A note on Almost Riemann Soliton and gradient almost Riemann soliton
aa r X i v : . [ m a t h . DG ] A ug A NOTE ON ALMOST RIEMANN SOLITON AND GRADIENTALMOST RIEMANN SOLITON
KRISHNENDU DE AND UDAY CHAND DE
Abstract.
The quest of the offering article is to investigate almost Riemannsoliton and gradient almost Riemann soliton in a non-cosymplectic normal almostcontact metric manifold M . Before all else, it is proved that if the metric of M isRiemann soliton with divergence-free potential vector field Z , then the manifold isquasi-Sasakian and is of constant sectional curvature - λ , provided α, β = constant.Other than this, it is shown that if the metric of M is ARS and Z is pointwisecollinear with ξ and has constant divergence, then Z is a constant multiple of ξ andthe ARS reduces to a Riemann soliton, provided α, β =constant. Additionally,it is established that if M with α, β = constant admits a gradient ARS ( γ, ξ, λ ),then the manifold is either quasi-Sasakian or is of constant sectional curvature − ( α − β ). At long last, we develop an example of M conceding a Riemannsoliton. Introduction
Since Einstein manifolds play out a huge job in Mathematics and material science,the examination of Einstein manifolds and their speculations is an intriguing pointin Riemannian and contact geometry. Lately, various generalizations of Einsteinmanifolds such as Ricci soliton, gradient Einstein soliton, gradient Ricci soliton,gradient m -quasi Einstein soliton etc. have researched. The notion of Ricci flowwas introduced by Hamilton [12] and defined by ∂∂ t g ( t ) = − S ( t ), where S denotesthe Ricci tensor.As a spontaneous generalization, the idea of Riemann flow ([19],[20]) is definedby ∂∂ t G ( t ) = − Rg ( t ), G = g ⊗ g , where R is the Riemann curvature tensor and ⊗ AMS 2010 Mathematics Subject Classification : 53C15, 53C25 53D15.Key words and phrases: 3-dimensional normal almost contact metric manifold, Almost Riemannsoliton, Gradient almost Riemann soliton. is Kulkarni-Nomizu product (executed as (see Besse [1], p. 47),( P ⊗ Q )( E, F, Z, W ) = P ( E, W ) Q ( F, U ) + P ( F, U ) Q ( E, W ) − P ( E, U ) Q ( F, W ) − P ( F, W ) Q ( E, U )) . Analogous to Ricci soliton, the entrancing thought of Riemann soliton was promotedby Hirica and Udriste [13]. As per Hirica and Udriste [13], a Riemannian metric g on a Riemannian manifold M is called a Riemann solitons if there exists a C ∞ vector field Z and a real scalar λ such that(1.1) 2 R + λg ⊗ g + g ⊗ £ Z g = 0 . Here we should see that, this new thought of Riemann soliton is nothing buta generalization of the space of constant sectional curvature. The soliton will betermed as expanding (if λ > steady (if λ = 0) or shrinking (if λ < gradient Riemann soliton if the vector field Z isgradient of the potential function γ . For this situation the forerunner condition canbe composed as(1.2) 2 R + λg ⊗ g + g ⊗ ∇ γ = 0 , where ∇ γ denotes the Hessian of γ . In the event that we fix the condition on theparameter λ to be a variable function, then the equation (1.1) and (1.2) turns into ARS and gradient ARS respectively. All through this paper the terminology“almost Riemann solitons” is composed as
ARS .Riemann solitons and gradient Riemann solitons on Sasakian manifolds have beeninvestigated in detail by Hirica and Udriste (see, [13]). Furthermore, Riemann’ssoliton concerning infinitesimal harmonic transformation was studied in [18]. In thisassociation, we notice that Sharma in [16] explored almost Ricci soliton in K -contactgeometry and in [17], with divergence-free soliton vector field. In [6], Riemannsoliton under the context of contact manifold has been studied and demonstrated afew intriguing outcomes.Quite a long while prior, in [14], Olszak explored the three dimensional normalalmost contact metric (briefly, acm ) manifolds mentioning several examples. Afterthe citation of [14], in recent years normal acm manifolds have been studied bynumerous eminent geometers (see, [7],[8],[9],[10] and references contained in those). NOTE ON SOLITONS 3
The above studies motivate us to investigate an
ARS and the gradient ARS ina 3-dimensional normal acm manifolds, since 3-dimensional normal acm manifoldcovers Sasakian manifold, Cosymplectic manifold, Kenmotsu manifold and Quasi-Sasakian manifold.The forthcoming article is structured as:In section 2, we reminisce about some facts and formulas of normal acm manifolds,which we will require in later sections. Beginning from Section 3, after giving theproof, we will engrave our main Theorems. After that, we develop an example of a3-dimensional normal acm manifold admitting a Riemann soliton. This expositionterminates with a concise bibliography that has been utilized during the formulationof the article. 2.
Preliminaries
Let M be an acm manifold endowed with a triplet of almost contact structure( η, ξ, φ ).In details, M is an odd-dimensional differentiable manifold equipped with a global1-form η , a unique characteristic vector field ξ and a (1 , φ, respectively, such that(2.1) φ E = − E + η ( E ) ξ, η ( ξ ) = 1 , φξ = 0 , η ◦ φ = 0 . A structure, named almost complex structure J on M × R is defined by(2.2) J ( E, λ dds ) = ( φE − λξ, η ( E ) dds ) , where ( E, λ dds ) indicates a tangent vector on M × R , E and λ dds being tangent to M and R respectively. After fulfilling the condition, the structure J is integrable, M ( η, ξ, φ ) is said to be normal (see, [2],[3]).The Nijenhuis torsion is defined by[ φ, φ ]( E, F ) = φ [ E, F ] + [ φE, φF ] − φ [ φE, F ] − φ [ E, φF ] . The structure ( η, ξ, φ ) is said to be normal if and only if(2.3) [ φ, φ ] + 2 dη ⊗ ξ = 0 . The Riemannian metric g on M is said to be compatible with ( η, ξ, φ ) if the con-dition(2.4) g ( φE, φF ) = g ( E, F ) − η ( E ) η ( F ) , K.DE AND U.C.DE holds for any
E, F ∈ X ( M ). In such case, the quadruple ( η, ξ, φ, g ) is termed asan acm structure on M and M is an acm manifold . The equation(2.5) η ( E ) = g ( E, ξ ) , is withal valid on such a manifold.Certainly, we can define the fundamental 2-form Φ by(2.6) Φ( E, F ) = g ( E, φF ) , where E, F ∈ X ( M ).For a normal acm manifold, we can write [14]:(2.7) ( ∇ E φ )( F ) = g ( φ ∇ E ξ, F ) − η ( F ) φ ∇ E ξ, (2.8) ∇ E ξ = α [ E − η ( E ) ξ ] − βφE, where α = divξ and β = tr ( φ ∇ ξ ), divξ is the divergent of ξ defined by divξ = trace { E −→ ∇ E ξ } and tr ( φ ∇ ξ ) = trace { E −→ φ ∇ E ξ } . Utilizing (2 .
8) in (2 .
7) welead(2.9) ( ∇ E φ )( F ) = α [ g ( φE, F ) ξ − η ( F ) φE ] + β [ g ( E, F ) ξ − η ( F ) E ] . Also in this manifold the subsequent relations hold [14]: R ( E, F ) ξ = [ F α + ( α − β ) η ( F )] φ E − [ Eα + ( α − β ) η ( E )] φ F (2.10) +[ F β + 2 αβη ( F )] φE − [ Eβ + 2 αβη ( E )] φF,S ( E, ξ ) = − Eα − ( φE ) β (2.11) − [ ξα + 2 ( α − β )] η ( E ) , (2.12) ξβ + 2 α β = 0 , (2.13) ( ∇ E η )( F ) = αg ( φE, φF ) − βg ( φE, F ) . NOTE ON SOLITONS 5
It is well admitted that in a 3-dimensional Riemannian manifold the Riemanncurvature tensor is always satisfies R ( E, F ) Z = S ( F, Z ) E − S ( E, Z ) F + g ( F, Z ) QE − g ( E, Z ) QF (2.14) − r g ( F, Z ) E − g ( E, Z ) F ] . By (2 . .
11) and (2 .
14) we infer S ( E, F ) = ( r ξα + α − β ) g ( φE, φF ) − η ( E )( F α + ( φF ) β ) − η ( F )( Eα + ( φE ) β )(2.15) − α − β ) η ( E ) η ( F ) . For α, β =constant, it follows from the above equation that a 3-dimensional normal acm manifold becomes an η -Einstein manifold.From (2 .
9) we conclude that the manifold is either α -Kenmotsu [11] or cosym-plectic [2] or β -Sasakian, provided α, β =constant. Also it is well known that a3-dimensional normal acm manifold reduces to a quasi-Sasakian manifold if andonly if α = 0 (see, [14],[15]). 3. Riemann Soliton
In this segment, we first write the subsequent result ([4],[5]):
Lemma 3.1.
In a Riemannian manifold if ( g, Z ) is a Ricci soliton, then we have k £ Z g k = dr ( Z ) + 2 div ( λZ − QZ ) . (3.1)Now, because of (2.8) we obtain(3.2) ( £ ξ g )( E, F ) = 2 α { g ( E, F ) − η ( E ) η ( F ) } . We consider a normal acm manifold M with α, β =constants admitting a Rie-mann soliton defined by(1.1). Using Kulkarni-Nomizu product in (1.1) we write2 R ( E, F, W, X ) + 2 λ { g ( E, X ) g ( F, W ) − g ( E, W ) g ( F, X ) } + { g ( E, X )( £ Z g )( F, W ) + g ( F, W )( £ Z g )( U, E ) − g ( E, W )( £ Z g )( F, X ) − g ( F, X )( £ Z g )( E, W ) } = 0 . (3.3)Contracting (3.3) over E and X , we infer K.DE AND U.C.DE ( £ Z g )( F, W ) + 2 S ( F, W ) + (4 λ + 2 divZ ) g ( F, W ) = 0 . (3.4)Thus Riemann soliton whose potential vector field is of vanishing divergence reducesto Ricci soliton.Hence we have ( £ Z g )( F, W ) + 2 S ( F, W ) + 4 λg ( F, W ) = 0 . (3.5)Setting Z = ξ and utilizing (3.2) we lead2 α { g ( F, W ) − η ( F ) η ( W ) } + 2 S ( F, W ) + 4 λg ( F, W ) = 0 , which implies that α { F − η ( F ) ξ } + QF + 2 λF = 0 . (3.6)Putting F = W = e i and taking divZ = 0 from (3.5) we get r = − λ . Hence in ourcase (3.1) takes the form12 k £ Z g k = dr ( Z ) + 2 div ( − λZ − QZ ) . (3.7)From (3.6) we get Qξ = − λξ . Therefore utilizing r = − λ and Qξ = − λξ weobtain from (3.7) ξ is a Killing vector. Hence (3.2) implies α = 0 that is the manifoldis quasi-Sasakian.Utilizing α = 0 in (3.6), we infer QF = − λF, . Hence from (2.14) we can write that the manifold is of constant sectional curvature − λ . Therefore we write: Theorem 3.1.
If the metric of a non-cosymplectic normal acm manifold M isRiemann soliton with a divergence-free potential vector field, then the manifold isquasi-Sasakian and is of constant sectional curvature − λ , provided α, β = constant . Almost Riemann Soliton
Here we consider a normal acm manifold M with α, β =constants admitting an ARS defined by(1.1).
NOTE ON SOLITONS 7
In particular, let the potential vector field Z be point-wise collinear with ξ (i.e., Z = cξ , where c is a function on M ) and has constant divergence. Then from (3.4)we lead(4.1) g ( ∇ E cξ, F ) + g ( ∇ F cξ, E ) + 2 S ( E, F ) + (4 λ + 2 divZ ) g ( E, F ) = 0 . Utilizing (2.5) and (2.8) in (4.1), we obtain2 αc [ g ( E, F ) − η ( E ) η ( F )] + ( Ec ) η ( F ) + ( F c ) η ( E )(4.2) +2 S ( E, F ) + (4 λ + 2 divZ ) g ( E, F ) = 0 . Replacing F by ξ in (4.2) and utilizing (2.1), (2.5) and (2.11) gives(4.3) ( Ec ) + ( ξc ) η ( E ) − α − β ) η ( E ) + (4 λ + 2 divZ ) η ( E ) = 0 . Putting E = ξ in (4 .
3) and utilizing (2 .
1) yields(4.4) ξc = [2( α − β ) − λ − divZ ] . Putting the value of ξc in (4 .
3) we infer(4.5) dc = [2( α − β ) − λ − divZ ] η. Applying d on (4 .
5) and using Poincare lemma d ≡
0, we lead(4.6) [2( α − β ) − λ − divZ ] dη + ( dλ ) η = 0 . Taking wedge product of (4 .
6) with η , we obtain(4.7) [2( α − β ) − λ − divZ ] η ∧ dη = 0 . Since η ∧ dη = 0 , we infer(4.8) [2( α − β ) − λ − divZ ] = 0 . Utilizing (4 .
8) in (4 .
5) gives dc = 0 i.e., c =constant. Also from (4.8) we have(4.9) λ = [( α − β ) − divZ ] = constant. K.DE AND U.C.DE
Hence we write the following:
Theorem 4.1.
If the metric of a non-cosymplectic normal acm manifold M is ARSand Z is pointwise collinear with ξ and has constant divergence, then Z is constantmultiple of ξ and the ARS reduces to a Riemann soliton, provided α, β = constant . Corollary 4.1.
If a non-cosymplectic normal acm manifold M with α, β = constantadmits an ARS of type ( g, ξ ), then the ARS reduces to a Riemann soliton. Gradient Almost Riemann Soliton
In this section we investigate a non-cosymplectic normal acm manifold M with α, β =constant, admitting gradient ARS . Now we prove the subsequent results:
Lemma 5.1.
For a non-cosymplectic normal acm manifold M with α, β = constant,we have ( ∇ E Q ) ξ = −{ r α − β ) } [ α { E − η ( E ) ξ } − βφE ] . (5.1) Proof.
For α, β =constants, we get from (2.15)(5.2) QF = { r α − β ) } F − { r α − β ) } η ( F ) ξ. Differentiating (5.2) covariantly in the direction of E and using (2.8) and (2.13),we get ( ∇ E Q ) F = dr ( E )2 ( F − η ( F ) ξ )(5.3) −{ r α − β ) } [ αg ( E, F ) ξ − αη ( E ) η ( F ) ξ + αη ( F ) E − βg ( φE, F ) ξ − βη ( F ) φE ] . Replacing F by ξ in (5.3) and utilizing (2.8), we get( ∇ E Q ) ξ = −{ r α − β ) } [ α { E − η ( E ) ξ } − βφE ] . (cid:3) NOTE ON SOLITONS 9
Lemma 5.2.
Let M ( η, ξ, φ, g ) be a non-cosymplectic normal acm manifold with α, β = constant. Then we have ξr = − α { r α − β ) } (5.4) Proof.
Recalling (5.3), we can write g (( ∇ E Q ) F, Z ) = dr ( E )2 [ g ( F, Z ) − η ( F ) η ( Z )](5.5) −{ r α − β ) } [ αg ( E, F ) η ( Z ) − αη ( E ) η ( F ) η ( Z )+ αη ( F ) g ( E, Z ) − βg ( φE, F ) η ( Z ) − βη ( F ) g ( φE, Z )] . Putting E = Z = e i (where { e i } be the orthonormal basis for the tangent spaceof M and taking P i , 1 ≤ i ≤ divQ = grad r , we obtain( ξr ) η ( F ) = − α { r α − β ) } η ( F ) . (5.6)Replacing F = ξ in the previous equation we have the required result. (cid:3) Lemma 5.3. (Lemma. 3.8 of [6] ) For any vector fields
E, F on M , in a gradientARS ( M, g, γ, m, λ ) , we infer R ( E, F ) Dγ = ( ∇ F Q ) E − ( ∇ E Q ) F + { F (2 λ + △ γ ) E − E (2 λ + △ γ ) F } , (5.7) where △ γ = div Dγ , △ is the Laplacian operator. Superseding F by ξ in (5.7) and utilizing Lemma 5.1, we get R ( E, ξ ) Dγ = dr ( ξ )2 [ E − η ( E ) ξ ] − { r α − β ) } [2 αE − αη ( E ) ξ − βφE ]+ { ξ (2 λ + △ γ ) E − E (2 λ + △ γ ) ξ } . (5.8)Then utilizing (2.10), we infer g ( E, ( α − β ) Dγ + D (2 λ + △ γ )) ξ = dr ( ξ )2 [ E − η ( E ) ξ ] −{ r α − β ) } [2 αE − αη ( E ) ξ − βφE ]+ { ( α − β )( ξγ ) + ξ (2 λ + △ γ ) } E. (5.9)Executing inner product of the foregoing equation with ξ yields E (( α − β ) γ + (2 λ + △ γ )) = { ( α − β )( ξγ ) + ξ (2 λ + △ γ ) } η ( E ) , (5.10) from which easily we lead d (( α − β ) γ + (2 λ + △ γ )) = { ( α − β )( ξγ ) + ξ (2 λ + △ γ ) } η, (5.11)where the exterior derivative is denoted by d . From the above equation we concludethat ( α − β ) γ + (2 λ + △ γ ) is invariant along the distribution D . In other terms, E (( α − β ) γ + (2 λ + △ γ )) = 0 for any E ∈ D . Hence utilizing (5 .
10) in (5.9), weget dr ( ξ )2 [ E − η ( E ) ξ ] − { r α − β ) } [2 αE − αη ( E ) ξ − βφE ] = 0 . (5.12)Contracting the previous equation and using (5.4), we lead α { r α − β ) } = 0 . (5.13)Now we split our study in the following cases:Case (i): If α = 0, then the manifold reduces to a quasi-Sasakian manifold.case (ii): If r = − α − β ), then from (2.15) we get S = − α − β ) g , thatis the manifold is an Einstein manifold and hence from (2.14) it follows that themanifold is of constant sectional curvature − ( α − β ). Hence we write: Theorem 5.1.
If a non-cosymplectic normal acm manifold M with α, β = constantadmits a gradient ARS ( γ, ξ, λ ) , then the manifold is either quasi-Sasakian or is ofconstant sectional curvature − ( α − β ) . Example
We consider the manifold M = { ( x, y, z ) ∈ R , z = 0 } and the linearly indepen-dent vector fields u = z ∂∂x , u = z ∂∂y , u = z ∂∂z . The Riemannian metric g is defined by g ( u , u ) = g ( u , u ) = g ( u , u ) = 0 ,g ( u , u ) = g ( u , u ) = g ( u , u ) = 1 . Let the 1-form η is given by η ( E ) = g ( E, u ) for any E ∈ X ( M ) and the tensorfield φ is given by φ ( u ) = − u , φ ( u ) = u , φ ( u ) = 0 . NOTE ON SOLITONS 11
Then utilizing the linearity of φ and g , we infer η ( u ) = 1 ,φ E = − E + η ( E ) u ,g ( φE, φF ) = g ( E, F ) − η ( E ) η ( F ) , for any E, F ∈ X ( M ) . Obviously, the structure ( η, ξ, φ, g ) admits an acm structureon M for u = ξ . Then we lead[ u , u ] = u u − u u = z ∂∂x ( z ∂∂z ) − z ∂∂z ( z ∂∂x )= z ∂ ∂x∂z − z ∂ ∂z∂x − z ∂∂x = − u . (6.1)Similarly [ u , u ] = 0 and [ u , u ] = − u . Utilizing Koszul’s formula for the Riemannian metric g, we can calculate ∇ u u = − u , ∇ u u = 0 , ∇ u u = u , ∇ u u = − u , ∇ u u = u , ∇ u u = 0 , (6.2) ∇ u u = 0 , ∇ u u = 0 , ∇ u u = 0 . From the above expression it is obvious that the manifold under consideration is anormal acm manifold with α , β =constants, since it satisfies (2.8) for α = − β = 0 and ξ = e . It can be easily verified that R ( u , u ) u = 0 , R ( u , u ) u = − u , R ( u , u ) u = − u ,R ( u , u ) u = − u , R ( u , u ) u = u , R ( u , u ) u = 0 ,R ( u , u ) u = u , R ( u , u ) u = 0 , R ( u , u ) u = u . In this example, it is easy to verify that the characteristic vector field ξ hasconstant divergence and obviously £ ξ g = 0 . Then equation (3.3) reduces to2 R ( E, F ) W + 2 λ { g ( F, W ) E − g ( E, W ) Y } = 0 , (6.3) for all vector field E, F, W . Also equation (6.3)holds for λ = 1. Thus the manifoldunder consideration admits a Riemann soliton ( g, ξ, λ ). References [1] Besse, A.,
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Krishnendu De,Assistant Professor of Mathematics,Kabi Sukanta Mahavidyalaya,Bhadreswar, P.O.-Angus, Hooghly,Pin 712221, West Bengal, India.
E-mail address : [email protected] Uday Chand DeDepartment of Pure MathematicsUniversity of Calcutta35, Ballygunge Circular RoadKol- 700019, West Bengal, India.
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