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Kyungpook Mathematical Journal 2022; 62(4): 737-749

Published online December 31, 2022 https://doi.org/10.5666/KMJ.2022.62.4.737

Copyright © Kyungpook Mathematical Journal.

Some Geometric Properties of η-Ricci Solitons on α-Lorentzian Sasakian Manifolds

Shashikant Pandey and Abhishek Singh, Rajendra Prasad

Department of Mathematics and Astronomy, University of Lucknow, Lucknow, 226007 Uttar Pradesh, India
e-mail : shashi.royal.lko@gmail.com and lkoabhi27@gmail.com

Department of Mathematics and Astronomy, University of Lucknow, Lucknow, 226007 Uttar Pradesh, India
e-mail : rp.manpur@rediffmail.com

Received: December 20, 2021; Revised: April 19, 2022; Accepted: May 3, 2022

We investigate the geometric properties of η-Ricci solitons on α-Lorentzian Sasakian (α-LS) manifolds, and show that a Ricci semisymmetric η-Ricci soliton on an α-LS manifold is an η-Einstein manifold. Further, we study φ-symmetric η-Ricci solitons on such manifolds. We prove that φ-Ricci symmetric η-Ricci solitons on an α-LS manifold are also η-Einstein manifolds and provide an example of a 3-dimensional α-LS manifold for the existence of such solitons.

Keywords: η*-Ricci solitons, φ*-Symmetric, Ricci semisymmetric, φ*Ricci symmetric, η* Einstein manifolds

The Ricci flow, which is used to compute the canonical metric based on the smooth manifold, was proposed by Hamilton [19] in 1982. The Ricci flow provides an evolution expression of metrics for a Riemannian manifold as follows:

tgij(t)=2Rij.

The Einstein metric can be naturally generalized to Ricci solitons which are defined on the Riemannian manifold (M,g) [6]. The triplet (g,V,ω1) is a Ricci soliton where g,V are the Riemannian metric or the pseudo Riemannian metric, and vector field (potential vector field), respectively. The real scalar ω1 is expressed in terms of a Ricci tensor S, and a Lie derivative operator £V, as

£Vg+2S+2ω1g=0,

If ω1>0, the Ricci solitons is called expanding while it is called shrinking if ω1<0. The case ω1=0 represents the steady Ricci soliton [20]. The Einstein equation can be recovered from Ricci solitons for V=0. The metric expressed by (2) is generally known as quasi-Einstein [7], [8] and is used frequently in physical systems. The fixed Ricci flow points for tg=2S which are projected from metrics space onto its diffeomorphic quotient, modulo scaling, are referred to as the compact Ricci solitons. These solitons frequently arise in Ricci flow on compact manifolds for larger limiting cases. Ricci solitons play an interesting role in string theory, the initial aspects of which were discussed by Friedman [17]. The similarity solution of Ricci flow in Riemannian geometry was introduced by [19] as it explores the concept of a singularity. Several authors have studied the geometric properties of Ricci solitons over different manifolds, for instance see [11], [12], [14] and [15],[23].

Cho and Kimura [9] proposed the concept of η-Ricci solitons as type of generalized Ricci solitons. Calin and Crasmareanu [5], [6] extended this concept for Hopf hypersurfaces in complex space. The tuple (g,V,ω1,ω2) with constants ω1 and ω2 denote the η-Ricci solitons with the condition

£Vg+2S+2ω1g+2ω2ηη=0,

In the current scenario, η-Ricci solitons are studied by various researchers have considered such η-Ricci solitons, and have found interesting geometric properties in many contexts: on Lorentzian para-Sasakian manifolds [2], [28], gradient η-Ricci solitons [3], on ϵ-para Sasakian manifolds [21] and [4], quasi-Sasakian 3-manifolds [22], 3-dimensional Kenmotsu manifolds [24], Sasakian 3-manifolds [25], para-Sasakian manifolds [26] and para Kenmotsu manifolds [29] and studied Lorentzian Sasakian manifold [27] etc.

The structure of the paper is as follows. The neccessary basic theory about α-LS manifolds is given in Section 2. In Section 3, the geometric properties of η-Ricci solitons on α-LS manifolds are investigated. In Section 4, we show that Ricci semisymmetric η-Ricci solitons on α-LS manifolds reduce to an η-Einstein manifold. In Section 5, we study φ-symmetric η-Ricci solitons on α-LS manifolds. In Section 6, we show that a φ-Ricci symmetric η-Ricci soliton on such a manifold is also an η-Einstein manifold. Finally, we provide an example of a 3-dimensional α-LS manifold for the existence of such solitons.

A (2n+1)-dimensional differentiable manifold M is said to be an α-LS manifold if it admits a (1,1)-tensor field φ, a vector field ζ and 1-form η and Lorentzian metric g* which satisfy the following conditions:

φ2=I+ηζ,η(ζ)=1,
φζ=0,ηφ=0,
g(φE,φF)=g(E,F)+η(E)η(F),
g(E,ζ)=η(E),

for any vector fields E,F on M.

Also, α-LS manifolds satisfy [16],

Eζ=αφE,
(Eη)F=αg(φE,F),
(Eφ)F=αg(E,F)ζαη(F)E,

where ∇ has the usual meaning.

Moreover, on α-LS manifolds the following relations hold (see [1]):

R(ζ,E)F=α2[g(E,F)ζη(F)E],
R(E,F)ζ=α2[η(F)Eη(E)F],
R(ζ,E)ζ=α2[η(E)ζ+E],
S(E,ζ)=2nα2η(E),
Qζ=2nα2ζ,
S(ζ,ζ)=2nα2,
S(φE,φF)=S(E,F)+2nα2η(E)η(F),

where R,S,Q are the Riemannian curvature, Ricci tensor and Ricci operator, respectively while α is a constant. S and Q are related by S(E,F)=g(QE,F) for all E,Fχ(M).

As per the definition of the Lie derivative, we have

(£ζg)(E,F)=(ζg)(E,F)+g(αφE,F)+g(E,αφF)    =2αg(φE,F),
(£ζφ)E=[ζ,φE]φ([ζ,E])    =ζφEφEζφ(ζE)+φ(Eζ)    =ζφEφ(ζE)    =(ζφ)E    =0.

Definition 2.1. ([18]) A (2n+1)-dimensional α-LS manifold with constants a, b and vector fields E,F defined on M satisfying the condition

S(E,F)=ag(E,F)+bη(E)η(F),

is called an η-Einstein manifold.

Definition 2.2. An α-LS manifold M with vector fields E,F defined on M satisfying the condition

R(E,F)S=0,

is called Ricci semisymmetric.

Definition 2.3. ([13]) An α-LS manifold M with vector fields E,F defined on M satisfying the condition

φ2((EQ)(F))=0,

is called φ-Ricci symmetric.

If E and F are orthogonal to ζ, then the manifold is said to be locally φ-Ricci symmetric.

Definition 2.4. ([30]) An α-LS manifold M is called φ-symmetric if

φ2((HR)(E,F)G)=0,

for all vector fields E,F,G,H on M.

Let (M,φ,ζ,η,g) be an α-LS manifold. The (potential) vector field V spans, and is orthogonal to, ζ, so we only consider the case V=ζ. Using equation (3), we obtain

(£ζg)(E,F)+2S(E,F)+2ω1g(E,F)+2ω2η(E)η(F)=0,

Because of (2.15), the (3.1) becomes

2αg(φE,F)+2S(E,F)+2ω1g(E,F)+2ω2η(E)η(F)=0,

for any E,Fχ(M), or equivalently:

S(E,F)=αg(φE,F)ω1g(E,F)ω2η(E)η(F),

for any E,Fχ(M).

The data (g,ζ,ω1,ω2) which satisfy the equation (3.1) is said to be an η-Ricci soliton on M (see [9]), in particular, if ω2=0, (g,ζ,ω1) is a Ricci soliton [20]. Ricci solitons is said to be expanding, shrinking and steady according as ω1 is positive, negative or zero [10].

From (3.2), we get

S(φE,φF)=αg(φE,F)ω1g(φE,φF).

Taking F=ζ in (3.2), we have

S(E,ζ)=(ω2ω1)η(E).

From (2.11) and (3.4), we obtain

ω2ω1=2nα2.

In this section, we investigate Ricci semisymmetric α-LS manifolds on η-Ricci solitons . According to Definition 2.2, we get

R(E,F)S=0,

which implies that

S(R(E,F)G,K)+S(G,R(E,F)K)=0.

Taking E=ζ in (4.1), we obtain

S(R(ζ,F)G,K)+S(G,R(ζ,F)K)=0.

Using (2.8) in (4.2), we infer

α2[g(F,G)S(ζ,K)+η(G)S(F,K)    +g(F,K)S(G,ζ)η(K)S(G,F)]=0.

Since α0, we obtain

g(F,G)S(ζ,K)+η(G)S(F,K)+g(F,K)S(G,ζ)η(K)S(G,F)=0.

Using (3.2) and (3.4) in (4.4), we infer

(ω2ω1)[g(F,G)η(K)+g(F,K)η(G)]+α[g(φF,K)η(G)+g(φF,G)η(K)] +ω1[g(F,K)η(G)+g(F,G)η(K)]  +2ω2η(F)η(G)η(K)  =0.  

Putting K=ζ in (4.5) and using (2.2), we infer

αg(φF,G)+ω2[g(F,G)+η(F)η(G)]=0,

which is equivalent to

αg(φF,G)+ω2g(φF,φG)=0,

Putting G=φG, we obtain

αg(φF,φG)+ω2g(φF,G)=0.

Subtracting (4.7) and (4.8), we get

(αω2)[g(φF,G)g(φF,φG)]=0.

for any F,G on M and follows ω2=α. From the relation (3.5), we have ω1=α2nα2.

Now from above, we are able to state our results.

Theorem 4.1. If (M,φ,ζ,η,g) is a Ricci semisymmetric α-LS manifold, (g,ζ,ω1,ω2) is an η-Ricci soliton on M, then ω2=α and ω1=α2nα2.

In case ω2=0, we derive the following.

Corollary 4.2. If α-LS manifolds (M,φ,ζ,η,g) satisfy the condition R(ζ,F)S=0, then there does not exist Ricci solitons with potential vector field ζ.

From (3.2), (3.5) and (4.7), we obtain

S=(ω2ω1){g(E,F)+η(E)η(F)}=2nα2{g(E,F)+η(E)η(F)}.

As a consequence, we can state following proposition.

Proposition 4.3. Let (M,φ,ζ,η,g) be an α-LS manifold. If M is Ricci semisymmetric and (g,ζ,ω1,ω2) is an η-Ricci soliton on M, then the manifold is an η-Einstein manifold.

Consider φ-symmetric η-Ricci solitons on α-LS manifolds. Then from definition 2.3, we have

φ2((EQ)(F))=0.

Using (2.1) and (5.1), we get

(EQ)(F)+η((EQ)(F))ζ=0.

Taking inner product in (5.2) with G, we have

g((EQ)(F),G)+η((EQ)(F))η(G)=0,

which implies

g(EQFQ(EF),G)+η((EQ)(F))η(G)=0.

After simplification, we obtain

g(EQF,G)S(EF,G)+η((EQ)(F))η(G)=0.

Putting F=ζ in (5.5), we have

g(EQζ,G)S(Eζ,G)+η((EQ)(ζ))η(G)=0.

Using (2.5) and (3.3) in (5.6), we infer

(ω2ω1)αg(φE,G)αS(φE,G)+η((EQ)(ζ))η(G)=0.

Taking G=φG in (5.7), we get

(ω2ω1)αg(φE,φG)αS(φE,φG)=0.

Since α0, we get

(ω2ω1)g(φE,φG)S(φE,φG)=0.

Now, using (3.3) in (5.9), we infer

ω2g(φE,φG)+αg(E,φG)=0.

Taking E=φE in (5.10), we have

ω2g(E,φG)+αg(φE,φG)=0.

Subtracting (5.10) from (5.11), we obtain

(ω2α)(g(φE,φG)g(E,φG))=0,

for any E,G and follows ω2=α. From the relation (3.5), we obtain ω1=α2nα2.

Now, we are ready to state the following results.

Theorem 5.1. If (M,φ,ζ,η,g) is φ-symmetric on α-LS manifolds, (g,ζ,ω1,ω2) is an η-Ricci soliton on M, then ω2=α and ω1=α2nα2.

In case ω2=0, we can state next result.

Corollary 5.2. If α-LS manifolds (M,φ,ζ,η,g) satisfy the condition φ2((EQ)(ζ))=0, then there does not exist Ricci solitons with potential vector field ζ.

From (3.2), (3.4) and (5.10) we have

S=(ω2ω1){g(E,F)+η(E)η(F)}=2nα2{g(E,F)+η(E)η(F)}.

This leads to the following proposition.

Proposition 5.3. Let (M,φ,ζ,η,g) be an α-LS manifold. If M is a φ-symmetric and (g,ζ,ω1,ω2) is an η-Ricci soliton on M, then the manifold is an η-Einstein manifold.

This section is devoted to the study of φ-Ricci symmetric η-Ricci solitons on α-LS manifolds.

Consider a φ-Ricci symmetric η-Ricci solitons on α-LS manifolds. Then from definition 2.4, we have

φ2((HR)(E,F)G)=0.

Using (2.1), we infer

(HR)(E,F)Gη((HR)(E,F)G)ζ=0,

Taking inner product with U in (57), we obtain

g((HR)(E,F)G,U)η((HR)(E,F)G)g(ζ,U)=0.

Let {σi}, i=1,2,...,n, be an orthonormal basis of the tangent space at any point of the manifold. Then by putting E=U=σi in (53) and taking summation over i , 1in, we get

(HS)(F,G)+ i=1nη((HR)(σi,F)G)g(ζ,σi)=0.

Putting G=ζ in (6.4), we get

(HS)(F,ζ)+ i=1nη((HR)(σi,F)ζ)g(ζ,σi)=0.

The second term of (6.5), takes the form

η((HR)(σi,F)ζ)=g(HR(σi,F)ζ,ζ)g(R(Hσi,F)ζ,ζ)        g(R(σi,HF)ζ,ζ)g(R(σi,F)Hζ,ζ),

and we obtain

η((HR)(σi,F)ζ)=0.

The equations (6.5) and (6.7) imply that

(HS)(F,ζ)=0,

which gives

H(S(F,ζ))S(HF,ζ)S(F,Hζ)=0.

In view of (2.5) and (3.4), we have

(ω2ω1)(Hη(F)η(HF))αS(F,φH)=0.

Putting F=φF in (6.8), we infer

αS(φF,φH)=(ω1ω2)g((Hφ)F,ζ).

Using (2.4), (2.7) and (3.3) in (6.9), we get

αω2g(φF,φH)+α2g(F,φH)=0.

Since α0, we infer

ω2g(φF,φH)+αg(F,φH)=0.

Putting F=φF, we have

ω2g(F,φH)+αg(φF,φH)=0.

Subtracting (6.11) from (6.12), we get

(ω2α)(g(φF,φH)g(F,φH))=0,

for any F,H it follows ω2=α. Using (3.5), we obtain ω1=α2nα2.

Hence, we can state the following results.

Theorem 6.1. If (M,φ,ζ,η,g) is a φ-Ricci symmetric on an α-LS manifold and (g,ζ,ω1,ω2) is an η-Ricci soliton, then ω2=α and ω1=α2nα2.

In case ω2=0, we deduce

Corollary 6.2. If α-LS manifolds (M,φ,ζ,η,g) satisfy the condition φ2((HR)(E,F)ζ)=0, then there does not exist Ricci solitons with potential vector field ζ.

Using (3.2), (3.5) and (6.11) we have

S=(ω2ω1){g(E,F)+η(E)η(F)}=2nα2{g(E,F)+η(E)η(F)}.

This leads to the following proposition.

Proposition 6.3. Let (M,φ,ζ,η,g) be an α-LS manifold. If M is a φ-Ricci symmetric and (g,ζ,ω1,ω2) is an η-Ricci soliton on M, then the manifold is an η-Einstein manifold.

Example 6.4. Now, we assume the 3-dimensional manifold

M={(p,q,r)R3:r0}

where p,q,r are the standard coordinates in R3.

The vector fields

σ1=erq,σ2=erp+q,σ3=αr,

are linearly independent at each point of M and α is a constant.

Let g be the Lorentzian metric defined as

g(σ1,σ3)=g(σ2,σ3)=g(σ1,σ2)=0,g(σ1,σ1)=g(σ2,σ2)=1,g(σ3,σ3)=1.

Let σ3=ζ. Then Lorentzian metric on M is given below

g=1(er)2{2(dp)2+(dq)22dpdq}1(α)2(dr)2.

Let η be the 1-form defined as

η(G)=g(G,σ3),

for any vector field G on M.

Let φ be the (1,1)-tensor field defined as

φ(σ1)=σ1, φ(σ2)=σ2,φ(σ3)=0.

Then, using the linearity of φ and g, we get

η(σ3)=1,φ2G=G+η(G)σ3,g(φG,φH)=g(G,H)+η(G)η(H),

for any vector field G,H on M.

It is easy to observe

η(σ1)=0,η(σ2)=0,η(σ3)=1.

Thus for σ3=ζ, the structure (φ,ζ,η,g) defines a Lorentzian almost contact metric structure on M.

Let ∇ be the Levi-Civita connection with respect to the Lorentzian metric g. Then we have

[σ1,σ2]=0, [σ1,σ3]=ασ1, [σ2,σ3]=ασ2.

Using Koszul's formula

2g(EF,G)=Eg(F,G)+Fg(G,E)Gg(E,F)    g(E,[F,G])g(F,[E,G])    +g(G,[E,F]),

One can easily obtain

σ1σ1=ασ3,σ1σ2=0,σ1σ3=ασ1,
σ2σ1=0,σ2σ2=ασ3,σ2σ3=ασ2,
σ3σ1=0,σ3σ2=0,σ3σ3=0.

Now, we see that the manifold is an α-LS manifold.

Also, the Riemannian curvature tensor R is given by

R(E,F)G=EFGFEG[E,F]G.

Then

R(σ1,σ2)σ2=α2σ1,R(σ1,σ3)σ3=α2σ1,R(σ2,σ1)σ1=α2σ2,
R(σ2,σ3)σ3=α2σ2,R(σ3,σ1)σ1=α2σ3,R(σ3,σ2)σ2=α2σ3.

Then, the Ricci tensor S is given by

S(σ1,σ1)=S(σ2,σ2)=2α2,S(σ3,σ3)=2α2.

From (3.2), we obtain S(σ1,σ1)=S(σ2,σ2)=αω1 and S(σ3,σ3)=ω1ω2, therefore ω1=α2α2 and ω2=α. The data (g,ζ,ω1,ω2) for ω1=α2α2 and ω2=α provides an η-Ricci soliton on an α-LS manifold.

  1. C. S. Baghewadi and G. Ingalahalli, Ricci solitons in Lorentzian α-Sasakian manifolds, Acta Math. Acad. Paedagog. Nyhzi., 28(2012), 59-68.
  2. A. M. Blaga, η-Ricci solitons on Lorentzian para-Sasakian manifolds, Filomat, 30(2)(2016), 489-496.
    CrossRef
  3. A. M. Blaga, On warped product gradient η-Ricci solitons, Filomat, 31(18)(2017), 5791-5801.
    CrossRef
  4. A. M. Blaga and S. Y. Perktas, Remarks on almost η-Ricci solitons in ε-para Sasakian manifolds, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., 68(2)(2019), 1621-1628.
    CrossRef
  5. C. Calin and M. Crasmareanu, η-Ricci solitons on Hopf hypersurfaces in complex space forms, Rev. Roumaine Math. Pures Appl., 57(1)(2012), 55-63.
  6. C. Calin and M. Crasmareanu, From the Eisenhart problem to Ricci solitons in f-Kenmotsu manifolds, Bull. Malays. Math. Sci. Soc., 33(3)(2010), 361-368.
  7. T. Chave and G. Valent, Quasi-Einstein metrics and their renormalizability properties, Helv. Phys. Acta., 69(1996), 344-347.
  8. T. Chave and G. Valent, On a class of compact and non-compact quasi-Einstein metrics and their renormalizability properties, Nuclear Phys. B., 478(3)(1996), 758-778.
    CrossRef
  9. J. T. Cho and M. Kimura, Ricci solitons and real hypersurfaces in a complex space form, Tohoku Math. J., 61(2)(2009), 205-212.
    CrossRef
  10. B. Chow, P. Lu and L. Ni, Hamilton's Ricci flow, Graduate Studies in Mathemtics, AMS, Providence, RI, USA, 77(2006).
    CrossRef
  11. U. C. De, Ricci solitons and gradient Ricci solitons in a P-Sasakian manifold, Aligarh Bull. Math., 29(1)(2010), 29-33.
  12. U. C. De and A. K. Mondal, 3-dimensional quasi-Sasakian manifolds and Ricci solitons, SUT J. Math., 48(1)(2012), 71-81.
    CrossRef
  13. U. C. De and A. Sarkar, On ϕ∗-Ricci symmetric Sasakian manifolds, Proc. Jangjeon Math. Soc., 11(1)(2008), 47-52.
  14. U. C. De, M. Turan, A. Yildiz and A. De, Ricci solitons and gradient Ricci solitons on 3-dimensional normal almost contact metric manifolds, Publ. Math. Debrecen, 80(1-2)(2012), 127-142.
    CrossRef
  15. S. Deshmukh, H. Alodan and H. Al-Sodais, A Note on Ricci Solitons, Balkan J. Geom. Appl., 16(1)(2011), 48-55.
  16. S. Dey, B. Pal and A. Bhattacharyya, Some classes of Lorentzian α-Sasakian manifolds admitting quarter symmetric metric connection, Acta Univ. Palacki. Olomuc. Fac. Rer. Nat., Mathematica, 55(2)(2016), 41-55.
  17. D. Friedan, Non linear models in 2 + ǫ dimensions, Ann. Physics, 163(1985), 318-419.
    CrossRef
  18. A. Gray, Einstein-like manifolds which are not Einstein, Geom. Dedicata, 7(1978), 259-280.
    CrossRef
  19. R. S. Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geometry., 17(2)(1982), 255-306.
    CrossRef
  20. R. S. Hamilton, The Ricci flow on surfaces, Contemp. Math., American Math. Soc., (1988).
  21. A. Haseeb, S. Pandey and R. Prasad, Some results on η-Ricci solitons in quasi-Sasakian 3-manifolds, Commun. Korean Math. Soc., 36(2)(2021), 377-387.
  22. A. Haseeb and R. Prasad, η-Ricci solitons in Lorentzian α-Sasakian manifolds, Facta Univ. Ser. Math. Inform., 35(3)(2020), 713-725.
    CrossRef
  23. T. Ivey, Ricci solitons on compact 3-manifolds, Differential Geom. Appl., 3(1993), 301-307.
    CrossRef
  24. S. Kishor and A. Singh, η-Ricci solitons on 3-dimensional Kenmotsu manifolds, Bull. Transilv. Univ. Braov Ser. III. Math. Comput. Sci., 13(62)(2020), 209-218.
    CrossRef
  25. P. Majhi, U. C. De and D. Kar, η-Ricci solitons on Sasakian 3-Manifolds, An. Univ. Vest Timi. Ser. Mat.-Inform., 55(2)(2017), 143-156.
    CrossRef
  26. D. G. Prakasha and B. S. Hadimani, η-Ricci solitons on para-Sasakian manifolds, J. Geom., 108(2017), 383-392.
    CrossRef
  27. R. Prasad, S. Pandey and A. Haseeb, On a Lorentzian Sasakian manifold endowed with a quarter-symmetric metric connection, An. Univ. Vest Timi. Ser. Mat.-Inform., 57(2)(2019), 61-76.
    CrossRef
  28. A. Singh and S. Kishor, Some types of η-Ricci solitons on Lorentzian para-Sasakian manifolds, Facta Univ. Ser. Math. Inform., 33(2)(2018), 217-230.
    CrossRef
  29. A. Singh and S. Kishor, Curvature properties of η-Ricci solitons on ParaKenmotsu manifolds, Kyungpook Math. J., 59(2019), 149-161.
  30. T. Takahashi, Sasakian ϕ-symmetric spaces, Tohoku Math. J., 29(1977), 91-113.
    CrossRef