Let M and N be two Riemannian manifolds, a harmonic map $\psi :M\to N$ is any critical point of the energy equation

where dv_g denotes the volume element of g, and the Euler-Lagrange equation corresponding to E(ψ ) is $\tau (\psi )=\text{trace}\nabla d\psi =0$.

In 1983, Eells and Lemaire [9] introduced the notion of biharmonic maps, which are a natural generalization of harmonic maps. A biharmonic map $\psi :M\to N$ is a critical point of the energy equation

where dv_g denotes the volume element of g, and the Euler-Lagrange equation [15] corresponding to E₂(ψ) is

Here Δ is the Laplacian operator given by $\Delta V=\text{tr}({\nabla }^{2}V)$, and R^N is the curvature tensor on the manifold N defined as $R^N(X,Y)=[{\nabla }_X,{\nabla }_Y]-{\nabla }_{[X,Y]}.$

Let M be the submanifold of the manifold $\overline{M}$, if the biharmonic map $\psi :M\to \overline{M}$ is an isometric immersion then M is biharmonic submanifold of $\overline{M}$. In the paper [2], Baird studied conformal and semi-conformal biharmonic maps. Oniciuc studied biharmonic submanifolds of CPⁿ in [10]. He studied explicit formula for biharmonic submanifolds in Sasakian space forms and deduced some conditions in [11]. He proved a gap theorem for the mean curvature of certain complete proper biharmonic pmc submanifolds and classified proper biharmonic pmc surfaces in $S^n\times R$ in [12]. In [16], Oniciuc studied biharmonic constant mean curvature surface in the sphere. Recently, Oniciuc proved several unique continuation results for biharmonic maps between Riemannian manifolds in [19]. He studied biharmonic maps between Riemannian manifolds in [18]. Over the last few years many authors have studied biharmonic submanifolds, for example see [5, 10, 18]. Recently, Ou studied biharmonic maps form tori into a 2-sphere in [27]. In the paper [1], Ou studied biharmonic Riemannian submanifolds.

The notion of f-biharmonic maps was introduced by Lu [17]; it is a natural generalization of biharmonic maps. In the papers [21, 22], Ou studied f-biharmonic maps and f-biharmonic submanifolds. In these papers he proved that a f-biharmonic map from a compact Riemannian manifold into a non-positively curved manifold with constant f-bienergy density is a harmonic map. In [20], Ou characterized harmonic maps and minimal submanifolds using the concept of f-biharmonic maps and proved that the set of all f-biharmonic maps from a 2-dimensional domain is invariant under the conformal change of the metric on the domain. In [24], Roth studied f-biharonic submanifolds of generalized space forms. He deduced some necessary and sufficient conditions for f-biharmonicity in the general case and many particular cases. In [2] Baird and Fardon studied conformal and semi conformal biharmonic maps.

Let us consider the $C^{\infty }$ differentiable function $f:M\to R$. Now, f-harmonic maps are the critical points of the f-energy functional $E_f(\psi )$ for the maps $\psi :M\to N$ between Riemannian manifolds, where

The Euler-Lagrange equation corresponding to $E_f(\psi )$ is given by

Analgously f-biharmonic maps are critical points of the f-bienergy functional $E_{2,f}(\psi )$ for maps $\psi :M\to N$ between Riemannian manifolds where

The Euler-Lagrange equation corresponding to $E_{2,f}(\psi )$ is given by

Clearly, we have the following relationship among these different types of harmonic maps:

Harmonic maps $\subset$ biharmonic maps $\subset$ f-biharmonic maps.

A f-biharmonic map is called a proper f-biharmonic map if it is neither a harmonic nor a biharmonic map. Also, we will call a f-biharmonic submanifold proper if it is neither minimal nor biharmonic.

The notion of trans-Sasakian Manifolds was introduced by Blair and Oubina [4, 23] as a generalization of Sasakian manifolds. Trans-Sasakian manifolds of type (α,β) are generalizations of α-Sasakian and β-Kenmotsu manifolds. It is known that a proper trans-Sasakian manifold exists only for dimension three and trans-Sasakian manifolds of type $(0,0),(0,\beta ),$ and (α,0) are known [14] as cosymplectic, β-Kenmotsu and α-Sasakian respectively. In higher dimension it is either α-Sasakian or β-Kenmotsu. In Differential Geometry of almost contact manifolds, submanifold theory has become an important topic of research. There are several works on invariant submanifolds. In [6], the authors studied invariant submanifolds of trans-Sasakian manifolds. Three dimensional trans-Sasakian Manifolds have been studied by the first author in the papers [8, 25, 26].

During last few years biharmonic maps on contact manifolds have become a popular area of research. So in the present paper we would like to study f-biharmonic maps on three dimensional trans-Sasakian manifolds. Precisely we study f-biharmonic submanifolds of three dimensional trans-Sasakian manifolds and find some conditions for the map f to be biharmonic or not.

The present paper is organized as follows: Section 1 is introductory. After the introduction we give some preliminaries in Section 2. In Section 3 we study f-biharmonic submanifolds of three-dimensional trans-Sasakian manifolds.

Let $\overline{M}$ be an odd dimensional smooth differential manifold with an almost contact metric structure (ϕ,ξ,η,g), where ϕ is a (1,1)-tensor field, ξ is a vector field, η is a one form and g is a Riemannian metric on $\overline{M}$. For such manifolds, we know [3]

for any $X,Y\in \chi (\overline{M})$, where $\chi (\overline{M})$ denotes the Lie algebra of all vector fields on $\overline{M}$.

For a contact metric manifold $(\overline{M},\varphi ,\xi ,\eta ,g)$, we define a (1,1) tensor field h by $h=\frac{1}{2}{\mathcal{L}}_{\xi }\varphi$ and $\mathcal{L}$ is the usual Lie derivative. Then h is symmetric and satisfies the following relations

for any $X,Y\in \chi (\overline{M})$.

Moreover, if $\overline{\nabla }$ denotes the Levi-Civita connection with respect to g, then the following relation holds

A connected manifold $\overline{M}$ with almost contact metric structure $(\varphi ,\xi ,\eta ,g)$ is called a trans-Sasakian manifold [23] if $(\overline{M}\times R,J,G)$ belongs to the class W₄ [13], where J is an almost complex structure on $\overline{M}\times R$ which is defined by

for any vector field X on $\overline{M}$ and the smooth function f on $\overline{M}\times R$, and G is the usual product metric on $\overline{M}\times R$. According to [4], an almost contact metric manifold is a trans-Sasakian manifold if and only if

for smooth functions $\alpha ,\beta$ on $\overline{M}$, where $\overline{\nabla }$ denote the covariant derivative with respect to g. Generally, $\overline{M}$, is said to be a trans-Sasakian manifold of type $(\alpha ,\beta )$.

In a three-dimensional trans-Sasakian manifold the curvature tensor with respect to the Levi-Civita connection $\overline{\nabla }$ is as follows [7]:

where r is the scalar curvature of the manifold.

Let M^m (m<;n) be the submanifold of a contact metric manifold ${\overline{M}}^n$. Let $\nabla$ and $\overline{\nabla }$ be the Levi-Civita connections of M and $\overline{M}$, respectively. Then for any vector fields $X,Y\in \chi (M)$, the second fundamental form σ is defined by

For any section of the normal bundle $T^{\perp }M$, we have

where ${\nabla }^{\perp }$ denotes the normal bundle connection of M. The second fundamental form σ and the shape operator A_N are related by

For any vector field $X\in \chi (M)$, we can right

where TX is the tangential component of ϕX and NX is the normal component of ϕX. Similarly, for any vector field V in normal bundle we have

where tV and nV are the tangential and normal components of $\varphi V$.

The submanifold M is said to be invariant if $\varphi X\in TM$ for any vector field X. On other hand M is said to be an anti-invariant submanifold if $\varphi X\in T^{\perp }M$ for any vector field X

We know for a isometric immersion ψ [24]

where H is the mean curvature. Now using the equation (1.1) in the above equation we have

By some classical and straightforward computations, we have

Using (3.3) in (3.2), we have

From the equation (1.3), we have the submanifold M is f-biharmonic if and only if

By simple calculation we have the above equation is equivalent to

For a f-biharmonic submanifold of a three-dimensional trans-Sasakian manifold we have the following:

Theorem 3.1. Let M be a submanifold of a three dimensional trans-Sasakian manifold $\overline{M}$. Then M is f-biharmonic if and only if the following equations hold

and

Proof. Form (2.7) we have

Let {e₁,e₂} be an orthogonal basis of the tangent space at a point of M. Then we have from above

Taking trace and using the equations (2.1), (2.11) and (2.12) we obtain

Using the equations (3.4) and (3.6) we can obtain

Therefore we have

Comparing the tangent and normal components we have the result of the theorem.

Now we have the following as particular cases of the above theorem.

Corollary 3.1. Let M be a submanifold of a three-dimensional trans-Sasakian manifold $\overline{M}$.

• (1) If M is anti-invariant, M is f-biharmonic if and only if

  $$\begin{array}{l}{\Delta }^{\perp }H+\text{tr}(\sigma (.,A_H.))+\frac{\Delta f}{f}H+2{\nabla }_{\text{grad}(\ln f)}^{\perp }H\\ =-2(\frac{r}{2}+2\xi \beta -2({\alpha }^2-{\beta }^2))H+2[(\frac{r}{2}+\xi \beta -3({\alpha }^2-{\beta }^2))\eta (H){\xi }^{\perp }\\ +\xi \beta H-\xi \alpha n(H)]+[2\xi \beta +(\frac{r}{2}+\xi \beta -3({\alpha }^2-{\beta }^2))]H,\end{array}$$

  and

  $$\begin{array}{l}\text{grad}|H{|}^2-2\text{tr}{A}_H\text{grad}(\ln f)+2\text{tr}(A_{{\nabla }^{\perp }H},.)\\ =2[(\frac{r}{2}+\xi \beta -3({\alpha }^2-{\beta }^2))\eta (H){\xi }^T-\eta (H)(T\text{grad}\alpha -T\text{grad }\beta )\\ +t(H)\xi \alpha ]-[{(\text{grad}\beta )}^T\eta (H)+g(\text{grad}\beta ,H){\xi }^T+g(\text{grad}\alpha ,\varphi H){\xi }^T\\ +(\frac{r}{2}+\xi \beta -3({\alpha }^2-{\beta }^2))\eta (H){\xi }^T].\end{array}$$

• (2) If M is invariant M is f-biharmonic if and only if

  $$\begin{array}{l}{\Delta }^{\perp }H+\text{tr}(\sigma (.,A_H.))+\frac{\Delta f}{f}H+2{\nabla }_{\text{grad}(\ln f)}^{\perp }H\\ =-2(\frac{r}{2}+2\xi \beta -2({\alpha }^2-{\beta }^2))H+2[(\frac{r}{2}+\xi \beta -3({\alpha }^2-{\beta }^2))\eta (H){\xi }^{\perp }\\ -\eta (H)(N\text{grad}\alpha -N\text{grad}\beta )+\xi \beta H-\xi \alpha n(H)]\\ +[2\xi \beta +(\frac{r}{2}+\xi \beta -3({\alpha }^2-{\beta }^2))]H,\end{array}$$

  and

  $$\begin{array}{l}\text{grad}|H{|}^2-2\text{tr}{A}_H\text{grad}(\ln f)+2\text{tr}(A_{{\nabla }^{\perp }H},.)\\ =2[(\frac{r}{2}+\xi \beta -3({\alpha }^2-{\beta }^2))\eta (H){\xi }^T+t(H)\xi \alpha ]-[{(\text{grad}\beta )}^T\eta (H)+\\ g(\text{grad}\beta ,H){\xi }^T+g(\text{grad}\alpha ,\varphi H){\xi }^T+(\frac{r}{2}+\xi \beta -3({\alpha }^2-{\beta }^2))\eta (H){\xi }^T].\end{array}$$

• (3) If ξ is normal to M, M is f-biharmonic if and only if

  $$\begin{array}{l}{\Delta }^{\perp }H+\text{tr}(\sigma (.,A_H.))+\frac{\Delta f}{f}H+2{\nabla }_{\text{grad}(\ln f)}^{\perp }H\\ =-2(\frac{r}{2}+2\xi \beta -2({\alpha }^2-{\beta }^2))H+2[(\frac{r}{2}+\xi \beta -3({\alpha }^2-{\beta }^2))\eta (H){\xi }^{\perp }\\ -\eta (H)(N\text{grad}\alpha -N\text{grad}\beta )+\xi \beta H-\xi \alpha n(H)]\\ +[2\xi \beta +(\frac{r}{2}+\xi \beta -3({\alpha }^2-{\beta }^2))]H,\end{array}$$

  and

  $$\begin{array}{l}\text{grad}|H{|}^2-2\text{tr}{A}_H\text{grad}(\ln f)+2\text{tr}(A_{{\nabla }^{\perp }H},.)\\ =2[-\eta (H)(T\text{grad}\alpha -T\text{grad}\beta )+t(H)\xi \alpha ]-[{(\text{grad}\beta )}^T\eta (H)].\end{array}$$

• (4) If ξ is tangent to M, M is f-biharmonic if and only if

  $$\begin{array}{l}{\Delta }^{\perp }H+\text{tr}(\sigma (.,A_H.))+\frac{\Delta f}{f}H+2{\nabla }_{\text{grad}(\ln f)}^{\perp }H\\ =-2(\frac{r}{2}+2\xi \beta -2({\alpha }^2-{\beta }^2))H+\xi \beta H-\xi \alpha n(H)]\\ +[2\xi \beta +(\frac{r}{2}+\xi \beta -3({\alpha }^2-{\beta }^2))]H,\end{array}$$

  and

  $$\begin{array}{l}\text{grad}|H{|}^2-2\text{tr}{A}_H\text{grad}(\ln f)+2\text{tr}(A_{{\nabla }^{\perp }H},.)\\ =2t(H)\xi \alpha -[g(\text{grad}\beta ,H){\xi }^T+g(\text{grad}\alpha ,\varphi H){\xi }^T],\end{array}$$

• (5) If M is a hypersurface, M is f-biharmonic if and only if

  $$\begin{array}{l}{\Delta }^{\perp }H+\text{tr}(\sigma (.,A_H.))+\frac{\Delta f}{f}H+2{\nabla }_{\text{grad}(\ln f)}^{\perp }H\\ =-2(\frac{r}{2}+2\xi \beta -2({\alpha }^2-{\beta }^2))H+2[(\frac{r}{2}+\xi \beta -3({\alpha }^2-{\beta }^2))\eta (H){\xi }^{\perp }\\ -\eta (H)(N\text{grad}\alpha -N\text{grad}\beta )+\xi \beta H-\xi \alpha n(H)]\\ +[2\xi \beta +(\frac{r}{2}+\xi \beta -3({\alpha }^2-{\beta }^2))]H,\end{array}$$

  and

  $$\begin{array}{l}\text{grad}|H{|}^2-2\text{tr}{A}_H\text{grad}(\ln f)+2\text{tr}(A_{{\nabla }^{\perp }H},.)\\ =2[(\frac{r}{2}+\xi \beta -3({\alpha }^2-{\beta }^2))\eta (H){\xi }^T-\eta (H)(T\text{grad}\alpha -T\text{grad}\beta ]\\ -[{(\text{grad}\beta )}^T\eta (H)+g(\text{grad}\beta ,H){\xi }^T+g(\text{grad}\alpha ,\varphi H){\xi }^T\\ +(\frac{r}{2}+\xi \beta -3({\alpha }^2-{\beta }^2))\eta (H){\xi }^T].\end{array}$$

Proof. Proof of the results is directly obtained from Theorem 3.1, using the following facts, respectively.

• (1) If M is invariant then N=0.

• (2) If M is anti-invariant then T=0.

• (3) If ξ is normal to M then ξ^T=0.

• (4) If ξ is tangent to M then η(H)=0 and ${\xi }^{\perp }=0$.

• (5) If M is a hypersurface then tH=0.

Theorem 3.2. Let M be a submanifold of a three dimensional trans-Sasakian manifold $\overline{M}$ with non zero constant mean curvature H and ξ is tangent to M, then M proper f-biharmonic if and only if

and $A_H\text{grad}(\ln f)=0$, or equivalent if and only if

Proof. Let M be a f biharmonic submanifold of $\overline{M}$ with constant mean curvature and ξ tangent to M then from the previous corollary we have

and

Since ξ is tangent to M then the equations are of the form

and $A_H\text{grad}(\ln f)=0$. Thus, the second equation is trivial and the first equation becomes

Now since $\text{tr}\sigma (.,A_H.)=|\sigma {|}^{2}H$ and H is non zero, so we have form above equation

Now from the Gauss formula we have

Using (2.7) in the above equation we have

Corollary 3.2. Let M be a submanifold of a three dimensional trans-Sasakian manifold $\overline{M}$ with non zero constant mean curvature H and ξ is tangent to M. If the functions α, β satisfy the inequality

then M is not f-biharmonic.

Proof. Form the Theorem 3.2 we know that M is f-biharmonic if and only if its second fundamental form σ satisfies the inequality

Since $|\sigma {|}^2\ge 0$, this is not possible if

Theorem 3.3. Let M be a submanifold of a three dimensional trans-Sasakian manifold $\overline{M}$ with non zero constant mean curvature H such that ξ and ϕ H are tangent to M. Define $F(f,\alpha ,\beta )$ on M by

• (1) if inf $F(f,\alpha ,\beta )$ is non-positive, M is not f-biharmonic.

• (2) if $F(f,\alpha ,\beta )$ is positive and M is proper f-biharmonic then

  $$0<|H{|}^2\le \frac{1}{2}F(f,\alpha ,\beta ).$$

Proof. M is proper f-biharmonic submanifold with constant mean curvature H and ξ is tangent to M, so we have form Corollary 3.1

and

Given that ϕ H is tangent to M, so tH=0. Therefore form the above equation we have

where

Taking inner product by H of the equation (??), we have

Now using the results $<\text{tr}(\sigma (.,A_H.)),H>=|A_H{|}^2,$ and $\Delta |H{|}^2=2(<{\Delta }^{\perp }H,H>-|{\nabla }^{\perp }H{|}^2)$, in the above equation we have

By using the Cauchy-Schwarz inequality $|A_H{|}^2\ge \frac{1}{2}\text{tr}(A_H)=2|H{|}^4,$ the equation reduces to

Therefore $F(f,\alpha ,\beta )\ge 2|H{|}^2,$ since |H| is positive. This proves the theorem.

The authors are thankful to the referee for his valuable suggestions towards the improvement of the paper.