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Kyungpook Mathematical Journal 2021; 61(2): 395-408

Published online June 30, 2021

Copyright © Kyungpook Mathematical Journal.

On f-biharmonic Submanifolds of Three Dimensional Trans-Sasakian Manifolds

Avijit Sarkar* and Nirmal Biswas

Department of Mathematics, University of Kalyani, Kalyani 741235, West Bengal, India
e-mail : avjaj@yahoo.co.in and nirmalbiswas.maths@gmail.com

Received: December 20, 2019; Revised: November 20, 2020; Accepted: November 23, 2020

The object of the present paper is to study f-biharmonic submanifolds of three dimensional trans-Sasakian manifolds. We find some necessary and sufficient conditions for such submanifolds to be f-biharmonic.

Keywords: trans-Sasakian manifolds, invariant submanifolds, anti-invariant submanifolds, f-biharmonic submanifolds

Let M and N be two Riemannian manifolds, a harmonic map ψ:MN is any critical point of the energy equation

E(ψ)=12M |dψ| 2dvg ,

where dvg denotes the volume element of g, and the Euler-Lagrange equation corresponding to E(ψ ) is τ(ψ)=tracedψ=0.

In 1983, Eells and Lemaire [9] introduced the notion of biharmonic maps, which are a natural generalization of harmonic maps. A biharmonic map ψ:MN is a critical point of the energy equation

E2(ψ)=12M |τψ| 2dvg ,

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

τ2(ψ)=Δτ(ψ)trace(RN(dψ,τ(ψ))dψ)=0.

Here Δ is the Laplacian operator given by ΔV=tr(2V), and RN is the curvature tensor on the manifold N defined as RN(X,Y)=[X,Y][X,Y].

Let M be the submanifold of the manifold M¯, if the biharmonic map ψ:MM¯ is an isometric immersion then M is biharmonic submanifold of 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 Sn×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 differentiable function f:MR. Now, f-harmonic maps are the critical points of the f-energy functional Ef(ψ) for the maps ψ:MN between Riemannian manifolds, where

Ef(ψ)=12M f|dψ|2dvg.

The Euler-Lagrange equation corresponding to Ef(ψ) is given by

τf(ψ)=fτ(ψ)+dψ(gradf)=0.

Analgously f-biharmonic maps are critical points of the f-bienergy functional E2,f(ψ) for maps ψ:MN between Riemannian manifolds where

E2,f(ψ)=12Mf|τψ|2dvg.

The Euler-Lagrange equation corresponding to E2,f(ψ) is given by

τ2,f(ψ)=fτ2(ψ)+(Δf)τ(ψ)+2(gradf)ψτ(ψ)=0.

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

Harmonic maps biharmonic maps 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,β), 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 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 M¯. For such manifolds, we know [3]

ϕ2X=X+η(X)ξ,  η(ξ)=1, η(X)=g(X,ξ),  g(ϕX,ϕY)=g(X,Y)η(X)η(Y), ϕξ=0,  ηoϕ=0,  g(X,ϕY)=g(ϕX,Y)

for any X,Yχ(M¯), where χ(M¯) denotes the Lie algebra of all vector fields on M¯.

For a contact metric manifold (M¯,ϕ,ξ,η,g), we define a (1,1) tensor field h by h=12Lξϕ and L is the usual Lie derivative. Then h is symmetric and satisfies the following relations

hξ=0,hϕ=ϕh,tr(h)=tr(ϕh)=0,η(hX)=0

for any X,Yχ(M¯).

Moreover, if ¯ denotes the Levi-Civita connection with respect to g, then the following relation holds

¯Xξ=ϕXϕhX.

A connected manifold M¯ with almost contact metric structure (ϕ,ξ,η,g) is called a trans-Sasakian manifold [23] if (M¯×R,J,G) belongs to the class W4 [13], where J is an almost complex structure on M¯×R which is defined by

J(X,fddt)=(ϕXfξ,η(X)ddt) 

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

(¯Xϕ)Y=α(g(X,Y)ξη(Y)X)+β(g(ϕX,Y)ξη(Y)ϕX)

for smooth functions α,β on M¯, where ¯ denote the covariant derivative with respect to g. Generally, M¯, is said to be a trans-Sasakian manifold of type (α,β).

In a three-dimensional trans-Sasakian manifold the curvature tensor with respect to the Levi-Civita connection ¯ is as follows [7]:

R(X,Y)Z=(r2+2ξβ2(α2β2))(g(Y,Z)Xg(X,Z)Y)    g(Y,Z)[(r2+ξβ3(α2β2))η(X)ξ    η(X)(ϕgradαϕgradβ)+(Xβ+(ϕX)α)ξ]    +g(X,Y)[(r2+ξβ3(α2β2))η(Y)ξ    η(Y)(ϕgradαϕgradβ)+(Yβ+(ϕY)α)ξ]    [(Zβ+(ϕZ)α)η(Y)+(Yβ+(ϕY)α)η(Z)    +(r2+ξβ3(α2β2))η(Y)η(Z)]X    +[(Zβ+(ϕZ)α)η(X)+(Xβ+(ϕX)α)η(Z)    +(r2+ξβ3(α2β2))η(X)η(Z)]Y,

where r is the scalar curvature of the manifold.

Let Mm (m<;n) be the submanifold of a contact metric manifold M¯n. Let and ¯ be the Levi-Civita connections of M and M¯, respectively. Then for any vector fields X,Yχ(M), the second fundamental form σ is defined by

¯XY=XY+σ(X,Y).

For any section of the normal bundle TM, we have

¯XN=ANX+N,

where denotes the normal bundle connection of M. The second fundamental form σ and the shape operator AN are related by

g(ANX,Y)=g(σ(X,Y),N).

For any vector field Xχ(M), we can right

ϕX=TX+NX,

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

ϕV=tV+nV,

where tV and nV are the tangential and normal components of ϕV.

The submanifold M is said to be invariant if ϕXTM for any vector field X. On other hand M is said to be an anti-invariant submanifold if ϕXTM for any vector field X

We know for a isometric immersion ψ [24]

τ(ψ)=trdψ=trσ=mH,

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

τ2(ψ)=mΔHtr(R(dψ,mH)dψ).

By some classical and straightforward computations, we have

ΔH=m2grad|H|2+tr(σ(.,AH.))+2tr(AH(.))+ΔH.

Using (3.3) in (3.2), we have

τ2(ψ)=m22grad|H|2+mtr(σ(.,AH.))+2mtr(AH(.))+mΔHtr(R(dψ,mH)dψ).

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

τ2,f(ψ)=fτ2(ψ)+(Δf)τ(ψ)+2(gradf)ψτ(ψ)=0.

By simple calculation we have the above equation is equivalent to

τ2(ψ)+mΔffH+2m(AHgrad(lnf)+grad(lnf)H)=0.

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 M¯. Then M is f-biharmonic if and only if the following equations hold

ΔH+tr(σ(.,AH.))+ΔffH+2grad(lnf)H=2(r2+2ξβ2(α2β2))H+2[(r2+ξβ3(α2β2))η(H)ξη(H)(NgradαNgradβ)+ξβHξαn(H)]+[2ξβ+(r2+ξβ3(α2β2))]H,

and

grad|H|22trAHgrad(lnf)+2tr(AH,.)=2[(r2+ξβ3(α2β2))η(H)ξTη(H)(TgradαTgrad β)+t(H)ξα][(gradβ)Tη(H)+g(gradβ,H)ξT+g(gradα,ϕH)ξT+(r2+ξβ3(α2β2))η(H)ξT].

Proof. Form (2.7) we have

R(X,Y)Z=(r2+2ξβ2(α2β2))(g(Y,Z)Xg(X,Z)Y)  g(Y,Z)[(r2+ξβ3(α2β2))η(X)ξ  η(X)(ϕgradαϕgradβ)+(Xβ+(ϕX)α)ξ]  +g(X,Y)[(r2+ξβ3(α2β2))η(Y)ξ  η(Y)(ϕgradαϕgradβ)+(Yβ+(ϕY)α)ξ]  [(Zβ+(ϕZ)α)η(Y)+(Yβ+(ϕY)α)η(Z)  +(r2+ξβ3(α2β2))η(Y)η(Z)]X  +[(Zβ+(ϕZ)α)η(X)+(Xβ+(ϕX)α)η(Z)  +(r2+ξβ3(α2β2))η(X)η(Z)]Y.

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

R(ei,Y)ei=(r2+2ξβ2(α2β2))(g(H,ei)eig(ei,ei)H)  g(H,ei)[(r2+ξβ3(α2β2))η(ei)ξ  η(ei)(ϕgradαϕgradβ)+(eiβ+(ϕei)α)ξ]  +g(ei,ei)[(r2+ξβ3(α2β2))η(H)ξ  η(H)(ϕgradαϕgradβ)+(Hβ+(ϕH)α)ξ]  [(eiβ+(ϕei)α)η(H)+(Hβ+(ϕH)α)η(ei)  +(r2+ξβ3(α2β2))η(H)η(ei)]ei  +[(eiβ+(ϕei)α)η(ei)+(eiβ+(ϕei)α)η(ei)  +(r2+ξβ3(α2β2))η(ei)η(ei)]H.

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

tr(R(.,H).)=2(r2+2ξβ2(α2β2))H+2[(r2+ξβ3(α2β2))η(H)ξη(H)(ϕgradαϕgradβ)+ξβHξαϕ(H)][(gradβ)Tη(H)+g(gradβ,H)ξT+g(gradα,ϕH)ξT+(r2+ξβ3(α2β2))η(H)ξT] +[2η(gradβ)+(r2+ξβ3(α2β2))]H.

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

tr(R(.,H).)=grad|H|2+tr(σ(.,AH.))+2tr(AH(.))    +ΔH+ΔffH2(AHgrad(lnf))+2grad(lnf)H.

Therefore we have

grad|H|2+tr(σ(.,AH.))+2tr(AH(.))+ΔH+ΔffH2(AHgrad(lnf))+2grad(lnf)H=2(r2+ξβ2(α2β2))H+2[(r2+ξβ3(α2β2))η(H)ξη(H)(ϕgradαϕgradβ)+ξβHξαϕ(H)][(gradβ)Tη(H)+g(gradβ,H)ξT+g(gradα,ϕH)ξT  +(r2+ξβ3(α2β2))η(H)ξT]+[2η(gradβ)+(r2+ξβ3(α2β2))]H.

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 M¯.

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

    ΔH+tr(σ(.,AH.))+ΔffH+2grad(lnf)H=2(r2+2ξβ2(α2β2))H+2[(r2+ξβ3(α2β2))η(H)ξ+ξβHξαn(H)]+[2ξβ+(r2+ξβ3(α2β2))]H,

    and

    grad|H|22trAHgrad(lnf)+2tr(AH,.)=2[(r2+ξβ3(α2β2))η(H)ξTη(H)(TgradαTgrad β)+t(H)ξα][(gradβ)Tη(H)+g(gradβ,H)ξT+g(gradα,ϕH)ξT+(r2+ξβ3(α2β2))η(H)ξT].

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

    ΔH+tr(σ(.,AH.))+ΔffH+2grad(lnf)H=2(r2+2ξβ2(α2β2))H+2[(r2+ξβ3(α2β2))η(H)ξη(H)(NgradαNgradβ)+ξβHξαn(H)]+[2ξβ+(r2+ξβ3(α2β2))]H,

    and

    grad|H|22trAHgrad(lnf)+2tr(AH,.)=2[(r2+ξβ3(α2β2))η(H)ξT+t(H)ξα][(gradβ)Tη(H)+g(gradβ,H)ξT+g(gradα,ϕH)ξT+(r2+ξβ3(α2β2))η(H)ξT].

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

    ΔH+tr(σ(.,AH.))+ΔffH+2grad(lnf)H=2(r2+2ξβ2(α2β2))H+2[(r2+ξβ3(α2β2))η(H)ξη(H)(NgradαNgradβ)+ξβHξαn(H)]+[2ξβ+(r2+ξβ3(α2β2))]H,

    and

    grad|H|22trAHgrad(lnf)+2tr(AH,.)=2[η(H)(TgradαTgradβ)+t(H)ξα][(gradβ)Tη(H)].

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

    ΔH+tr(σ(.,AH.))+ΔffH+2grad(lnf)H=2(r2+2ξβ2(α2β2))H+ξβHξαn(H)]+[2ξβ+(r2+ξβ3(α2β2))]H,

    and

    grad|H|22trAHgrad(lnf)+2tr(AH,.)=2t(H)ξα[g(gradβ,H)ξT+g(gradα,ϕH)ξT],

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

    ΔH+tr(σ(.,AH.))+ΔffH+2grad(lnf)H=2(r2+2ξβ2(α2β2))H+2[(r2+ξβ3(α2β2))η(H)ξη(H)(NgradαNgradβ)+ξβHξαn(H)]+[2ξβ+(r2+ξβ3(α2β2))]H,

    and

    grad|H|22trAHgrad(lnf)+2tr(AH,.)=2[(r2+ξβ3(α2β2))η(H)ξTη(H)(TgradαTgradβ][(gradβ)Tη(H)+g(gradβ,H)ξT+g(gradα,ϕH)ξT+(r2+ξβ3(α2β2))η(H)ξT].

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 ξT=0.

  • (4) If ξ is tangent to M then η(H)=0 and ξ=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 M¯ with non zero constant mean curvature H and ξ is tangent to M, then M proper f-biharmonic if and only if

|σ|2=3r27ξβ+7(α2β2)Δff,

and AHgrad(lnf)=0, or equivalent if and only if

ScalM=3r2+9ξβ8(α2β2)+Δff3|H|2.

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

ΔH+tr(σ(.,AH.))+ΔffH+2grad(lnf)H=2(r2+2ξβ2(α2β2))H+2[(r2+ξβ3(α2β2))η(H)ξη(H)(NgradαNgradβ)+ξβHξαn(H)]+[2ξβ+(r2+ξβ3(α2β2))]H,

and

grad|H|22trAHgrad(lnf)+2tr(AH,.)=2[η(H)(TgradαTgradβ)+t(H)ξα][(gradβ)Tη(H)].

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

tr(σ(.,AH.))=2(r2+2ξβ2(α2β2))H+2[(r2+ξβ3(α2β2))η(H)ξη(H)(NgradαNgradβ)+ξβHξαn(H)]+[2ξβ+(r2+ξβ3(α2β2))]HΔffH,

and AHgrad(lnf)=0. Thus, the second equation is trivial and the first equation becomes

trσ(.,AH.)=[3r27ξβ+7(α2β2)Δff]H.

Now since trσ(.,AH.)=|σ|2H and H is non zero, so we have form above equation

|σ|2=3r27ξβ+7(α2β2)Δff.

Now from the Gauss formula we have

ScalM= i,jg(R(ei,ej)ej,ei)|σ|22H2.

Using (2.7) in the above equation we have

ScalM=3r29ξβ8(α2β2)+Δff3|H|2.

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

3r27ξβ+7(α2β2)Δff

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

|σ|2=3r27ξβ+7(α2β2)Δff,

Since |σ|20, this is not possible if

3r27ξβ+7(α2β2)Δff.

Theorem 3.3. Let M be a submanifold of a three dimensional trans-Sasakian manifold M¯ with non zero constant mean curvature H such that ξ and ϕ H are tangent to M. Define F(f,α,β) on M by

F(f,α,β)=2r9ξβ+9(α2β2)Δff.
  • (1) if inf F(f,α,β) is non-positive, M is not f-biharmonic.

  • (2) if F(f,α,β) is positive and M is proper f-biharmonic then

    0<|H|212F(f,α,β).

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

ΔH+tr(σ(.,AH.))+ΔffH+2grad(lnf)H=2(r2+2ξβ2(α2β2))H+ξβHξαn(H)]+[2ξβ+(r2+ξβ3(α2β2))]H

and

grad|H|22trAHgrad(lnf)+2tr(AH,.)=2t(H)ξα[g(gradβ,H)ξT+g(gradα,ϕH)ξT].

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

ΔH+tr(σ(.,AH.))=[2r9ξβ+9(α2β2)Δff]        =F(f,α,β)H,

where

F(f,α,β)=2r9ξβ+9(α2β2)Δff.

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

<ΔH,H>+<tr(σ(.,AH.)),H>=F(f,α,β)|H|2.

Now using the results <tr(σ(.,AH.)),H>=|AH|2, and Δ|H|2=2(<ΔH,H>|H|2), in the above equation we have

|AH|2+|ΔH|2=F(f,α,β)|H|2.

By using the Cauchy-Schwarz inequality |AH|212tr(AH)=2|H|4, the equation reduces to

F(f,α,β)|H|2=|AH|2+|H|22|H|4+|H|22|H|4.

Therefore F(f,α,β)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.

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