Almost η -Ricci solitons on Kenmotsu manifolds
aa r X i v : . [ m a t h . DG ] A ug Almost η -Ricci solitons on Kenmotsu manifolds Dhriti Sundar Patra ∗ and Vladimir Rovenski † Abstract
In this paper we characterize the Einstein metrics in such broader classes of metrics as almost η -Ricci solitons and η -Ricci solitons on Kenmotsu manifolds, and generalize some results of otherauthors. First, we prove that a Kenmotsu metric as an η -Ricci soliton is Einstein metric if eitherit is η -Einstein or the potential vector field V is an infinitesimal contact transformation or V is collinear to the Reeb vector field. Further, we prove that if a Kenmotsu manifold admits agradient almost η -Ricci soliton with a Reeb vector field leaving the scalar curvature invariant,then it is an Einstein manifold. Finally, we present new examples of η -Ricci solitons and gradient η -Ricci solitons, which illustrate our results. Keywords : Almost contact metric structure, Einstein manifold, Ricci tensor, η -Ricci soliton,infinitesimal contact transformation Mathematics Subject Classifications (2010)
Ricci soliton , defined on a Riemannian manifold (
M, g ) by the partial differential equation12 L V g + Ric + λ g = 0 , (1)is a natural generalization of Einstein metric (i.e., the Ricci tensor is a constant multiple of theRiemannian metric g ). In (1), L V denotes the Lie-derivative in the direction of V , Ric is the Riccitensor of g and λ is a constant. A Ricci soliton is trivial if V is either zero or Killing on M . ARicci soliton is said to be shrinking, steady, and expanding, as λ is negative, zero, and positive,respectively (and there are many examples of each of them, e.g., [10]). Otherwise, it will be calledindefinite. Actually, a Ricci soliton can be considered as a generalized fixed point of the normalizedversion of Hamilton’s Ricci flow [20]: ∂g∂t = − λ to be a C ∞ -regular (i.e., smooth) function and named it asRicci almost soliton, later on studied by Barros et. al. [1]. Some applications of (semi-)Riemannianalmost Ricci solitons to general relativity were discussed in [13]. In [8], Cho-Kimura studied realhypersurfaces in a complex space form and generalized the notion of Ricci soliton to η - Ricci soliton ,defined on (
M, g ) by 12 L V g + Ric + λ g + µ η ⊗ η = 0 , (2)where the tensor product notation ( η ⊗ η )( X, Y ) = η ( X ) η ( Y ) is used and λ , µ are real constants.Later on (2) was studied by Calin-Crasmareanu [7], Blaga et. al. [2, 3, 4] and Naik-Venkatesha [24].An η -Ricci soliton is said to be almost η -Ricci soliton if λ and µ are smooth functions on M (seedetails in [2, 4]). When the potential vector field V is a gradient of a smooth function f : M → R ∗ Department of Mathematics, Birla Institute of Technology Mesra, Ranchi: 835 215, Indiae-mail: [email protected] and [email protected] † Mathematical Department, University of Haifa, Mount Carmel, 31905 Haifa, Israele-mail: [email protected] η -Ricci soliton, and (3)reads as Hess f + Ric + λ g + µ η ⊗ η = 0 , (3)where Hess f is the Hessian of f . As a generalization of a Ricci soliton, it is also said to be shrinking,steady, and expanding, as λ is negative, zero, and positive, respectively, e.g., [2, 24].Many authors studied Ricci solitons, η -Ricci solitons and their generalizations in the frameworkof almost contact and paracontact geometries, e.g., contact metrics as Ricci solitons by Cho-Sharma[9], K -contact and ( k, µ )-contact metrics as Ricci solitons by Sharma [27], contact metrics as Riccialmost solitons by Ghosh-Sharma [16, 28], contact metrics as h -almost Ricci solitons by Ghosh-Patra[19], almost contact B -metrics as Ricci-like soliton by Manev [23], para-Sasakian and Lorentzianpara-Sasakian metrics as η -Ricci solitons by Naik-Venkatesha [24] and Blaga [3], etc. Based on theabove results in a modern and active field of research, a natural question can be posed: Are there almost contact metric manifolds, whose metrics are η -Ricci solitons? The notion of a warped product is very popular in differential geometry as well as in general rela-tivity, e.g., [6]. Some solutions of Einstein field equations are warped products and some spacetimemodels, e.g., Robertson-Walker spacetime, Schwarzschild spacetime, Reissner-Nordstr¨om spacetimeand asymptotically flat spacetime, are warped product manifolds (see [21] for more details). In [22],Kenmotsu first introduced and studied a special class of almost contact metric manifolds, knownas Kenmotsu manifolds, which is characterized in terms of warped products. The warped product R × f N (of the real line R and a K¨ahler manifold N ) with the warping function f ( t ) = ce t , (4)given on an open interval J = ( − ε, ε ), admits Kenmotsu structure, and conversely, every pointof a Kenmotsu manifold has a neighbourhood, which is locally a warped product J × f N , where f is given by (4). First, Ghosh considered an almost contact metric, in particular, a Kenmotsumetric, as a Ricci soliton and proved that a 3-dimensional Kenmotsu metric as a Ricci solitonis of constant negative curvature − η -Einstein [15] or the potential vector field V is contact,see [18]. Shanmukha-Venkatesha [26] studied Ricci semi-symmetric Kenmotsu manifolds with η -Ricci solitons, and Sabina et. al. [14] studied η -Ricci solitons on a Kenmotsu manifold satisfyingsome curvature conditions.Our goal in this paper is to answer the question posed above using methods of local Riemanniangeometry and to characterize Einstein metrics in some of the broader classes of metrics mentionedabove. Thus, we study almost η -Ricci solitons, in particular, η -Ricci solitons, on a Kenmotsumanifold. This paper is organized as follows. In Section 2, the basic facts about contact metricmanifolds and Kenmotsu manifolds are given. In Section 3, we consider Kenmotsu metrics as η -Ricci solitons and find some important conditions, when a Kenmotsu metric as an η -Ricci soliton isEinstein, and Theorems 3.1 and 3.2 and Corollary 3.1 generalize some results of the above mentionedauthors. In Section 4, we consider almost η -Ricci solitons on a Kenmotsu manifold and find some η -Einstein and Einstein manifolds, using concept of almost η -Ricci solitons. We also present examplesof Kenmotsu manifolds that admit η -Ricci solitons and gradient η -Ricci solitons, which illustrateour results. Here, we give some definitions and basic facts (see details in [5, 22]), which are used in the paper.An almost contact structure on a smooth manifold M n +1 of dimension 2 n + 1 is a triple ( ϕ, ξ, η ),where ϕ is a (1 , ξ is a vector field (called Reeb vector field) and η is a 1-form, satisfying ϕ = − I + η ⊗ ξ, η ( ξ ) = 1 , (5)2here I denotes the identity endomorphism. It follows from (5) that ϕ ( ξ ) = 0, η ◦ ϕ = 0 andrank ϕ = 2 n (see [5]). A smooth manifold M endowed with an almost contact structure is calledan almost contact manifold. A Riemannian metric g on M is said to be compatible with an almostcontact structure ( ϕ, ξ, η ) if g ( ϕX, ϕY ) = g ( X, Y ) − η ( X ) η ( Y )for any X ∈ X ( M ), where X ( M ) is the Lie algebra of all vector fields on M . An almost contactmanifold endowed with a compatible Riemannian metric is said to be an almost contact metricmanifold and is denoted by M ( ϕ, ξ, η, g ). The fundamental 2-form Φ on M ( ϕ, ξ, η, g ) is defined byΦ( X, Y ) = g ( X, ϕY ) for any X ∈ X ( M ). An almost contact metric manifold satisfying dη = 0 and d Φ = 2 η ∧ Φ is said to be an almost Kenmotsu manifold (e.g., [12, 11]). An almost contact metricstructure is said to be normal if the tensor N ϕ = [ ϕ, ϕ ]+2 d η ⊗ ξ vanishes on M , where [ ϕ, ϕ ] denotesthe Nijenhuis tensor of ϕ . A normal almost Kenmotsu manifold is said to be a Kenmotsu manifold(see [22]); equivalently, an almost contact metric manifold is said to be a Kenmotsu manifold (see[22]) if ( ∇ X ϕ ) Y = g ( ϕX, Y ) ξ − η ( Y ) ϕX for any X, Y ∈ X ( M ), where ∇ is the Levi-Civita connection of g . The following formulas hold onKenmotsu manifolds, see [18, 22]: ∇ X ξ = X − η ( X ) ξ, (6) R ( X, Y ) ξ = η ( X ) Y − η ( Y ) X, (7)( ∇ ξ Q ) X = − QX − nX, (8) Qξ = − nξ, (9)for all X, Y ∈ X ( M ), where R and Q denote, respectively, the curvature tensor and the Ricci operatorof g associated with the Ricci tensor and given by Ric( X, Y ) = g ( QX, Y ) for
X, Y ∈ X ( M ). Notethat (6)–(8) are simple and (9) is proven in [18]. Recall the commutation formula, see p. 23 in [29],( L V ∇ Z g − ∇ Z L V g − ∇ [ V,Z ] g )( X, Y ) = − g (( L V ∇ )( Z, X ) , Y ) − g (( L V ∇ )( Z, Y ) , X ) . (10)A contact metric manifold M n +1 is called η - Einstein , if its Ricci tensor has the following form:Ric = α g + β η ⊗ η, (11)where α and β are smooth functions on M . For an η -Einstein K -contact manifold of dimension >
3, these α and β are constant [30]), but for an η -Einstein Kenmotsu manifold this is not true [22]. η -Ricci solitons Here, we study η -Ricci solitons on a Kenmotsu manifold. Lemma 3.1.
Let M n +1 ( ϕ, ξ, η, g ) be a Kenmotsu manifold. If g represents an η -Ricci soliton withthe potential vector field V then ( L V R )( X, ξ ) ξ = 0 for all X ∈ X ( M ) .Proof. Taking the covariant derivative of (2) along Z ∈ X ( M ) and using (6), we have( ∇ Z L V g )( X, Y ) = − ∇ Z Ric)(
X, Y ) − µ { g ( X, Z ) η ( Y ) + g ( Y, Z ) η ( X ) − η ( X ) η ( Y ) η ( Z ) } (12)for all X, Y ∈ X ( M ). Since Riemannian metric is parallel, it follows from (10) that( ∇ Z L V g )( X, Y ) = g (( L V ∇ )( Z, X ) , Y ) + g (( L V ∇ )( Z, Y ) , X ) . g (( L V ∇ )( Z, X ) , Y ) + g (( L V ∇ )( Z, Y ) , X ) = − ∇ Z Ric)(
X, Y ) − µ { g ( X, Z ) η ( Y ) + g ( Y, Z ) η ( X ) − η ( X ) η ( Y ) η ( Z ) } (13)for all X, Y, Z ∈ X ( M ). Cyclically rearranging X, Y and Z in (13), we obtain g (( L V ∇ )( X, Y ) , Z ) = ( ∇ Z Ric)(
X, Y ) − ( ∇ X Ric)(
Y, Z ) − ( ∇ Y Ric)(
Z, X ) − µ { g ( X, Y ) η ( Z ) − η ( X ) η ( Y ) η ( Z ) } . (14)Taking the covariant derivative of (9) along X ∈ X ( M ) and using (6), we obtain( ∇ X Q ) ξ = − QX − nX. (15)Substituting ξ for Y ∈ X ( M ) in (14) and applying (8) and (15), we obtain( L V ∇ )( X, ξ ) = 2 QX + 4 nX (16)for any X ∈ X ( M ). Next, using (6), (16) in the covariant derivative of (16) along Y , yields( ∇ Y L V ∇ )( X, ξ ) + ( L V ∇ )( X, Y ) = 2( ∇ Y Q ) X + 2 η ( Y )( QX + 2 nX )for any X ∈ X ( M ). Plugging this in the following commutation formula (see [29], p. 23):( L V R )( X, Y ) Z = ( ∇ X L V ∇ )( Y, Z ) − ( ∇ Y L V ∇ )( X, Z ) , we deduce( L V R )( X, Y ) ξ = 2 { ( ∇ X Q ) Y − ( ∇ Y Q ) X } + 2 { η ( X ) QY − η ( Y ) QX } + 4 n { η ( X ) Y − η ( Y ) X } (17)for all X, Y ∈ X ( M ). Substituting Y by ξ in (17) and using (8), (9) and (15), yields our result.Now, we consider an η -Einstein Kenmotsu metric as an η -Ricci soliton and characterize theEinstein metrics in such a wider class of metrics. Theorem 3.1.
Let M n +1 ( ϕ, ξ, η, g ) , n > , be an η -Einstein Kenmotsu manifold. If g representsan η -Ricci soliton with the potential vector field V , then it is Einstein with constant scalar curvature r = − n (2 n + 1) .Proof. First, tracing (11) gives r = (2 n + 1) α + β and putting X = Y = ξ in (11) and using (9),yields α + β = − n and r = 2 n ( α − (cid:0) r n (cid:1) g − (cid:0) n + 1 + r n (cid:1) η ⊗ η. By (5), the foregoing equation entails that( ∇ Y Q ) X = Y ( r )2 n ϕ X − (cid:0) n + 1 + r n (cid:1)(cid:0) g ( X, Y ) ξ + η ( X )( Y − η ( Y ) ξ ) (cid:1) for all X, Y ∈ X ( M ). By virtue of this, (17) provides( L V R )( X, Y ) ξ = (cid:0) X ( r ) ϕ Y − Y ( r ) ϕ X (cid:1) /n (18)for all X, Y ∈ X ( M ). Inserting Y = ξ in (18) and applying Lemma 3.1, we get ξ ( r ) ϕ X = 0 for any X ∈ X ( M ). This implies ξ ( r ) = 0. Using this in the trace of (15), gives r = − n (2 n + 1).4he curvature tensor of a 3-dimensional Riemannian manifold has well-known form R ( X, Y ) Z = g ( Y, Z ) QX − g ( X, Z ) QY + g ( QY, Z ) X − g ( QX, Z ) Y − r (cid:0) g ( Y, Z ) X − g ( X, Z ) Y (cid:1) (19)for all X, Y, Z ∈ X ( M ). Inserting Y = Z = ξ in (19) and using (9), we obtain QX = (1 + r X − (cid:0) r (cid:1) η ( X ) ξ (20)for any X ∈ X ( M ). Thus, proceeding in the same way as in Theorem 3.1 and applying Lemma 3.1we conclude that r = −
6. Hence, from (20) we have QX = 2 nX . Plugging this in (19) implies that( M, g ) is of constant negative curvature −
1. Thus, from Theorem 3.1 we obtain the following.
Corollary 3.1.
If a -dimensional Kenmutsu metric g represents an η -Ricci soliton with the po-tential vector field V , then it is of constant negative curvature − . Remark 3.1.
Ghosh proved Theorem 3.1 in [15] and Corollary 3.1 in [17] for Ricci solitons. In thisarticle, using different technique, we prove two above results in a short and direct way for an η -Riccisoliton on a Kenmotsu manifold. Since Ricci soliton is a particular case of an η -Ricci soliton, wecan also derive similar results for Ricci solitons using this method.A vector field X on a contact metric manifold M is called a contact or infinitesimal contacttransformation if it preserves the contact form η , i.e., there is a smooth function ρ : M → R satisfying L X η = ρ η, (21)and if ρ = 0 then the vector field X is called strict. We consider a Kenmotsu metric as an η -Riccisoliton, whose potential vector field V is contact or V collinear to ξ . First, we derive the following. Lemma 3.2.
Let M n +1 ( ϕ, ξ, η, g ) be a Kenmotsu manifold. If g represents an η -Ricci soliton witha potential vector field V then we have λ + µ = 2 n .Proof. Equation (7) gives us R ( X, ξ ) ξ = − X + η ( X ) ξ . Taking its Lie derivative along V , we obtain( L V R )( X, ξ ) ξ + R ( X, L V ξ ) ξ + R ( X, ξ ) L V ξ = { ( L V η ) X } ξ + η ( X ) L V ξ. (22)Putting ( L V R )( X, ξ ) ξ = 0 (by Lemma 3.1) and applying (7), equation (22) becomes( L V g )( X, ξ ) + 2 η ( L V ξ ) X = 0 (23)for any X ∈ X ( M ). Using (2) in the Lie derivative of g ( ξ, ξ ) = 1, we obtain η ( L V ξ ) = λ + µ − n. (24)Plugging the values of ( L V g )( X, ξ ) and η ( L V ξ ) from (2) and (24) respectively, into (23), we obtain(2 n − λ − µ ) ϕ X = 0. Using (5) and then tracing, we achieve the required result. Theorem 3.2.
Let M n +1 ( ϕ, ξ, η, g ) , n > , be a Kenmotsu manifold. If g represents an η -Riccisoliton and the potential vector field V is an infinitesimal contact transformation, then V is strictand ( M, g ) is an Einstein manifold with constant scalar curvature r = − n (2 n + 1) .Proof. First, recall the known formula (see p. 23 of [29]): L V ∇ X Y − ∇ X L V Y − ∇ [ V,X ] Y = ( L V ∇ )( X, Y ) (25)for all
X, Y ∈ X ( M ). Now, setting Y = ξ in (25) and using (6), we find( L V ∇ )( X, ξ ) = L V ( X − η ( X ) ξ ) − L V X + η ( L V X ) ξ = − ( L V η )( X ) ξ − η ( X ) L V ξ (26)for any X ∈ X ( M ). Taking Lie-derivative of η ( X ) = g ( X, ξ ) along V and using (2), (21) andLemma 3.2, we obtain L V ξ = ρξ . Thus, (24) and Lemma 3.2 gives ρ = 0, therefore, L V ξ = 0 and V is strict. Also, (21) gives L V η = 0. It follows from (26) that ( L V ∇ )( X, ξ ) = 0. Thus, from (16)we prove the rest part of this theorem. 5 heorem 3.3.
Let M n +1 ( ϕ, ξ, η, g ) be a Kenmotsu manifold. If g represents a η -Ricci soliton withnon-zero potential vector field V is collinear to ξ , then g is Einstein with constant scalar curvature r = − n (2 n + 1) .Proof. By hypothesis: V = σ ξ for some smooth function σ on M . Using (6) in the covariantderivative of V = σ ξ along arbitrary X ∈ X ( M ) yields ∇ X V = X ( σ ) ξ + σ ( X − η ( X ) ξ ) for any X ∈ X ( M ). Taking this into account, the soliton equation (2) transforms into2Ric( X, Y ) + X ( σ ) η ( Y ) + Y ( σ ) η ( X ) + 2( σ + λ ) g ( X, Y ) − σ − µ ) η ( X ) η ( Y ) = 0 (27)for all X, Y ∈ X ( M ). Inserting X = Y = ξ in (27) and using (9) and Lemma 3.2, we get ξ ( σ ) = 0.It follows from (27) that X ( σ ) = 0. Putting it into (27) givesRic = − ( σ + λ ) g + ( σ − µ ) η ⊗ η. (28)This shows that ( M, g ) is an η -Einstein manifold, therefore, from Theorem 3.1 we conclude that( M, g ) is an Einstein manifold. Thus, from (28) we have σ = µ , therefore, σ + λ = 2 n (follows fromLemma 3.2). This implies, using (28), that Ric = − ng , hence, r = − n (2 n + 1), as required. Remark 3.2.
In the above theorem we got X ( σ ) = 0 for all vector fields X , therefore, the smoothfunction σ reduces to a constant and it equals to the constant soliton µ , hence, V = µ ξ .In [22], Kenmotsu found the following necessary and sufficient condition for a Kenmotsu manifold M n to have constant ϕ -holomorphic sectional curvature H :4 R ( X, Y ) Z = ( H − { g ( Y, Z ) X − g ( X, Z ) Y } + ( H + 1) { η ( X ) η ( Z ) Y − η ( Y ) η ( Z ) X + η ( Y ) g ( X, Z ) ξ − η ( X ) g ( Y, Z ) ξ + g ( X, ϕZ ) ϕY − g ( Y, ϕZ ) ϕX + 2 g ( X, ϕY ) ϕZ } (29)for all X, Y, Z ∈ X ( M ). The following corollary of Theorem 3.1 illustrates Lemma 3.2 and gives anexample of a Kenmotsu manifold that admits an η -Ricci soliton. Corollary 3.2.
If a metric g of a warped product R × f N n for n > , where f ( t ) = ce t and N is aK¨ahler manifold represents an η -Ricci soliton with the potential vector field V , then it is of constantcurvature H = − .Proof. Idea of the following construction is that if we choose V = n ξ , i.e., σ = n , then R × f N n isan η -Ricci soliton for n > λ = n and µ = n . It follows from (29) that4 QX = ((2 n − H − n + 1)) X − (2 n − H + 1) η ( X ) ξ (30)for any X ∈ X ( M ). Also, a warped product R × f N n , where f ( t ) = ce t on the real line R , see(4), and N n is a K¨ahler manifold, admits the Kenmotsu structure. Thus, it follows from (30)that R × f N n is an η -Einstein Kenmotsu manifold, and by Theorem 3.1, we conclude that it is anEinstein manifold. Thus, it follows from (30) that β = 0, i.e., H = −
1, hence, α = − n . η -Ricci solitons Here, we study almost η -Ricci solitons on a Kenmotsu manifold. It is known that an almost η -Riccisoliton (i.e., satisfying (2) for some smooth functions λ and µ ) is the generalization of a Ricci almostsoliton. In [27], Sharma proved that if a K -contact metric represents a gradient Ricci soliton, thenit is an Einstein manifold, and Ghosh [16] generalized this result for Ricci almost solitons. Recently,Ghosh [18] considered Ricci almost solitons on a Kenmotsu manifold and proved that if a Kenmotsumetric is a gradient Ricci almost soliton and the Reeb vector field ξ leaves the scalar curvature r invariant, then it is an Einstein manifold. To generalize the above results, we consider gradientalmost η -Ricci solitons on a Kenmotsu manifold and prove the following.6 heorem 4.1. If a Kenmotsu manifold M n +1 ( ϕ, ξ, η, g ) admits a gradient almost η -Ricci solitonand ξ leaves the scalar curvature r invariant, then ( M, g ) is an Einstein manifold with constantscalar curvature r = − n (2 n + 1) .Proof. The gradient version of the soliton equation (3) can be exhibited for any X ∈ X ( M ) as ∇ X Df + QX + λX + µη ( X ) ξ = 0 . (31)The curvature tensor, obtained from (31) and the definition R ( X, Y ) = [ ∇ X , ∇ Y ] − ∇ [ X,Y ] , satisfies R ( X, Y ) Df = ( ∇ Y Q ) X − ( ∇ X Q ) Y + Y ( λ ) X − X ( λ ) Y + Y ( µ ) η ( X ) ξ − X ( µ ) η ( Y ) ξ + µ (cid:0) η ( Y ) X − η ( X ) Y (cid:1) (32)for all X, Y ∈ X ( M ). Now, replacing Y by ξ in (32) and using (8) and (15), we obtain R ( X, ξ ) Df = − QX − nX + ξ ( λ ) X − X ( λ ) ξ − X ( µ ) ξ + ξ ( µ ) η ( X ) ξ + µϕ X (33)for any X ∈ X ( M ). By virtue of (7), equation (33) reduces to X ( λ + µ + f ) ξ = − QX + (cid:0) ξ ( λ + f ) + µ − n (cid:1) X + ( ξ ( µ ) − µ ) η ( X ) ξ (34)for any X ∈ X ( M ). Next, taking inner product of (34) with ξ and using (7), we obtain X ( λ + µ + f ) = ξ ( λ + µ + f ) η ( X ). Putting this into (34), we achieve QX = (cid:0) ξ ( λ + f ) − µ − n (cid:1) X − (cid:0) ξ ( λ + f ) + µ (cid:1) η ( X ) ξ (35)for any X ∈ X ( M ). This shows that ( M, g ) is an η -Einstein manifold. Further, contracting (32)over X with respect to an orthonormal basis { e i } ≤ i ≤ n +1 , we computeRic( Y, Df ) = − X n +1 i =1 g (( ∇ e i Q ) Y, e i ) + Y ( r ) + 2 n Y ( λ ) + Y ( µ ) − η ( Y ) ξ ( µ ) + 2 nµ η ( Y ) . (36)The following formula for Riemannian manifolds is well known: trace g { X −→ ( ∇ X Q ) Y } = Y ( r ).Applying this formula in (36), we achieveRic( Y, Df ) = 12 Y ( r ) + 2 n Y ( λ ) + Y ( µ ) − η ( Y ) ξ ( µ ) + 2 nµ η ( Y ) (37)for any X ∈ X ( M ). From (7) one can compute Ric( ξ, Df ) = − n ξ ( f ). Plugging it into (37), wefind ξ ( r ) + 4 n { ξ ( f + λ ) + µ } = 0. Using this in the trace of (8), we get ξ ( λ + f ) = 2 n + 1 − µ + r n .By virtue of this, equation (35) reduces to QX = (1 + r n ) X − (cid:0) (2 n + 1) + r n (cid:1) η ( X ) ξ (38)for any X ∈ X ( M ). By our assumptions, ξ ( r ) = 0, therefore, the trace of (8) gives r = − n (2 n + 1).Thus, from (38) the required result follows.Next, considering a Kenmotsu metric as an almost η -Ricci soliton, whose non-zero potentialvector field V is pointwise collinear to the Reeb vector field ξ , we extend Theorem 4.1 from gradientalmost η -Ricci solitons to almost η -Ricci solitons by proving the following. Theorem 4.2.
If a Kenmotsu manifold M n +1 ( ϕ, ξ, η, g ) admits an almost η -Ricci soliton withnon-zero potential vector field V collinear to the Reeb vector field ξ and ξ leaves the scalar curvature r invariant, then ( M, g ) is an Einstein manifold with constant scalar curvature r = − n (2 n + 1) . roof. Since V = τ ξ for some smooth function τ on M , it follows that( L V g )( X, Y ) = X ( τ ) η ( Y ) + Y ( τ ) η ( X ) + 2 τ (cid:0) g ( X, Y ) − η ( X ) η ( Y ) (cid:1) for any X, Y ∈ X ( M ). By virtue of this, the soliton equation (2) transforms into2Ric( X, Y ) + X ( τ ) η ( Y ) + Y ( τ ) η ( X ) + 2( τ + λ ) g ( X, Y ) = 2( τ − µ ) η ( X ) η ( Y ) (39)for any X, Y ∈ X ( M ). Now, putting X = Y = ξ in (39) and using (9), yields ξ ( τ ) = 2 n − λ − µ .Thus, (39) yields X ( τ ) = 2 n − λ − τ . Using this in (39) implies thatRic = − ( τ + λ ) g − (2 n − τ − λ ) η ⊗ η. (40)Hence, ( M, g ) is η -Einstein. Moreover, if ξ leaves the scalar curvature r invariant, i.e., ξ ( r ) = 0,again, tracing (15) gives ξ ( r ) = − { r + 2 n (2 n + 1) } , and therefore, r = − n (2 n + 1). Using this inthe trace of (40) yields τ + λ = 2 n . By (40), QX = − nX ; thus, ( M, g ) is an Einstein manifold.If τ is constant (instead of being a function) and V = τ ξ , then (39) and (40) also hold. Setting X = Y = ξ in (39) and using (9), gives ξ ( τ ) = 2 n − λ − µ . This yields λ + τ = 2 n when τ is constant.Hence, without assuming that ξ leaves the scalar curvature r invariant, from (40) we conclude that( M, g ) is an Einstein manifold with the Einstein constant − n , therefore, we derived the following. Theorem 4.3.
If a Kenmotsu manifold M n +1 ( ϕ, ξ, η, g ) admits a non-trivial almost η -Ricci solitonwith V = τ ξ for some constant τ , then it is an Einstein manifold with constant scalar curvature − n (2 n + 1) . Now, we construct explicit example of a five-dimensional Kenmotsu manifold that admits an η -Ricci soliton and a gradient η -Ricci soliton. Example 4.1.
Let M = { ( x, y, z, u, v ) ∈ R } be a 5-dimensional manifold, where ( x, y, z, u, v ) areCartesian coordinates in R . Let e = v ∂∂x , e = v ∂∂y , e = v ∂∂z , e = v ∂∂u , e = − v ∂∂v . Clearly, ( e i )form an orthonormal basis of vector fields on M . Define the structure ( ϕ, ξ, η, g ) as follows: ϕ ( e ) = e , ϕ ( e ) = − e , ϕ ( e ) = e , ϕ ( e ) = − e , ϕ ( e ) = 0 , ξ = e , η = dv, and g ij = δ ij – the Kronecker symbol. By Koszul’s formula, we compute the non-trivial componentsof the Levi-Civita connection ∇ in the following form: ∇ e i e i = − e , ∇ e i e = e i ( i = 1 , , , . (41)Using this, we can verify that M ( ϕ, ξ, η, g ) is a Kenmotsu manifold and compute the followingnon-zero components of its curvature tensor: R ( e , e ) e = e R ( e , e ) e = − e R ( e , e ) e = e R ( e , e ) e = − e R ( e , e ) e = e R ( e , e ) e = − e R ( e , e ) e = e R ( e , e ) e = e R ( e , e ) e = e R ( e , e ) e = e R ( e , e ) e = − e R ( e , e ) e = e R ( e , e ) e = − e R ( e , e ) e = − e R ( e , e ) e = e R ( e , e ) e = − e R ( e , e ) e = e R ( e , e ) e = − e R ( e , e ) e = e R ( e , e ) e = − e Thus, the nonzero components of the Ricci tensor are as follows: Ric( e i , e i ) = − i = 1 , . . . , − g. (42)Hence, ( M, g ) is an Einstein manifold with constant scalar curvature r = −
20 = − n (2 n + 1) for n = 2. If we consider the potential vector field V = σξ for some constant σ , then, using (41), we get L V g = 2 σ g − σ η ⊗ η. (43)8ence, using (42) in (43) and remembering the soliton equation (2), we can say that the metric g is an η -Ricci soliton with the potential vector field V = σξ and the constants λ = 4 − σ and µ = σ .Again, λ + µ = 4 = 2 n , where n = 2. This illustrates Lemma 3.2.In general, for the potential vector field V = 2 x ∂∂x + 2 y ∂∂y + 2 z ∂∂z + 2 u ∂∂u + v ∂∂v on M , using(41), we find ( L V g )( e i , e j ) = n , if i = j = 1 , , , , , elsewhere; therefore, we achieve L V g = 2 g − η ⊗ η. (44)So, combining (42) and (44), one can see that soliton equation (2) is satisfied by λ = 3 and µ = 1, i.e., the metric g is an η -Ricci soliton with the above potential vector field V and theconstants λ = 3 and µ = 1, which also satisfy λ + µ = 2 n , for n = 2. Choosing the function f ( x, y, z ) = x + y + z + u + v on M , from (42) and (44) we conclude that the metric g is agradient η -Ricci soliton with the potential function f . In this article, we use methods of local Riemannian geometry to study solutions of (3) and char-acterize Einstein metrics in such broader classes of metrics as almost η -Ricci solitons and η -Riccisolitons on Kenmotsu manifolds, which compose a special class of almost contact manifolds. Ourresults generalize some results of other authors and are important not only for differential geometry,but also for theoretical physics. Following [13], we can think about physical applications of (almost) η -Ricci solitons. We delegate for further study the following questions:1: Under what conditions is an η -Ricci soliton with the potential vector field on a Kenmotsumanifold trivial?2: Are Theorems 1–3 true without assuming the η -Einstein condition or that the potential vectorfield V is an infinitesimal contact transformation or V is collinear to the Reeb vector field?3: Is the Kenmotsu metric, admitting a gradient almost η -Ricci soliton with a potential vectorfield, trivial?4: Which of the results of this paper are also true for generalized quasi-Einstein Kenmotsumanifolds? References [1] Barros, A. and Ribeiro, Jr.E.: Some characterizations for compact almost Ricci solitons, Proc.AMS, 140(3), 1033-1040 (2012)[2] Blaga, Adara-M.: Almost η -Ricci solitons in ( LCS ) n -manifolds, Bull. Belg. Math. Soc. SimonStevin, 25(5), 641–653 (2018)[3] Blaga, Adara-M.: η -Ricci solitons on Lorentzian para-Sasakian manifolds, Filomat, 30(2), 489–496 (2016)[4] Blaga, Adara-M. and ¨Ozg¨ur, C.: Almost η -Ricci and almost η -yamabe solitons with torseform-ing potential vector field. Preprint, arXiv:2003.12574, 2020[5] Blair, D.E.: Riemannian Geometry of Contact and Symplectic Manifolds , Birkh¨auser, Boston,2002[6] Chen, B.-Y.:
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