $\gamma$ is
harmonic in $\gamma^{*}TM$ ($\Delta\mathbb{H}=0$) if and only if

$\nabla_{\gamma^{\prime}}\nabla_{\gamma^{\prime}}\nabla_{\gamma^{\prime}}\gamma^{\prime}=0.$

When $M$ is the Euclidean space $\mathbf{E}^{m}$, a curve $\gamma$ satisfies
$\Delta\mathbb{H}=0$ if and only if $\gamma$ is biharmonic, i.e.,
$\Delta\Delta\gamma=0$ since $\Delta\gamma=-\mathbb{H}$.

The following general result is essentially obtained in [24].

###### Theorem 1.1

Let $\gamma$ be a Frenet curve in a Riemannian $3$-manifold $(M,g)$. Then
$\gamma$ satisfies $\Delta\mathbb{H}=\lambda\mathbb{H}$ in $\gamma^{*}TM$ if
and only if $\gamma$ is a geodesic $(\lambda=0)$ or a helix satisfying
$\lambda=\kappa^{2}+\tau^{2}$.

Proof. Let $I$ be an open interval and $\gamma=\gamma(s):I\to M$ be a curve
parametrised by the arclength parameter $s$ with Frenet frame field
$P=(\mathbf{p}_{1},\mathbf{p}_{2},\mathbf{p}_{3})$. Direct computation shows
that

(9)
$\nabla_{\gamma^{\prime}}\mathbb{H}=-\kappa^{2}\mathbf{p}_{1}+\kappa^{\prime}\>\mathbf{p}_{2}+\kappa\tau\mathbf{p}_{3}.$

Let us compute the Laplacian of $\mathbb{H}$:

$-\Delta\mathbb{H}=\nabla_{\gamma^{\prime}}\nabla_{\gamma^{\prime}}\mathbb{H}=-3\kappa\kappa^{\prime}\>\mathbf{p}_{1}+(\kappa^{\prime\prime}-\kappa^{3}-\kappa\tau^{2})\mathbf{p}_{2}+(2\kappa^{\prime}\tau+\kappa\tau^{\prime})\mathbf{p}_{3}.$

Hence $\Delta\mathbb{H}=\lambda\mathbb{H}$ if and only if

$\kappa\>\tau^{\prime}=0,\ \ \kappa^{3}+\kappa\>\tau^{2}=\lambda\kappa.$

These formulae imply that $\gamma$ is a geodesic or a helix satisfying
$\lambda=\kappa^{2}+\tau^{2}$.

Conversely every geodesic satisfies $\Delta\mathbb{H}=0$. Helices satisfy
$\Delta\mathbb{H}=\lambda\mathbb{H}$ with $\lambda=\kappa^{2}+\tau^{2}$.
$\Box$

###### Corollary 1.1

([19]) Let $\gamma$ be a curve in Euclidean $3$-space $\mathbf{E}^{3}$. Then
$\gamma$ is biharmonic if and only if $\gamma$ is a straight line.

On the contrary, in indefinite semi-Euclidean space, there exist nongeodesic
biharmonic curves. Chen and Ishikawa [15] classified biharmonic spacelike
curves in $\mathbf{E}^{m}_{\nu}$. (See also [28]).

###  1.5 Curves with normal-harmonic mean curvature

The results in the preceding subsection say that to characterise curves which
are non geodesics we need to use another differential operator for our
purpose.

In this subsection we use the normal Laplacian.

Let $\gamma:I\to M$ be a Frenet curve in an oriented Riemannian $3$-manifold
$M$ parametrised by the arclength. Denote by
$P=(\mathbf{p}_{1},\mathbf{p}_{2},\mathbf{p}_{3})$ the Frenet frame field of
$\gamma$ as before. Then the normal bundle $T^{\perp}\gamma$ of the curve
$\gamma$ is given by

$T^{\perp}\gamma=\bigcup_{s\in I}T^{\perp}_{s}\gamma,\
T^{\perp}_{s}\gamma=\mathbf{R}\>\mathbf{p}_{2}(s)\oplus\mathbf{R}\>\mathbf{p}_{3}(s).$

The normal connection $\nabla^{\perp}$ is a connection of $T^{\perp}\gamma$
defined by

$\nabla_{\gamma^{\prime}}^{\perp}X=\mathrm{normal}\ \mathrm{component}\
\mathrm{of}\ \nabla_{\gamma^{\prime}}X$

for any section $X$ of the normal bundle $T^{\perp}\gamma$.

By using the Frenet frame field, $\nabla^{\perp}$ can be represented as

$\nabla^{\perp}_{\gamma^{\prime}}X=\nabla_{\gamma^{\prime}}X-g(\nabla_{\gamma^{\prime}}X,\mathbf{p}_{1})\mathbf{p}_{1}.$

Let us denote by $\Delta^{\perp}$ the Laplace operator acting on the space
$\Gamma(T^{\perp}\gamma)$ of all smooth sections of the normal bundle
$T^{\perp}\gamma$. The operator $\Delta^{\perp}$ is called the normal
Laplacian of $\gamma$ in $M$. The normal Laplacian $\Delta^{\perp}$ is given
by

$\Delta^{\perp}X=-\nabla^{\perp}_{\gamma^{\prime}}\nabla^{\perp}_{\gamma^{\prime}}X,\
\ X\in\Gamma(T^{\perp}\gamma).$

Now we compute $\Delta^{\perp}\mathbb{H}$. From (9), we have

$\nabla^{\perp}_{\gamma^{\prime}}\mathbb{H}=\kappa^{\prime}\>\mathbf{p}_{2}+\kappa\tau\mathbf{p}_{3}.$

From this equation, we get

$-\Delta^{\perp}\mathbb{H}=(\kappa^{\prime\prime}-\kappa\tau^{2})\mathbf{p}_{2}+(2\kappa^{\prime}\tau+\kappa\tau^{\prime})\mathbf{p}_{3}.$

###### Theorem 1.2

(cf. [24]) A curve $\gamma$ satisfies
$\Delta^{\perp}\mathbb{H}=\lambda\mathbb{H}$ if and only if

$\kappa^{\prime\prime}-\kappa\tau^{2}=-\lambda\kappa,\ \
2\kappa^{\prime}\tau+\kappa\tau^{\prime}=0.$

###### Corollary 1.2

A curve $\gamma$ satisfies $\Delta^{\perp}\mathbb{H}=0$ if and only if

$\kappa^{\prime\prime}-\kappa\tau^{2}=0,\ \
2\kappa^{\prime}\tau+\kappa\tau^{\prime}=0.$

We shall apply these general results for curves in Sasakian $3$-manifolds in
the next section. Note that Barros and Garay classified curves, which satisfy
$\Delta^{\perp}\mathbb{H}=\lambda\mathbb{H}$ in space forms [4],[5].

##  2 Curves and surfaces in $3$-dimensional
Sasaki manifolds

###  2.1 Curves in $3$-dimensional Sasaki manifolds

Now let $M^{3}=(M,\eta,\xi,\varphi,g)$ be a contact Riemannian $3$-manifold
with an associated metric $g$. A curve $\gamma=\gamma(s):I\to M$ parametrised
by the arclength parameter is said to be a Legendre curve if $\gamma$ is
tangent to the contact distribution $D$ of $M$. It is obvious that $\gamma$ is
Legendre if and only if $\eta(\gamma^{\prime})=0$.

Let $\gamma$ be a Legendre curve in $M^{3}$. Then we can take a Frenet frame
field $P=(\mathbf{p}_{1},\mathbf{p}_{2},\mathbf{p}_{3})$ so that
$\mathbf{p}_{1}=\gamma^{\prime}$ and $\mathbf{p}_{3}=\xi$. (See [2]).

Now we assume that $M$ is a Sasaki manifold. Then by definition, the Frenet-
Serret formula of $\gamma$ is given explicitly by

$\nabla_{\gamma^{\prime}}P=P\left(\begin{array}[]{ccc}0&-\kappa&0\\\
\kappa&0&-1\\\ 0&1&0\end{array}\right).$

Namely every Legendre curve has constant torsion $1$ [2].

Now we investigate curves with harmonic or normal-harmonic mean curvature
vector field in Sasakian 3-manifolds.

The following two results are direct consequence of Theorem 1.1 and Theorem
1.2, respectively.

###### Corollary 2.1

Let $\gamma$ be a Legendre curve in $3$-dimensional Sasaki manifold. Then
$\gamma$ satisfies $\Delta\mathbb{H}=\lambda\mathbb{H}$ in $\gamma^{*}TM$ if
and only if $\gamma$ is a Legendre geodesic $(\lambda=0)$ or a Ledendre helix
satisfying $\lambda=\kappa^{2}+1$ $(\lambda\not=0)$.

###### Remark 2.1

Sasaki manifolds together with compatible Lorentz metric are called Sasakian
spacetimes ([20],[40]). On Sasakian spacetimes, the Reeb vector fields are
timelike. Every $3$-dimensional Sasakian spacetime contains proper biharmonic
Legendre curves. In fact, in a $3$-dimensional Sasakian spacetime biharmonic
Legendre curves are Legendre geodesics or Legendre helices with curvature $1$.
(cf. [28]).

###### Proposition 2.1

Let $\gamma$ be a Legendre curve in a Sasakian $3$-manifold. Then
$\Delta^{\perp}\mathbb{H}=\lambda\mathbb{H}$ if and only if $\gamma$ is a
Legendre geodesic $(\lambda=0)$ or a Legendre helix with constant nonzero
curvature $(\lambda\not=0)$. In the latter case, $\lambda=1$.

###  2.2 Biharmonic Hopf cylinders

In this section we study harmonicity and normal-harmonicity of the mean
curvature of Hopf cylinders.

Let $M^{3}$ be a regular Sasaki manifold with Boothby-Wang fibration
$\pi:M\to\bar{M}$.

Take a curve $\bar{\gamma}=\bar{\gamma}(s)$ parametrised by the arclength $s$
in the base space form $\bar{M}$. Let us denote by $S=S_{\bar{\gamma}}$ the
Hopf cylinder of $\bar{\gamma}$. (See Section 1.3)

Let $\mathbf{t}=(\bar{\mathbf{p}}_{1})^{*}$ be the horizontal lift of
$\bar{\mathbf{p}}_{1}$ with respect to the Boothby-Wang fibering. Then
$(\mathbf{t},\xi)$ gives an orthonormal frame field of $M$. The unit normal
vector field $\mathbf{n}$ is the horizontal lift of $\bar{\mathbf{p}}_{2}$.
Note that $\mathbf{n}=\varphi\mathbf{t}$.

The mean curvature vector field $\mathbb{H}$ of $S$ is
$\mathbb{H}=H\>\mathbf{n}=(\bar{\kappa}\circ\pi)\mathbf{n}/2$.

Now we study harmonicity and normal-harmonicity of $\mathbb{H}$. Denote by
$\iota$ the inclusion map of $S$ into $M$. Then the Laplace operator $\Delta$
acting on the space $\Gamma(\iota^{*}TM)$ and the normal Laplacian
$\Delta^{\perp}$ of $S$ are given by

$\Delta=-\left(\nabla_{\mathbf{t}}\nabla_{\mathbf{t}}+\nabla_{\xi}\nabla_{\xi}\right),\
\
\Delta^{\perp}=-\left(\nabla^{\perp}_{\mathbf{t}}\nabla^{\perp}_{\mathbf{t}}+\nabla^{\perp}_{\xi}\nabla^{\perp}_{\xi}\right),$

respectively. Direct computation shows that

$\nabla_{\mathbf{t}}\mathbb{H}=-2H^{2}\mathbf{t}+H^{\prime}\mathbf{n}+H\xi,\ \
\nabla^{\perp}_{\mathbf{t}}\mathbb{H}=H^{\prime}\mathbf{n},\ \
\nabla_{\xi}\>\mathbb{H}=H\>\mathbf{t},\ \
\nabla^{\perp}_{\xi}\>\mathbb{H}=0,$
$\nabla_{\xi}\nabla_{\xi}\mathbb{H}=-H\mathbf{n}.$

Thus we get

$-\Delta\mathbb{H}=-6HH^{\prime}\mathbf{t}+(H^{\prime\prime}-4H^{3}-2H)\mathbf{n}+2H^{\prime}\xi,$
$-\Delta^{\perp}\mathbb{H}=H^{\prime\prime}\mathbf{n}.$

###### Theorem 2.1

A Hopf cylinder $S_{\bar{\gamma}}$ in a $3$-dimensional regular Sasaki
manifold satisfies $\Delta\mathbb{H}=\lambda\mathbb{H}$ in $\iota^{*}TM$ if
and only if $\bar{\gamma}$ is a geodesic $(\lambda=0)$ or a Riemannian circle
$(\lambda\not=0)$. In case that $\lambda\not=0$, the eigenvalue $\lambda$ is
$\lambda=4H^{2}+2>2$.

###### Remark 2.2

Every Hopf cylinder in a $3$-dimensional regular Sasaki manifold is anti
invariant. Sasahara showed that an anti invariant surface in
$\mathbf{R}^{3}(-3)$ satisfies $\Delta\mathbb{H}=\lambda\mathbb{H},\
\lambda\not=0$ if and only if it is a Hopf cylinder over a circle with
$\lambda>2$. See Proposition 11 in [37].

###### Lemma 2.1

A Hopf cylinder $S_{\bar{\gamma}}$ satisfies
$\Delta^{\perp}\mathbb{H}=\lambda\mathbb{H}$ if and only if $\gamma$ is
defined by one of the following natural equations:

  1. (1)

$\bar{\kappa}(s)=as+b,\ a,b\in\mathbf{R},\ \lambda=0;$

  2. (2)

$\bar{\kappa}(s)=a\cos(\sqrt{\lambda}s)+b\sin(\sqrt{\lambda}s),\ \lambda>0;$

  3. (3)

$\bar{\kappa}(s)=a\exp(\sqrt{-\lambda}s)+b\exp(-\sqrt{-\lambda}s),\
\lambda<0.$

Proof. The Hopf cylinder $S_{\bar{\gamma}}$ satisfies
$\Delta^{\perp}\mathbb{H}=\lambda\mathbb{H}$ if and only if $\bar{\gamma}$
satisfies

${\bar{\kappa}}^{\prime\prime}+\lambda{\bar{\kappa}}=0.$

Thus the result follows. $\Box$

###### Theorem 2.2

A Hopf cylinder $S_{\bar{\gamma}}$ satisfies $\Delta^{\perp}\mathbb{H}=0$ if
and only if $\bar{\gamma}$ is one of the following:

  1. (1)

a geodesic;

  2. (2)

a Riemannian circle or;

  3. (3)

a Riemannian clothoid ( Cornu spiral ).

Here a Riemannian clothoid is a curve in $\bar{M}^{2}$ whose curvature is a
linear function of the arclength.

###### Remark 2.3

On curves in Riemannian 2-space forms, the following result is obtained [24]:

###### Theorem 2.3

Let $\bar{\gamma}$ be a curve in Riemannian $2$-manifold $\bar{{M}}^{2}$. To
avoid the confusion, let us denote by $\Delta^{\perp}_{\bar{\gamma}}$ and
$\mathbb{H}_{\bar{\gamma}}$ the normal Laplacian of $\bar{\gamma}$ and the
mean curvature vector in $\bar{{M}}^{2}$ respectively. Then
$\Delta_{\bar{\gamma}}^{\perp}\mathbb{H}_{\bar{\gamma}}=\lambda\mathbb{H}_{\bar{\gamma}}$
if and only if

  1. (1)

$\bar{\gamma}$ is a geodesic, Riemannian circle or a Riemannian clothoid;

  2. (2)

$\bar{\kappa}(s)=a\cos(\sqrt{\lambda}s)+b\sin(\sqrt{\lambda}s),\ \lambda>0$;

  3. (3)

$\bar{\kappa}(s)=a\exp(\sqrt{-\lambda}s)+b\exp(-\sqrt{-\lambda}s),\
\lambda<0$.

###### Corollary 2.2

Let $M$ be a $3$-dimensional regular Sasaki manifold
with Boothby-Wang fibering $\pi:M\to\bar{M}$. Let $\bar{\gamma}$ be a curve in
$\bar{M}$. Then the Hopf cylinder $S=S_{\bar{\gamma}}$ satisfies
$\Delta^{\perp}\mathbb{H}=\lambda\mathbb{H}$ if and only if $\bar{\gamma}$
satisfies
$\Delta_{\bar{\gamma}}^{\perp}\mathbb{H}_{\bar{\gamma}}=\lambda\mathbb{H}_{\bar{\gamma}}$.

Theorem 2.2 is a generalisation of a result obtained by Barros and Garay [3].
In fact, if we choose $M^{3}=S^{3}$ then we obtain the following.

###### Theorem 2.4

([3]) A Hopf cylinder $S_{\bar{\gamma}}$ in the unit $3$-sphere $S^{3}$
satisfies $\Delta^{\perp}\mathbb{H}=0$ if and only if $\gamma$ is one of the
following:

  1. (1)

a geodesic,

  2. (2)

a Riemannian circle or

  3. (3)

a Riemannian clothoid.

Here a Riemannian clothoid is a curve in the $2$-sphere $S^{2}(1/2)$ of radius
$1/2$ whose curvature is a linear function of the arclength.

Riemannian clothoids are called “Cornu spirals” in [3].

## Part II

##  3 Polyharmonic maps

Let $(M^{m},g)$ and $(N^{n},h)$ be Riemannian manifolds and $\phi:M\to N$ a
smooth map. The tension field $\mathscr{T}(\phi)$ is a section of the vector
bundle $\phi^{*}(TN)$ defined by

$\mathscr{T}(\phi):=\mathrm{tr}(\nabla d\phi).$

A smooth map $\phi$ is said to be a harmonic map if its tension field
vanishes. It is well known that $\phi$ is harmonic if and only if $\phi$ is a
critical point of the energy:

$E(\phi)=\int\frac{1}{2}|d\phi|^{2}dv_{g}$

over every compact supported region of $M$.

Now let $\phi$ be a harmonic map. Then the Hessian $\mathcal{H}_{\phi}$ of the
energy is given by the following second variation formula:

$\mathcal{H}_{\phi}(V,W)=\int h(\mathcal{J}_{\phi}(V),W)dv_{g},\ \
V,W\in\Gamma(\phi^{*}TN).$

Here the operator $\mathcal{J}_{\phi}$ is the Jacobi operator of the harmonic
map $\phi$ defined by

$\mathcal{J}_{\phi}(V):=\bar{\Delta}_{\phi}V-\mathcal{R}_{\phi}(V),\ \
V\in\Gamma(\phi^{*}TN),$
$\bar{\Delta}_{\phi}:=-\\{\sum_{i=1}^{m}(\nabla^{\phi}_{e_{i}}\nabla^{\phi}_{e_{i}}-\nabla^{\phi}_{\nabla_{e_{i}}e_{i}}\\},\
\ \mathcal{R}_{\phi}(V)=\sum_{i=1}^{m}R^{N}(V,d\phi(e_{i}))d\phi(e_{i}).$

Here $\nabla^{\phi},\ R^{N}$ and $\\{e_{i}\\}$ denote the induced connection
of $\phi^{*}TN$, curvature tensor of $N$ and a local orthonormal frame field
of $M$, respectively.

For general theory of harmonic maps and their Jacobi operators, we refer to
[21] and [42].

J. Eells and J. H. Sampson suggested to study polyharmonic maps (See [23] and
[21], p. 77 (8.7)). Let $\phi:M\to N$ be a smooth map as before. Then $\phi$
is said to be a polyharmonic map of order $k$ if it is an extremal of the
functional:

$E_{k}(\phi)=\int|(d+d^{*})^{k}\phi|^{2}dv_{g}.$

Here $d^{*}$ is the codifferential operator. In particular, if $k=2$, we have

$E_{2}(\phi)=\int|\mathscr{T}(\phi)|^{2}dv_{g}.$

The Euler-Lagrange equation of the functional $E_{2}$ was computed by Caddeo
and Oproiu (See [9], p. 867) and G. Y. Jiang [29]–[30], independently. The
Euler-Lagrange equation of $E_{2}$ is

$\mathscr{T}_{2}(\phi):=-\mathcal{J}_{\phi}(\mathscr{T}(\phi))=0.$

###### Remark 3.1

Let $\phi:M\to N$ be an isometric immersion. Then its tension field is
$m\mathbb{H}$. Thus the functional $E_{2}$ is given by

$E_{2}(\phi)=m^{2}\int|\mathbb{H}|^{2}dv_{g}.$

In case that $M$ is $2$-dimensional, $E_{2}(\phi)$ the total mean curvature of
$M$ up to constant multiple. See [11], Section 5.3.

In particular, if $N=\mathbf{E}^{n}$ and $\phi$ an isometric immersion, then

$\mathscr{T}_{2}(\phi)=-\Delta_{M}\Delta_{M}\phi,$

since $\Delta_{M}\phi=m\mathbb{H}$. Here $\Delta_{M}$ is the Laplacian of
$(M,g)$. Thus the polyharmonicity (of order $2$) for an isometric immersion
into Euclidean space is equivalent to the biharmonicity in the sense of Chen.
On this reason, polyharmonic maps of order $2$ are frequently called
biharmonic maps (or $2$-harmonic maps) [9], [29], [30], [34].

Obviously, the notion of $p$-harmonic map in the sense of [22], p. 397 is
different from that of polyharmonic map of order $p$.

Hereafter we call polyharmonic maps of order $2$ by the name “polyharmonic
maps” in short.

Caddeo, Montaldo and Oniciuc classified polyharmonic curves in
$3$-dimensional Riemannian space forms. More precisely they showed the
following two results.

###### Theorem 3.1

([9]) Let $N$ be a $3$-dimensional Riemannian space form of nonpositive
curvature. Then all the polyharmonic curves are geodesics.

Next for the study of polyharmonic curves in positively curved space forms, we
may assume that $N^{3}$ is the unit $3$-sphere.

###### Theorem 3.2

([9]) Let $\gamma:I\to S^{3}$ be a polyharmonic curve parametrised by the
arclength. Then $0\leq\kappa\leq 1$ and $\gamma$ is one of the following:

  1. (1)

$\kappa$ is a geodesic ($\kappa=0$) .

  2. (2)

If $k=1$ then $\gamma$ is a Riemannian circle of curvature $1;$

  3. (3)

If $0<\kappa<1$ then $\gamma$ is a geodesic of the Clifford minimal torus of
$S^{3}$.

The preceding theorem implies the following result:

###### Corollary 3.1

Let $\gamma:I\to S^{3}$ be a Legendre curve parametrised by the arclength.
Then $\gamma$ is polyharmonic if and only if $\gamma$ is a Legendre geodesic.

In fact, curves in the the latter two classes can not be Legendre. (Recall
that every Legendre curve has constant torsion $1$).

Now we study polyharmonic Legendre curves in contact Riemannian $3$-manifolds.

Let $M^{3}$ be a contact Riemannian $3$-manifold and $\gamma:I\to M$ a Frenet
curve framed by $(\mathbf{p}_{1},\mathbf{p}_{2},\mathbf{p}_{3})$. Then direct
computation shows that

$\mathscr{T}_{2}(\gamma)=-3\kappa\kappa^{\prime}\mathbf{p}_{1}+(\kappa^{\prime\prime}-\kappa^{3}-\kappa\tau^{2})\mathbf{p}_{2}+(2\kappa^{\prime}\tau+\kappa\tau^{\prime})\mathbf{p}_{3}+\kappa
R(\mathbf{p}_{2},\mathbf{p}_{1})\mathbf{p}_{1}.$

Now assume that $M$ is a Sasakian space form of constant holomorphic sectional
curvature $c$ then

$\displaystyle R(\mathbf{p}_{2},\mathbf{p}_{1})\mathbf{p}_{1}$
$\displaystyle=$ $\displaystyle\frac{c+3}{4}\mathbf{p}_{2}$ $\displaystyle+$
$\displaystyle\frac{c-1}{4}\\{\eta(\mathbf{p}_{2})\eta(\mathbf{p}_{1})\mathbf{p}_{1}$
$\displaystyle-$
$\displaystyle\eta(\mathbf{p}_{1})^{2}\mathbf{p}_{2}-\eta(\mathbf{p}_{2})\xi+3g(\mathbf{p}_{2},\varphi\mathbf{p}_{1})\varphi\mathbf{p}_{1}\\}.$

In particular, if $\gamma$ is Legendre, then
$R({\mathbf{p}}_{2},{\mathbf{p}}_{1}){\mathbf{p}}_{1}=c\>\mathbf{p}_{2}$. Thus
a Legendre curve $\gamma$ in $M$ is polyharmonic if and only if

$\kappa=\mathrm{constant},\ \kappa^{3}-(c-1)\kappa=0,\ \tau=1.$

If we look for nongeodesic polyharmonic Legendre curves, we obtain

$\kappa=\mathrm{constant},\ \kappa^{2}=c-1,\ \tau=1.$

Thus we obtain the following result which is a generalisation of Corollary
3.1.

###### Theorem 3.3

Let $M^{3}(c)$ be a Sasakian space form of constant holomorphic sectional
curvature $c$ and $\gamma:I\to M$ a polyharmonic Legendre curve parametrised
by the arclength.

  1. (1)

If $c\leq 1$, then $\gamma$ is a Legendre geodesic;

  2. (2)

If $c>1$, then $\gamma$ is a Legendre geodesic or a Legendre helix of
curvature $\sqrt{c-1}$.

Let $\phi:M\to N$ be an isometric immersion. Then $\phi$ is a critical point
of the volume functional if and only if $\phi$ is minimal. The Jacobi operator
$\mathscr{J}$ of a minimal immersion $\phi$ (with respect to the volume
functional) is appeared in the second variation formula of the volume and
given by [39]

$\mathscr{J}V=\Delta^{\perp}V-\mathscr{S}V+\mathscr{R}(V),\ \
V\in\Gamma(T^{\perp}M).$

Here the operators $\mathscr{S}$ and $\mathscr{R}$ are defined by

$h(\mathscr{S}V,W)=\mathrm{tr}(\mathcal{A}_{V}\circ\mathcal{A}_{W}),\ \
\mathscr{R}(V)=\sum_{i=1}^{m}(R^{N}(d\phi(e_{i}),V)d\phi(e_{i}))^{\perp}.$

Here $\mathcal{A}_{V}$ denotes the Weingarten operator with respect to $V$.

Arroyo, Barros and Garay studied submanifolds in $S^{3}$ whose mean curvature
vector fields are eigen-section of the Jacobi operator with respect to the
volume functional [1], [4], [5]. Such study for surfaces in $5$-dimensional
Sasakian space forms can be found in [37].

It seems to be interesting to study similar problems for submanifolds in space
forms or Sasakian space forms with respect to the energy functional.

In [9], all the polyharmonic surfaces in $S^{3}$ are classified. More
precisely, the only non-minimal polyharmonic surfaces are totally umbilical
$2$-spheres.

Based on this result, we would like to propose the following problem:

Are there non-minimal and non totally umbilical polyharmonic submanifolds in
homogeneous Riemannian manifolds ?

To close this paper, we study polyharmonic Hopf cylinders in
$3$-dimensional Sasakian space forms. Moreover we show the existence of non-
minimal and non totally umbilical polyharmonic surfaces in Sasakian space
forms.

First we recall the following result which is a consequence of the main result
in [9]:

###### Proposition 3.1

There are no non minimal polyharmonic Hopf cylinders in the unit $3$-sphere
$S^{3}$.

Now we generalise this result to Sasakian space forms.

Let $S=S_{\bar{\gamma}}$ be a Hopf cylinder and $\iota:S\subset M^{3}(c)$ its
inclusion map into a Sasakian space form $M^{3}(c)$. Then the bitension field
$\mathscr{T}_{2}(\iota)$ is given by

$\mathscr{T}_{2}(\iota)=-\mathcal{J}_{\iota}(\mathscr{T}(\iota))=-2\mathcal{J}_{\iota}(\mathbb{H}).$

We use the orthonormal frame field $\\{{\mathbf{t}},\xi\\}$ as before. Then
since $S$ is flat, we have

$\bar{\Delta}_{\iota}\mathbb{H}=\Delta\mathbb{H},\ \
\mathcal{R}(\mathbb{H})=H(R(\mathbf{n},\mathbf{t})\mathbf{t}+R(\mathbf{n},\xi)\xi).$

Using the curvature formula of Sasakian space form, we get

$\mathcal{R}(\mathbb{H})=(c+1)H\>\mathbf{n}.$

Hence

$\mathcal{J}_{\iota}(\mathbb{H})=6HH^{\prime}\mathbf{t}-(H^{\prime\prime}-4H^{3}+(c-1)H)\mathbf{n}-2H^{\prime}\xi.$

Thus $\mathcal{J}_{\iota}(\mathbb{H})=\lambda\>\mathbb{H}$ if and only if

$H^{\prime}=0,\ 4H^{3}=(c-1+\lambda)H$

and hence $H=0$ or $\lambda=4H^{2}+1-c,\ H\not=0$.

###### Theorem 3.4

Let $S$ be a Hopf cylinder in a Sasakian space form $M^{3}(c)$. Then $S$
satisfies $\mathcal{J}_{\iota}(\mathbb{H})=\lambda\>\mathbb{H}$ if and only if
the base curve of $S$ is a Riemannian circle or a geodesic. In case that the
base curve is not a geodesic, then $\lambda=4H^{2}+1-c$.

###### Corollary 3.2

Let $\iota:S_{\bar{\gamma}}\to M^{3}(c)$ be a polyharmonic Hopf cylinder in a
Sasakian space form.

  1. (1)

If $c\leq 1$ then $\bar{\gamma}$ is a geodesic;

  2. (2)

If $c>1$ then $\bar{\gamma}$ is a geodesic or a Riemannian circle of curvature
$\bar{\kappa}=\sqrt{c-1}$.

In particular, there exist nonminimal polyharmonic Hopf cylinders in
Sasakian space forms of holomorphic sectional curvature greater than $1$.

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Department of Mathematics Education, Faculty of Education,

Utsunomiya University, Minemachi 350, Utsunomiya, 321-8505, Japan

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Department of Mathematical Sciences, Faculty of Science,

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