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### Article

Kyungpook Mathematical Journal 2022; 62(2): 333-345

Published online June 30, 2022

### ∗-Ricci Soliton on (κ<0,μ)-almost Cosymplectic Manifolds

Savita Rani and Ram Shankar Gupta*

University School of Basic and Applied Sciences, Guru Gobind Singh Indraprastha University, Sector-16C, Dwarka, New Delhi-110078, India
e-mail : mansavi.14@gmail.com and ramshankar.gupta@gmail.com

Received: June 22, 2021; Revised: October 10, 2021; Accepted: November 15, 2021

We study *-Ricci solitons on non-cosymplectic (κ,μ)-acs (almost cosymplectic) manifolds M. We find *-solitons that are steady, and such that both the scalar curvature and the divergence of the potential field is negative. Further, we study concurrent, concircular, torse forming and torqued vector fields on M admitting Ricci and *-Ricci solitons. Also, we provide some examples.

Keywords: *-Ricci soliton, (κ, μ)-almost cosymplectic manifolds, Nullity distribution, Torse forming vector field

In the framework of Riemannian geometry, Blair et al. [2] introduced a (κ,μ)-space in contact geometry as contact manifold M whose curvature tensor R satisfies

R(U,W)ξ=κ(η(W)Uη(U)W)+μ(η(W)hUη(U)hW),

for any U and WTM and for h a symmetric operator given by h=12Lξψ, where ψ is a (1,1) tensor field and κ,μ are constants. If κ = 1 and h = 0, then (κ,μ)-spaces reduces to the Sasakian manifolds. Non-Sasakian manifolds have proven to be more interesting in this context. The unit tangent sphere bundle of a flat Riemannian manifold with the usual contact metric structure is an example of non-Sasakian spaces of this type. Moreover, this type of spaces is invariant under D-homothetic transformations. These factors drive the study of this type of manifold. Boeckx proved that a non-Sasakian contact metric manifold satisfying (1.1) is completely determined locally by its dimension for the constant values of κ,μ [3].

If κ,μ are functions, then a contact metric manifold satisfying (1.1) is called a generalized (κ,μ)-space [16]. Koufogiorgos et al. introduced a (κ,μ,ν)-contact metric manifold which satisfies [17]

R(U,W)ξ=κ(η(W)Uη(U)W)+μ(η(W)hUη(U)hW)            +ν(η(W)ψhUη(U)ψhW),

for smooth functions κ,μ,ν on M2n+1 and they proved that it reduces to a (κ,μ)-manifold in dimension 2n+15. Later, the generalized (κ,μ)-space with divided R5 was introduced in [5]. This further generalizes generalized (κ,μ)-spaces. Here R5=R5,1R5,2 is divided into R5,1 and R5,2 such that

R5,1(U,W)X=g(hW,X)hUg(hU,X)hW,R5,2(U,W)X=g(ψhW,X)ψhUg(ψhU,X)ψhW,

for any U, W, XTM. Sharma and Vrancken [19] studied (κ,μ)-contact manifolds with non-Killing conformal vector fields. Chen [7] examined a closed Einstein-Weyl structure and two Einstein-Weyl structures on an acs (κ<0,μ)-manifold. Ghosh and Sharma [12] investigated a (κ,μ)-contact manifold with a divergence free Cotton tensor. De and Sardar [10], studied Bach-flat (κ,μ)-almost co-Kähler manifolds.

In the last few decades there has been extensive study about Ricci solitons and *-Ricci solitons on manifolds. The notion of Ricci solitons was introduced by Hamilton as a natural generalization of Einstein metrics.

A Ricci soliton on a Riemannian manifold satisfies the following equation [8]

Ric+12LYg=ρg,

where LY is the Lie-derivative along a smooth potential field Y, g is the Riemannian metric, ρ is a real scalar and Ric is the Ricci tensor. Ricci solitons serve as solutions to the Ricci flow of Hamilton \cite{18, which evolve along the symmetries of the flow. The soliton is steady, expanding or shrinking if ρ = 0, <0 or >0, respectively.

In 1959, Tachibana [21] introduced the notion of *-Ricci tensors on almost Hermitian manifolds. Later, Hamada [13] defined *-Ricci tensors of real hypersurfaces in non-flat complex space forms, and then Kaimakamis et al. [15] introduced the notion of *-Ricci solitons in non-flat complex space forms.

The *-Ricci tensor on an almost contact metric (a.c.m) manifold M ([13]) is defined by

Ric(U,W)=12trace(XR(U,ψW)ψX),U,WTM,

where ψ is a (1, 1)-tensor field and R is a Riemann curvature tensor.

A *-Ricci soliton on a Riemannian manifold (M, g) is a generalisation of the *-Einstein manifold and defined as [15]:

Ric+12LYg=ρg.

Many authors have studied solitons on a.c.m manifolds: Sharma initiated the study of Ricci solitons in contact geometry as a K-contact and (κ,μ)-contact metric [18]. Suh et al. studied Ricci solitons on almost co-Kähler manifolds [20]. Dai [9] investigated *-Ricci soliton on a (κ<0,μ)-acs manifold and proved that there do not exist *-Ricci soliton on a (κ<0,μ)-acs manifold.

In view of the above, we study the existence of *-Ricci solitons on a non-cosymplectic (κ,μ)-acs manifold. Contrary to the non-existence of *-Ricci soliton in [9] we show that there exists a steady *-Ricci soliton. Further, we study the existence/non-existence of some particular type of potential vector field Y on a non-cosymplectic (κ,μ)-acs manifold admitting Ricci solitons and *-Ricci solitons.

This paper is organised as follows: in Section 2, we give some background which is necessary to understand the subsequent sections. In Section 3, we study *-Ricci solitons on non-cosymplectic (κ,μ)-almost cosymplectic manifolds. Section 4 deals with the study of a potential vector field Y as concurrent, concircular, torse forming and torqued vector field. In Section 5, we give some examples.

A smooth Riemannian manifold M2n+1 is called an a.c.m manifold if there exists structure tensors (ψ,ξ,η,g) satisfying [1]

ψ2=id+ηξ,ηoψ=0,ψξ=0,η(ξ)=1,
g(ψU,ψW)=g(U,W)η(U)η(W),

for any U,WTM, where ψ is a tensor field of type (1,1), ξ a global vector field and η a 1-form. We denote by 𝚽 the fundamental 2-form which is defined as Φ(U,W)=g(U,ψW). An a.c.m manifold M2n+1 with dη=Φ is called contact manifold. An a.c.m manifold with η and 𝚽 closed is called an almost cosymplectic manifold. A normal almost cosymplectic manifold is called cosymplectic manifold.

On an acs manifold, we have [4]

h=12Lξψ,h=hoψ,
hξ=0,hψ+ψh=0,trh=trh=0,
ψlψl=2h2,
Uξ=hU,

where l=R(.,ξ)ξ and both h, h' are symmetric operators with respect to metric g.

Also, on an acs manifold, we have [11]

R(U,W)ξ=(Wψh)U(Uψh)W,

for any U,WTM.

The (κ,μ)-nullity distribution of an acs manifold M2n+1 for (κ,μ)R2 is a distribution [11]

N(κ,μ):pNp(κ,μ)={ZTp(M)|R(U,W)X=κ(g(W,X)U          g(U,X)W)+μ(g(W,X)hUg(U,X)hW)}.

Endo [11] introduced (κ,μ)-acs manifolds with the Reeb vector ξ in the (κ,μ)-nullity distribution, which satisfy (1.1). If ξ is in the (κ0,μ)-nullity distribution, then such manifolds are called non-cosymplectic (κ,μ)-acs manifolds. For more work about non-cosymplectic (κ,μ)-acs manifolds, please see [11].

On (κ<0,μ)-acs manifold using (1.1) and (2.7), we have

(Wψh)U(Uψh)W=κ(g(W,X)Ug(U,X)W)        +μ(g(W,X)hUg(U,X)hW).

Also, on a Riemannian manifold M we have following [23]:

(XLYg)(U,W)=g((LY)(X,U),W)+g((LY)(X,W),U),
(LYR)(U,W)X=(ULY)(W,X)(WLY)(U,X),

U,W,XTM.

Let M2n+1 be a (κ,μ)-acs manifold. Then, from (1.1), we obtain

l=κψ2+μh.

Using (3.1) in (2.5), we get

h2=κψ2.

Let U be an eigenvector of h for eigenvalue θ with Uξ, then using (2.3), (2.4) and (3.2), we get

θ2=κ.

From (3.3), we find that κ0. However, κ = 0 if and only if h=0. Here, we study non-cosymplectic (κ<0,μ)-acs manifolds.

On an acs manifold M2n+1(κ<0,μ), we have([4, 9])

Q=μh+2nκηξ,
Ric(U,W)=κg(ψU,ψW),

where Q denotes Ricci operator and U,WTM.

Lemma 3.1. Let M2n+1(κ<0,μ) be an acs manifold satisfying (1.5), then

(LYR)(U,W)X=2κ2(η(U)g(W,X)η(W)g(U,X))ξ      +2κμ(η(U)g(hW,X)η(W)g(hU,X))ξ      +2κ(g(hU,X)hWg(hW,X)hU),

for any U,W,XTM.

Proof. Using (3.5) in (1.5), we obtain

(LYg)(W,X)=2ρg(W,X)+2κg(ψW,ψX).

Differentiating (3.7) with respect to U on M and using (2.6), we find

(ULYg)(W,X)=2κ(g(hU,W)η(X)+g(hU,X)η(W)).

Using (3.8) in (2.9), we obtain

g((LY)(U,W),X)+g((LY)(U,X),W)=2κ(g(hU,W)η(X)                  +g(hU,X)η(W)).

Similarly, we get

g((LY)(W,X),U)+g((LY)(W,U),X)=2κ(g(hW,X)η(U)                  +g(hW,U)η(X)),
g((LY)(X,U),W)+g((LY)(X,W),U)=2κ(g(hX,U)η(W)                  +g(hX,W)η(U)).

Adding (3.9) and (3.10) and subtracting (3.11), we get

g((LY)(U,W),X)=2κg(hU,W)η(X),

which gives

(LY)(W,X)=2κg(hW,X)ξ.

Differentiating (3.12) along U on M, we find

(ULY)(W,X)=2κ(g((Uh)W,X)ξ+g(hW,X)hU).

Using (3.13) in (3.10), we obtain

(LYR)(U,W)X=2κ(g((Wψh)U,X)ξg(ψhU,X)ψhW)      +2κ(g((Uψh)W,X)ξg(ψhW,X)ψhU).

Using (2.8) in (3.14), we get (3.6). Hence, the proof of Lemma is complete.

Theorem 3.2. Let M2n+1(κ<0,μ) be an acs manifold satisfying (1.5). Then *-soliton is steady.

Proof. Let {ei}i=02n be a local orthonormal basis for TM. From (3.6), we obtain

g((LYR)(U,W)X,U)=2κ2(η(U)g(W,X)η(W)g(U,X))η(U)        +2κμ(η(U)g(hW,X)η(W)g(hU,X))η(U)        +2κ(g(hU,X)g(hW,U)g(hW,X)g(hU,U)),

for U,W,XTM.

Contracting (3.15) over U, we get

(LYS)(W,X)=2κ2g(ψW,ψX)+2κμg(hW,X)+2κg(hX,hW),

which gives

(LYg)(QW,X)+g((LYQ)W,X)=2κ2g(ψW,ψX)+2κμg(hW,X)                  +2κg(ψhX,ψhW).

Using (3.7) in (3.16), we obtain

2(κ+ρ)g(QW,X)2κη(QW)η(X)+g((LYQ)W,X)=2κ2g(ψW,ψX)+2κμg(hW,X)+2κg(hW,hX).

Using (3.2) and (3.4) in (3.17), we find

(LYQ)W=2μρhW4nκρη(W)ξ.

From (3.4), we get

QW=μhW+2nκη(W)ξ.

Lie-derivative of (3.19) along Y gives

(LYQ)W=μ(LYh)W+2nκg(WY,ξ)ξ+2nκg(W,hY)ξ+2nκη(W)LYξ.

Comparing (3.18) and (3.20), we obtain

2μρhW4nκρη(W)ξ=μ(LYh)W+2nκg(WY,ξ)ξ        +2nκg(W,hY)ξ+2nκη(W)LYξ.

Putting W=ξ in (3.21), we get

4nκρξ=μhLYξ+2nκg(ξY,ξ)ξ+2nκLYξ.

Taking inner product of (3.22) with ξ, we get

κρ=0,

which implies ρ=0 as κ<0. Hence the proof of the Theorem.

Using (1.5), (2.4), (3.4), (3.5), and Theorem 3.2, we get r=2κn, and divY=2κn, where r and div denote the scalar curvature and divergence, respectively. Therefore, we have

Corollary 3.3. Let M2n+1(κ<0,μ) be an acs manifold satisfying (1.5), then divY=r<0.

Corollary 3.4. Let M2n+1(κ<0,μ) be an acs manifold satisfying (1.5), then Y cannot be ξ.

Proof. Suppose Y=ξ. Then, putting potential vector field Y=ξ in (1.5) and using (2.6), (3.5) and Theorem 3.2, we get

g(hU,W)=κg(ψU,ψW)

Taking trace of (3.23) and using (2.4), we obtain 2κn=0, which is not possible as κ<0. Hence the result.

Remark 3.5. In [9] the author proved that there do not exist *-Ricci soliton on non-cosymplectic (κ,μ)-acs manifolds. Unfortunately, there is a crucial error in their proofs. In page 4 of [9], the equation (3.5) should be corrected as

(LVR)(X,Y)Z=2κ2(η(X)g(Y,Z)η(Y)g(X,Z))ξ      +2κμ(η(X)g(hY,Z)η(Y)g(hX,Z))ξ      +2κ(g(hX,Z)hYg(hY,Z)hX).

In page 5 of [9], the equation (3.12) should be corrected as

(LVQ)X=2μλhX4nκλη(X)ξ.

Thereafter, the argument given in [9] is not useful to obtain correct result.

### 4. *-Ricci Solitons on (κ<0,μ)-acs Manifolds with Some Particular Potential Vector Fields

In 1944, Yano [22] introduced a torse-forming vector field as a generalization of concircular, concurrent and parallel vector fields.

Definition 4.1. A vector field V is called torse forming if

UV=fU+ω(U)V,

where fC(M), UTM and ωis a 1-form.

The vector field V is called concircular if ω in (4.1) vanishes identically. Concircular vector field is also known as geodesic vector field as its integral curve forms geodesics. It has interesting applications in general relativity and in the theory of conformal and projective transformation.

If V satisfies

UV=U,UTM,

then V is called concurrent. If V satisfies (4.1) with f=0, then V is called recurrent. Also, if f=ω=0 in (4.1), then V is called parallel.

Recently, in 2017, Chen [6] introduced a torqued vector field. If a non-vanishing V satisfies (4.1) with ω(V)=0, then V is called torqued, f the torqued function and ω torqued form of V. For more details about these vector fields (please see [6, 22]) and references therein.

Now, we have

Theorem 4.2. Let M2n+1(κ<0,μ) be an acs manifold satisfying (1.3). Then the potential vector field Y cannot be concurrent.

Proof. Suppose Y be a concurrent field, then using (3.4) and (4.2) in (1.3), we have

μg(hU,W)+2nκη(U)η(W)=(ρ1)g(U,W).

Contracting (4.3) over U and W, we obtain

2nκ=(ρ1)(2n+1).

From (4.4), we find that

κ=(ρ1)(2n+1)2n.

Using ρ=2nκ (cf. [20], Theorem 5.1) in (4.5), we get κ=2n+14n2>0, a contradiction as κ<0. Thus proof is complete.

Theorem 4.3. Let M2n+1(κ<0,μ) be an acs manifold satisfying (1.5), then the potential vector field Y cannot be concurrent.

Proof. Suppose Y be a concurrent field. Using (3.5), (4.2) and Theorem 3.2 in (1.5), we get

(1κ)g(U,W)+κη(U)η(W)=0.

Contracting (4.6) over U and W, we obtain κ=2n+12n>0, which is not possible as κ<0. Hence the result.

Theorem 4.4. Let M2n+1(κ<0,μ) be an acs manifold satisfying (1.5). If Y be a torse forming vector field, then

f+ω(Y)2n+1=2nκ2n+1,

where fC(M) satisfying (4.1).

Proof. Let Y be a torse forming field, then using (3.5), (4.1) and Theorem 3.2 in (1.5), we get

ω(U)g(Y,W)+ω(W)g(Y,U)+(2f2κ)g(U,W)=2κη(U)η(W).

Putting U=ei and W=ei and tracing i=1to2n+1, we obtain (4.7).

Corollary 4.5. Let M2n+1(κ<0,μ) be an acs manifold satisfying (1.5). If Y be a torqued vector field, then torqued function f is a constant and ξ cannot be torqued.

Proof. Suppose Y be a torqued field then ω(Y)=0. Using this condition in (4.7) we obtain f=2nκ2n+1. Whereby, we get f is a constant. As ω(ξ)0 so ξ cannot be torqued.

Theorem 4.6. Let M2n+1(κ<0,μ) be an acs manifold satisfying (1.5). If Y be a concircular vector field, then f is a constant given by

f=2nκ2n+1.

Proof. Let Y be a concircular field, then

UY=fU,fC(M).

Using (3.5), (4.10) and Theorem 3.2 in (1.4), we get

(fκ)g(U,W)+κη(U)η(W)=0.

Contracting (4.11) over U and W, we obtain (4.9).

Theorem 4.7. Let M2n+1(κ<0,μ) be an acs manifold satisfying (1.3). If Y be a torse forming vector field, then

ρ=f+ω(Y)2n+1+2nκ2n+1.

Proof. Let Y be a torse forming field. Using (4.1) and (3.4) in (1.3), we get

ω(U)g(Y,W)+ω(W)g(Y,U)+2μg(hU,W)+4nκη(U)η(W)                =2(ρf)g(U,W).

Taking trace of (4.13) over U and W, we obtain (4.12).

Theorem 4.8. Let M2n+1(κ<0,μ) be an acs manifold satisfying (1.3). If Y be a concircular vector field, then f is a constant given by

f=ρ2nκ2n+1.

Proof. Let Y be a concircular field. Using (3.4) and (4.10) in (1.3), we get

(fρ)g(U,W)+μg(hU,W)+2nκη(U)η(W)=0.

Contracting (4.15) over U and W, we obtain (4.14).

Corollary 4.9. Let M2n+1(κ<0,μ) be an acs manifold satisfying (1.3). If Y be a torqued vector field, then torqued function f is a constant.

Proof. Suppose Y be a torqued field then ω(Y)=0. Using this condition in (4.12) we obtain f=ρ2nκ2n+1. Hence f is a constant.

### 5. Examples of *-Ricci Soliton on (κ<0,μ)-acs Manifolds

In the following examples, the (1,1)-tensor ψ is defined as

ψ(e1)=e2,ψ(e2)=e1,ψ(e3)=0.

Example 5.1. Consider M={(x,y,z)R3:x0} with structure tensors

g=dxdx+dydyxdydzxdzdy+(x2+1)dzdz,e1=x,e2=y,e3=ξ=z+xy,η=dz,h=1200012x2000.

From (5.1), we have

eq,e2=0,q=1,3;e1,e3=e2.

The Koszul's formula with Riemannian connection ∇ is given by

2g(UW,X)=Ug(W,X)+Wg(X,U)Xg(U,W)g(U,W,X)    g(W,U,X)+g(X,U,W),

U,W,XTM.

From (5.2) and (5.3), we find

eqeq=0,q=1,2,3;e2e1=12e3=e1e2,e3e1=12e2=e1e3,e3e2=e2e3=12e1.

Computing Riemann curvature tensors using (5.4), we get

R(e2,e1)ξ=0,R(e2,ξ)ξ=14e2,R(e1,ξ)ξ=34e1.

Further, from (1.1), we obtain

R(e2,e1)ξ=0,R(eq,ξ)ξ=κeq+μheq,q=1,2.

From (2.4) and (3.3), we find that h on M satisfies

he1=κe1,he2=κe2,he3=0.

Using (5.1), (5.5), (5.6) and (5.7), we find that M is a (κ<0,μ)-acs manifold with κ=14,μ=1.

Further, from (3.5), we obtain

S(ei,ei)=14,i=1,2;S(e3,e3)=0,S(eq,ep)=0,qp;q,p=1,2,3.

Also, we have

Y=(zx4)x+(z2x22y4)yxz,[Y,e1]=(e1/4)+e3,[Y,e2]=e2/4,[Y,e3]=e1.

Now, we can see that

(LYg)(ep,eq)+2S(ep,eq)=2ρg(ep,eq),

for ρ=0 and p,q=1,2,3.

Hence M is a non-cosymplectic (14,1)-acs manifold admitting steady *-Ricci soliton. Also, divY=12.

Example 5.2. Consider M={(x,y,z)R3:x,y0} with

g=dxdxy2dxdz+dydyxdydzy2dzdxxdzdy+(x2+y24+1)dzdz,e1=x,e2=y,e3=ξ=z+xy+y2x,η=dz,h=34000343y83x4000,Y=9x16x9y16y2c1z,

where Y is potential field and c1 is an arbitrary constant. Then similar to Example 5.1, computation can be done to show that M is a non-cosymplectic (916,12)-acs manifold admitting steady *-Ricci soliton.

Example 5.3. Consider M={(x,y,z)R3:x,y0} with structure tensors

g=dxdxydxdz+dydyxdydzydzdxxdzdy+(x2+y2+1)dzdz,η=dz,e1=x,e2=y,e3=ξ=z+xy+yz,h=10y01x000.

Then, M is a non-cosymplectic (-1, 0)-acs manifold. Further, M admits steady *-Ricci soliton with potential vector field

Y=(x+ez+ez)x+(yez+ez)yz.

Moreover, M admits expanding Ricci soliton with potential vector field

Y=(2x+ez+ez)x+(2y+ezez)yz.

The authors are thankful to the referees for their valuable suggestions for improvement of the article. The first author is thankful to GGSIP University for research fellowship F.No. GGSIPU/DRC/2021/685.

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