1. Introduction In the same way like the Ricci solitons generate self-similar solutions to Ricci flow

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1 Kragujevac Journal of Mathematics Volume 4) 018), Pages ON GRADIENT η-einstein SOLITONS A. M. BLAGA 1 Abstract. If the potential vector field of an η-einstein soliton is of gradient type, using Bochner formula, we derive from the soliton equation a nonlinear second order PDE. Under certain conditions, the existence of an η-einstein soliton forces the manifold to be of constant scalar curvature. 1. Introduction In the same way like the Ricci solitons generate self-similar solutions to Ricci flow 1.1) t g = S, the notion of Einstein solitons, which generate self-similar solutions to Einstein flow 1.) t g = S scal ) g, was introduced by G. Catino and L. Mazzieri [4]. The interest in studying this equation from different points of view arises from concrete physical problems. On the other hand, gradient vector fields play a central role in Morse-Smale theory [16], aspects of gradient η-ricci solitons being discussed in [3]. In what follows, after characterizing the manifold of constant scalar curvature via the existence of η-einstein solitons, we focus on the case when the potential vector field ξ is of gradient type i.e., ξ = gradf), for f a nonconstant smooth function on M) and give the Laplacian equation satisfied by f. Under certain assumptions, the existence of an η-einstein soliton implies that the manifold is quasi-einstein. Remark that quasi-einstein manifolds arose during the study of exact solutions of Einstein field equations. Key words and phrases. Gradient η-einstein soliton, gradient vector field, Laplace equation. 010 Mathematics Subject Classification. Primary: 53C1, 53C5. Secondary: 53B50, 53C15. Received: September 15, 016. Accepted: February 7,

2 30 A. M. BLAGA. η-einstein soliton equation In the study of the η-einstein soliton equation we will consider certain assumptions, one essential condition being ξ = I χm) η ξ which naturally arises in different geometries: Kenmotsu [10], Lorenzian-Kenmotsu [1], para-kenmotsu [1] etc. Immediate properties of this structure, which will be used later, are given in the next proposition. Proposition.1. [3] Let M, g) be an m-dimensional Riemannian manifold and η be the g-dual 1-form of the nonzero vector field ξ. If ξ satisfies ξ = I χm) η ξ, where is the Levi-Civita connection associated to g, then: a) L ξ g = g η η); b) RX, Y )ξ = ηx)y ηy )X, for any X, Y χm); c) Sξ, ξ) = 1 m) ξ. Consider the equation.1) L ξ g + S + λ scal)g + µη η = 0, where L ξ is the Lie derivative operator along the vector field ξ, S is the Ricci curvature tensor field, scal is the scalar curvature of the Riemannian metric g, and λ and µ are real constants. For µ 0, the data g, ξ, λ, µ) will be called η-einstein soliton. Remark that if the scalar curvature scal of the manifold is constant, then the η-einstein soliton g, ξ, λ, µ) reduces to an η-ricci soliton g, ξ, λ scal, µ) and, moreover, if µ = 0, to a Ricci soliton ) g, ξ, λ scal. Therefore, the two concepts of η-einstein soliton and η-ricci soliton are distinct on manifolds of nonconstant scalar curvature. Writing now L ξ g in terms of the Levi-Civita connection, we obtain.) SX, Y ) = g X ξ, Y ) gx, Y ξ) λ scal)gx, Y ) µηx)ηy ), for any X, Y χm). An important geometrical object in studying Ricci solitons is a symmetric 0, )- tensor field which is parallel with respect to the Levi-Civita connection [], [7]. The existence of such tensors on smooth manifolds carrying different structures such as contact [14], K-contact [15], P -Sasakian [11], α-sasakian [8] etc. was investigated by many authors. The starting point was the Eisenhart problem of finding symmetric and skew symmetric) parallel tensors on various spaces. He proved in [9] that if a positive definite Riemannian manifold admits a second order parallel symmetric covariant tensor field other than a constant multiple of the metric, then it is reducible. In our case, we show that if we ask to be satisfied the condition ξ = I χm) η ξ, then any symmetric -parallel 0, )-tensor field must be a constant multiple of the metric. If we take α := L ξ g + S + µη η, the above mentioned result leads us to characterize the existence of the soliton g, ξ, λ, µ) in terms of a scalar curvature property. Following the ideas of Călin and Crasmareanu [5] we shall study the equation.1), applying similar techniques.

3 ON GRADIENT η-einstein SOLITONS 31 Let α be such a symmetric 0, )-tensor field which is parallel with respect to the Levi-Civita connection α = 0). From the Ricci identity αx, Y ; Z, W ) αx, Y ; W, Z) = 0, one obtains for any X, Y, Z, W χm) [13].3) αrx, Y )Z, W ) + αz, RX, Y )W ) = 0. In particular, for Z = W := ξ from the symmetry of α follows αrx, Y )ξ, ξ) = 0, for any X, Y χm). If ξ = I χm) η ξ, from Proposition.1 we have RX, Y )ξ = ηx)y ηy )X and replacing this expression in α we get.4) αy, ξ) ηy )αξ, ξ) = 0, for any Y χm), equivalent to.5) αy, ξ) αξ, ξ)gy, ξ) = 0, for any Y χm). Differentiating the equation.5) covariantly with respect to the vector field X χm) we obtain α X Y, ξ) + αy, X ξ) = αξ, ξ)[g X Y, ξ) + gy, X ξ)], and substituting the expression of X ξ = X ηx)ξ we get.6) αy, X) = αξ, ξ)gy, X), for any X, Y χm). As α is -parallel, follows αξ, ξ) is constant and we conclude the following. Proposition.. Under the hypotheses above, any parallel symmetric 0, )-tensor field is a constant multiple of the metric. Applying these results, we conclude the following theorem. Theorem.1. Let η be the g-dual 1-form of the unitary vector field ξ on the Riemannian manifold M, g) such that ξ satisfies ξ = I χm) η ξ, where is the Levi- Civita connection associated to g. Assume that the symmetric 0, )-tensor field α := L ξ g +S +µη η is parallel with respect to. Then g, ξ, λ := 1 [αξ, ξ) scal], µ) satisfies equation.1) if and only if M is of constant scalar curvature. Proof. Compute αξ, ξ) and from.1) we get αξ, ξ) = L ξ g)ξ, ξ) + Sξ, ξ) + µηξ)ηξ) = λ + scal, and taking into account that α = 0, we deduce that scal = c a real constant), so λ = 1 [αξ, ξ) c]. Conversely, if scal = c a real constant), from.6) and α = 0 we get αx, Y ) = αξ, ξ)gx, Y ) = λ c)gx, Y ), for any X, Y χm). Therefore, L ξ g + S + µη η = λ c)g, i.e. g, ξ, λ, µ) satisfies equation.1).

4 3 A. M. BLAGA The above condition we asked to be satisfied by the potential vector field ξ, namely X ξ = X ηx)ξ, naturally arises if M, ϕ, ξ, η, g) is for example, Kenmotsu manifold [10]. Example.1. Let M = {x, y, z) R 3, z > 0}, where x, y, z) are the standard coordinates in R 3. Set ϕ := y dx + x dy, ξ := z z, η := 1 z dz, g := 1 dx dx + dy dy + dz dz). z Then ϕ, ξ, η, g) is a Kenmotsu structure on M. Consider the linearly independent system of vector fields: Follows E 1 := z x, E := z y, E 3 := z z. ϕe 1 = E, ϕe = E 1, ϕe 3 = 0, ηe 1 ) = 0, ηe ) = 0, ηe 3 ) = 1, [E 1, E ] = 0, [E, E 3 ] = E, [E 3, E 1 ] = E 1, and the Levi-Civita connection is deduced from Koszul s formula precisely g X Y, Z) =XgY, Z)) + Y gz, X)) ZgX, Y )) gx, [Y, Z]) + gy, [Z, X]) + gz, [X, Y ]), E1 E 1 = E 3, E1 E = 0, E1 E 3 = E 1, E E 1 = 0, E E = E 3, E E 3 = E, E3 E 1 = 0, E3 E = 0, E3 E 3 = 0. Then the Riemann and the Ricci curvature tensor fields are given by: RE 1, E )E = E 1, RE 1, E 3 )E 3 = E 1, RE, E 1 )E 1 = E, RE, E 3 )E 3 = E, RE 3, E 1 )E 1 = E 3, RE 3, E )E = E 3, SE 1, E 1 ) = SE, E ) = SE 3, E 3 ) =, and the scalar curvature scal = 6. From.1) computed in E i, E i ) follows [ge i, E i ) ηe i )ηe i )] + SE i, E i ) + λ scal)ge i, E i ) + µηe i )ηe i ) = 0, for all i {1,, 3}, and we have 1 δ i3 ) 4 + λ µδ i3 = 0 λ µ 1)δ i3 = 0, for all i {1,, 3}. Therefore, µ = 1 and λ = define an η-einstein soliton on M, ϕ, ξ, η, g).

5 ON GRADIENT η-einstein SOLITONS 33 The condition X ξ = X ηx)ξ implies L ξ g = g η η) and the equation.) becomes.7) SX, Y ) = λ + 1 scal ) gx, Y ) µ 1)ηX)ηY ). Recall that the manifold is called quasi-einstein if the Ricci curvature tensor field S is a linear combination with real scalars λ and µ respectively, with µ 0) of g and the tensor product of a nonzero 1-form η satisfying ηx) = gx, ξ), for ξ a unit vector field [6] and respectively, Einstein if S is collinear with g. Sufficient conditions for M, g) to be quasi-einstein or Einstein are given in the next two propositions. Proposition.3. Let η be the g-dual 1-form of the nonzero vector field ξ on the Riemannian manifold M, g) such that ξ satisfies ξ = I χm) η ξ, where is the Levi-Civita connection associated to g. If g, ξ, λ, µ) satisfy the equation.1), then M, g) is quasi-einstein. Proof. It follows from.7). If we ask for certain curvature conditions, namely, Rξ, X) S = 0 and Sξ, X) R = 0, we deduce that M is either Einstein or get its scalar curvature depending on the constants λ, µ) that define the η-einstein soliton on M respectively, where by we denote the derivation of the tensor algebra at each point of the tangent space: Rξ, X) S)Y, Z) := ξ R X) S)Y, Z) := Sξ R X)Y, Z)+SY, ξ R X)Z), for X R Y )Z := RX, Y )Z; Sξ, X) R)Y, Z)W := ξ S X) R)Y, Z)W := ξ S X)RY, Z)W +Rξ S X)Y, Z)W + RY, ξ S X)Z)W + RY, Z)ξ S X)W, for X S Y )Z := SY, Z)X SX, Z)Y. Proposition.4. Let η be the g-dual 1-form of the nonzero and nonunitary vector field ξ on the Riemannian manifold M, g) such that ξ satisfies ξ = I χm) η ξ, where is the Levi-Civita connection associated to g. If g, ξ, λ, µ) satisfy the equation.1) and Rξ, X) S = 0, then M, g) is Einstein manifold. Proof. The condition that must be satisfied by S is:.8) SRξ, X)Y, Z) + SY, Rξ, X)Z) = 0, for any X, Y, Z χm). Replacing the expression of S from.7) and from the symmetries of R we get.9) µ 1)[ηY )gx, Z) + ηz)gx, Y ) ηx)ηy )ηz)] = 0, for any X, Y, Z χm). For X = Y = Z := ξ we have.10) µ 1)[ηξ)] [ηξ) 1] = 0, which implies µ = 1.

6 34 A. M. BLAGA Remark.1. Under the hypotheses of Proposition.4, there is no Ricci soliton with the potential vector field ξ. Proposition.5. Let η be the g-dual 1-form of the nonzero and nonunitary vector field ξ on the Riemannian manifold M, g) such that ξ satisfies ξ = I χm) η ξ, where is the Levi-Civita connection associated to g. If g, ξ, λ, µ) satisfy the equation.1) and Sξ, X) R = 0, then λ, µ) satisfy λ + 1) + ξ µ 1) = scal. Proof. The condition that must be satisfied by S is:.11) SX, RY, Z)W )ξ Sξ, RY, Z)W )X + SX, Y )Rξ, Z)W Sξ, Y )RX, Z)W + SX, Z)RY, ξ)w Sξ, Z)RY, X)W +SX, W )RY, Z)ξ Sξ, W )RY, Z)X = 0, for any X, Y, Z, W χm). Taking the inner product with ξ, the relation.11) becomes.1) SX, RY, Z)W ) ξ Sξ, RY, Z)W )ηx) + SX, Y )ηrξ, Z)W ) Sξ, Y )ηrx, Z)W ) + SX, Z)ηRY, ξ)w ) Sξ, Z)ηRY, X)W ) +SX, W )ηry, Z)ξ) Sξ, W )ηry, Z)X) = 0, for any X, Y, Z, W χm). For W := ξ and from the symmetries of R we get.13) SX, RY, Z)ξ) ξ Sξ, RY, Z)ξ)ηX) + Sξ, ξ)gry, Z)ξ, X) = 0, for any X, Y, Z χm). Replacing the expression of S from.7), we get.14) ξ [λ + scal +µ 1) ξ ][ηy )gx, Z) ηz)gx, Y )] = 0, for any X, Y, Z χm). For Z := ξ we have.15) ξ [λ + scal +µ 1) ξ ][ηx)ηy ) ξ gx, Y )] = 0, for any X, Y χm) and we obtain.16) λ + scal +µ 1) ξ = 0, which is stated. Corollary.1. Let η be the g-dual 1-form of the nonzero and nonunitary vector field ξ on the Riemannian manifold M, g) such that ξ satisfies ξ = I χm) η ξ, where is the Levi-Civita connection associated to g. If g, ξ, λ, 0) satisfy the equation.1) and Sξ, X) R = 0, then λ = ξ +scal 1.

7 ON GRADIENT η-einstein SOLITONS Gradient η-einstein solitons We are interested in gradient η-einstein solitons, as solutions of the equation 3.1) L ξ g + S + λ scal)g + µη η = 0, where g is a Riemannian metric, S is the Ricci curvature, scal is the scalar curvature, η is a 1-form whose g-dual vector field ξ is of gradient type, ξ := gradf), for f a smooth function on M, and λ and µ are real constants µ 0). The data g, ξ, λ, µ) which satisfy the equation 3.1) is said to be a gradient η-einstein soliton on M. Taking the trace of the relation 3.1), applying then ξ and observing that if ξ = m i=1 ξ i E i, for {E i } 1 i m a local orthonormal frame field with Ei E j = 0 in a point, m traceη η) = [dfe i )] = m ξ j ξ k ge i, E j )ge i, E k ) = ξ i ) = i=1 = 1 i,j m 1 i,j,k m ξ i ξ j ge i, E j ) = ξ, we get 3.) ξdivξ)) + 1 m ) ξ scal) + µξ ξ ) = 0, and because L ξ g = Hessf), taking the divergence of the same relation and computing it in ξ we obtain 3.3) divl ξ g))ξ)+divs))ξ) tracedscal) dscal))+µdivdf df))ξ) = 0. From 3.1) we deduce 3.4) Sξ, ξ) = 1 ξ ξ ) λ ξ + scal ξ µ ξ 4. Multiplying 3.3) by 1 m, substracting 3.), using the fact that scal) = divs), we get 3.5) 3.6) divl ξ g))ξ) = ξ ) ξ, ξ ) ξ =Sξ, ξ) + ξdivξ)), i=1 3.7) divdf df))ξ) = 1 tracedf d ξ )) + ξ divξ), and with 3.4), we get 3.8) ) 1 m ξ ) =1 m) ξ + 1 ξ ξ ) + λ ξ scal ξ + 1 m ) tracedscal) dscal)) { + µ ξ 4 1 m ) tracedf d ξ )) 1 m ) } ξ divξ) + ξ ξ ).

8 36 A. M. BLAGA Theorem 3.1. Let M, g) be an m-dimensional Riemannian manifold m > ) and η be the g-dual 1-form of the gradient vector field ξ := gradf). Assume that 3.1) defines an η-einstein soliton on M and ξ satisfies ξ = I χm) η ξ, where is the Levi-Civita connection associated to g. If µ = 1, then M is of constant scalar curvature; if µ 1, then M is of constant scalar curvature if and only if ξ is of constant length, and in this case, the Laplacian equation becomes 3.9) f) = m 1 µ. Proof. We have 3.10) ξ ξ ) = ξ ξ 4 ), and 3.11) ξ ξ 4 ) = 4 ξ 4 ξ 6 ), and from.7) we get 3.1) Sξ, ξ) = Also from Proposition.1: λ + 1 scal 3.13) Sξ, ξ) = ξ m ξ, therefore 3.14) ξ = We obtain m 1 λ + scal ) ) ξ µ 1) ξ 4. ξ µ 1) ξ ) ξ µ 1) = m λ + scal. If µ = 1, then from 3.15) we obtain scal = λ + m), i.e. M is of constant scalar curvature. Let µ 1. If the scalar curvature is constant, then ξ is constant. Conversely, if ξ is of constant length, from 3.10) follows ξ = 1 and from 3.15) we obtain scal = λ + µ + 1 m), i.e. M is of constant scalar curvature. We also have m m 3.16) ξ := g Ei ξ, Ei ξ) = {1 + ξ )[ηe i )] } = m + ξ ξ ), i=1 i=1 for {E i } 1 i m a local orthonormal frame field with Ei E j = 0 in a point. Now using the relations above, 3.8) becomes 3.9). Remark that in this case, the soliton 3.1) is completely determined by f, m and scal. Example 3.1. The soliton considered in Example.1 is a gradient η-einstein soliton, as the potential vector field ξ is of gradient type, ξ = gradf), where fx, y, z) := ln z.

9 ON GRADIENT η-einstein SOLITONS 37 Acknowledgements. The author thanks to the referees for their valuable suggestions that definitely improved the paper. References [1] N. S. Basavarajappa, C. S. Bagewadi and D. G. Prakasha, Some results on Lorentzian β-kenmotsu manifolds, An. Univ. Craiova Ser. Mat. Inform ), [] C. L. Bejan and M. Crasmareanu, Ricci solitons in manifolds with quasi-constant curvature, Publ. Math. Debrecen 781) 011), [3] A. M. Blaga, On warped product gradient η-ricci solitons, arxiv: ). [4] G. Catino and L. Mazzieri, Gradient Einstein solitons, Nonlinear Anal ), [5] C. Călin and M. Crasmareanu, Eta-Ricci solitons on Hopf hypersurfaces in complex space forms, Rev. Roumaine Math. Pures Appl. 571) 01), [6] M. C. Chaki and R. K. Maity, On quasi Einstein manifolds, Publ. Math. Debrecen ), [7] M. Crasmareanu, Parallel tensors and Ricci solitons in Nk)-quasi Einstein manifolds, Indian J. Pure Appl. Math. 434) 01), [8] L. Das, Second order parallel tensor on α-sasakian manifold, Acta Math. Acad. Paedagog. Nyházi. N.S.) 31) 007), [9] L. P. Eisenhart, Symmetric tensors of the second order whose first covariant derivatives are zero, Trans. Amer. Math. Soc. 5) 193), [10] K. Kenmotsu, A class of almost contact Riemannian manifolds, Tohoku Math. J ), [11] Z. Li, Second order parallel tensors on P -Sasakian manifolds with a coefficient k, Soochow J. Math. 31) 1997), [1] Z. Olszak, The Schouten-van Kampen affine connection adapted to an almost para) contact metric structure, Publ. Inst. Math. Beograd) N.S.) 94108) 013), [13] R. Sharma, Second order parallel tensor in real and complex space forms, Int. J. Math. Math. Sci. 14) 1989), [14] R. Sharma, Second order parallel tensor on contact manifolds, Algebras Groups Geom ), [15] R. Sharma, Second order parallel tensor on contact manifolds II, C. R. Math. Acad. Sci. Soc. R. Can. 136) 1991), [16] S. Smale, On gradient dynamical systems, Ann. Math. 741) 1961), Department of Mathematics, West University of Timişoara, Bld. V. Pârvan nr. 4, 3003, Timişoara, România address: adarablaga@yahoo.com