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Abstract
In the Present paper we study Ricci solitons in trans-sasakian manifolds. In particular we consider Ricci solitons in f-Kenmotsu manifolds and we provethe conditionsfor the Ricci solitons to be shrinking, steady and expanding.
Key words
Ricci solitons; f-Kenmotsu; Trans-Sasakian; Shrinking; Steady; Expanding
1.INTRODUCTION
In [10], Ramesh Sharma started the study of the Ricci solitons in contact geometry. Later Mukut Mani Tripathi [11], Cornelia Livia Bejan and Mircea Crasmareanu [3] and others extensively studied Ricci solitons in contact metric manifolds. A Ricci soliton is a generalization of an Einstein metric and is defined on a Riemannian manifold (M,g) by LVg + 2Ric + 2λg = 0,(1.1)
where V is a complete vectorfield on M andλis a constant. The Ricci soliton is said to be shrinking,steady and expandingaccordingasλis negative, zero and positive respectively. If the vectorfield V is the gradient of a potential function f then g is called a gradient Ricci soliton and (1.1) takes the form,
??f = Ric +λg.
Perelman [9] proved that a Ricci soliton on a compact n-manifold is a gradient Ricci soliton. In [11], Ramesh Sharma studied Ricci solitons in K-contact manifolds, where the structurefieldξis killing and he proved that a complete K-contact gradient soliton is compact Einstein and Sasakian. M. M. Tripathi [11] studied Ricci solitons in N(K)-contact metric and (k,μ) manifolds. Motivated by the above studies on Ricci solitons, in this paper, we study Ricci solitons in an important class of manifolds introduced by Kenmotsu in [6].
2.PRELIMINARIES
A (2n+1) dimensional smooth manifold M is said to be an almost contact metric manifold if it admits an almost contact metric structure (φ,ξ,η,g) consisting of a tensorfieldφof type (1,1), a vectorfieldξ, a 1-formηand Riemannian metric g compatible with (φ,ξ,η) satisfyingΦ2=?I +η?ξ,η(ξ) = 1,φξ= 0,η?φ= 0(2.1) and g(φX,φY) = g(X,Y)?η(X)η(Y).(2.2) An almost contact metric manifold is said to be an f-Kenmotsu manifold if(?Xφ)Y = f[g(φX,Y)ξ?φ(X)η(Y)],(2.3) where f∈C∞(M) is strictly positive and df?η= 0 holds. From (2.3) we have?Xξ= f(X?η(X)ξ).(2.4) An almost contact metric manifold is called a trans-Sasakian manifold [4] [8] if(?Xφ)Y =α(g(X,Y)ξ?η(Y)X) +β(g(φX,Y)ξ?η(Y)φX),(2.5) for some smooth functionsαandβon M.
3.RICCI SOLITONS IN F-KENMOTSU MANIFOLDS
Let M be an n dimensional f-Kenmotsu manifold and let (g,V,λ)be a Ricci soliton in M. Let {ei},1≤i≤n be an orthonormal basis of TPM at P∈M. Then from (1.1), we have
S =?(λg +1 2LVg).(3.1) From (2.4), we have(Lξg)(X,Y) = f[g(X,Y)?η(X)η(Y)].(3.2) From (3.1) and (3.2), we have S(X,Y) =?λg(X,Y)?f[g(X,Y)?η(X)η(Y)].(3.3) It is easy to verify from (3.3) that S(φX,Y) =?S(X,φY)(3.4) and
S(ξ,ξ) =?λ.(3.5)
From (2.3) and (2.4), wefind that R(X,Y)ξ= f2[η(X)Y?η(Y)X] + (Y f)φ2X?(Xf)φ2Y(3.6) and S(X,ξ) =?[(n?1)f2+ξf]η(X)?(n?2)X(f).(3.7) From (3.7), we obtain S(ξ,ξ) =?(n?1)[f2+ξf].(3.8)
H.G. Nagaraja; C.R. Premalatha/Progress in Applied Mathematics Vol.3 No.2, 2012 Comparing (3.5) and (3.8), we obtainλ= (n?1)(f2+ξf)(3.9)
From (3.9), it is clear thatλis positive if f is a constant. Thus we have
Ricci soliton in a f-Kenmotsu manifold is expanding, provided f is a constant.
Suppose f is not a constant. If {ei} is an orthonormal basis of TPM at P∈M, then taking X = Y = eiin(3.3) and summing over 1≤i≤n, we get r =?λn?f(n?1),(3.10)
where r is the scalar curvature.
Differentiating (3.10) covariantly w.r.to X, we get Xr=?(n?1)Xf,(3.11) where
Xr=?Xr, Xf=?Xf.
From (3.3), we have QX =?λX?f(φ2X).(3.12) In view of (2.5), differentiation of (3.12) yields
(?YQ)X = Y f(φ2X)?f2η(X)φ2Y + fΦ(X,Y)ξ.
Contracting the above equation with respect to Y, we get(divQ)X = (φ2X) + f2(n?1)η(X).(3.13) Using (3.11) and the identity
(divQ)X =
As it is well knownthat for a Kenmotsumanifildthe curvaturer is negative. Henceλis positive forconstant r. Thus we have,
Theorem 3.3. A Ricci soliton in a Kenmotsu manifold of constant curvature is expanding.
4.RICCI SOLITONS IN 3-DIMENSIONAL TRANS-SASAKIAN MANIFOLDS
Suppose (Mn,g) is a 3-dimensional trans-Sasakian manifold and (g,V,λ) is a Ricci soliton in (Mn,g). If V is a conformal killing vectorfield, then LVg =ρg,(4.1)
2)[η(Y)η(W)?g(Y,W)] = 0. This implies r = 2ξβ?2(α2?β2)(4.8) From (4.4) and (4.8), we have 6λ=ρ?4[ξβ?(α2?β2)].(4.9)
From (4.9), we have
Theorem 4.1.In a 3-dimensionaltrans-Sasakianmanifold, a Ricci Soliton (g,V,λ),where V is conformal killing is
i) expanding forρ> 4(ξβ?(α2?β2)) ii) shrinking forρ< 4(ξβ?(α2?β2)) and iii) is steady forρ= 4(ξβ?(α2?β2))
Takingβ= 0 in (4.9), we getρ=?4α2if and only ifλ= 0. Sinceρis positive,λcannot be zero. Thus we have
Theorem 4.2.A Ricci soliton (g,V,λ) in anα-Sasakian manifold, where V is conformal killing cannot be steady.
Let (Mn,g) be a f-Kenmotsu manifold. Then from (4.2), we have
R.S = S(R(X,Y)Z,W)+ S(Z,R(X,Y)W)
= (?λ+
H.G. Nagaraja; C.R. Premalatha/Progress in Applied Mathematics Vol.3 No.2, 2012 Conversely suppose R.S = 0, i.e S(R(X,Y)Z,W)+ S(Z,R(X,Y)W) = 0.(4.10) Taking f = 1 in (3.6) and (3.7), we get R(X,Y)ξ=η(X)Y?η(Y)X,(4.11) S(X,ξ) =?(n?1)η(X).(4.12) Taking W =ξin (4.10) and using (4.11) and (4.12), we obtain S(Y,Z) =?(n?1)g(Y,Z). Substituting this in (3.1), we get
(LVg)(Y,Z) =ρg(Y,Z)
whereρ= 2((n?1)?λ). i.e V is confirmal killing. Thus we have
Theorem 4.3.Let (g,V,λ) be a Ricci soliton in a Kenmotsu manifold (Mn,g). Then (Mn,g) is Ricci-semi symmetric if and only if V is conformal killing.
REFERENCES
[1]Binh T.Q., Tamassy, L., U.C.De & M.Tarafdar (2002). Some Remarks on Almost Kenmotsu Manifolds. Math. Pannon, 13(1), 31-39.
[2]Constantin Calin & Mircea Crasmareanu (2010). From the Eisenhart Problem to Ricci Solitons in f-Kenmotsu Manifolds. Bull.Malays.Math.Sci.Soc.(2),33(3), 361-368.
[3]Cornelia Livia Bejan & Mircea Crasmareanu(2011).Ricci Solitons in Manifoldswith Quasi-Constant Curvature. Publ. Math. Debrecen, 78/1, 235-243.
[4]De U.C. & Tripathi M.M.(2003). Ricci Tensor in 3-Dimensional Trans-Sasakian Manifolds, Kyungpook Math. J, 43(2), 247-255, MR198228.
[5]De U.C. & Mondal A.K. (2009). On 3-Dimensional Normal Almost Contact Metric Manifolds Satisfying Certain Curvature Conditions. Commun, Korean Math.Soc. 24(2), 265-275.
[6]Kenmotsu K. (1972).A Class of Almost Contact RiemannianManifolds.Tohoku Math. J., 21, 93-103.
[7]Nagaraja H.G. (2010). On N(K)-Mixed Quasi Einstein Manifolds. European Journal of Pure and Applied Mathematics, 3(1), 16-25.
[8]H.G.Nagaraja (2011). Recurrent Trans-Sasakian Manifolds. Mathematicki Vesnik, 63(2), 79-86.
[9]Perelman G. (2002). The Entropy Formula for the Ricci Flow and Its Geometric Applications, arXiv: math.DG/0211159v1.
[10] Sinha B.B. & Ramesh Sharma (1983). On Para-A-Einstein manifolds, Publications De L’Institut Mathematique. Nouvelle Serie, Tome, 34(48), 211-215.
[11] Tripathi M.M. (2008). Ricci Solitons in Contact Metric Manifolds. arXiv:0801.4222v1, [math.DG], 28.
In the Present paper we study Ricci solitons in trans-sasakian manifolds. In particular we consider Ricci solitons in f-Kenmotsu manifolds and we provethe conditionsfor the Ricci solitons to be shrinking, steady and expanding.
Key words
Ricci solitons; f-Kenmotsu; Trans-Sasakian; Shrinking; Steady; Expanding
1.INTRODUCTION
In [10], Ramesh Sharma started the study of the Ricci solitons in contact geometry. Later Mukut Mani Tripathi [11], Cornelia Livia Bejan and Mircea Crasmareanu [3] and others extensively studied Ricci solitons in contact metric manifolds. A Ricci soliton is a generalization of an Einstein metric and is defined on a Riemannian manifold (M,g) by LVg + 2Ric + 2λg = 0,(1.1)
where V is a complete vectorfield on M andλis a constant. The Ricci soliton is said to be shrinking,steady and expandingaccordingasλis negative, zero and positive respectively. If the vectorfield V is the gradient of a potential function f then g is called a gradient Ricci soliton and (1.1) takes the form,
??f = Ric +λg.
Perelman [9] proved that a Ricci soliton on a compact n-manifold is a gradient Ricci soliton. In [11], Ramesh Sharma studied Ricci solitons in K-contact manifolds, where the structurefieldξis killing and he proved that a complete K-contact gradient soliton is compact Einstein and Sasakian. M. M. Tripathi [11] studied Ricci solitons in N(K)-contact metric and (k,μ) manifolds. Motivated by the above studies on Ricci solitons, in this paper, we study Ricci solitons in an important class of manifolds introduced by Kenmotsu in [6].
2.PRELIMINARIES
A (2n+1) dimensional smooth manifold M is said to be an almost contact metric manifold if it admits an almost contact metric structure (φ,ξ,η,g) consisting of a tensorfieldφof type (1,1), a vectorfieldξ, a 1-formηand Riemannian metric g compatible with (φ,ξ,η) satisfyingΦ2=?I +η?ξ,η(ξ) = 1,φξ= 0,η?φ= 0(2.1) and g(φX,φY) = g(X,Y)?η(X)η(Y).(2.2) An almost contact metric manifold is said to be an f-Kenmotsu manifold if(?Xφ)Y = f[g(φX,Y)ξ?φ(X)η(Y)],(2.3) where f∈C∞(M) is strictly positive and df?η= 0 holds. From (2.3) we have?Xξ= f(X?η(X)ξ).(2.4) An almost contact metric manifold is called a trans-Sasakian manifold [4] [8] if(?Xφ)Y =α(g(X,Y)ξ?η(Y)X) +β(g(φX,Y)ξ?η(Y)φX),(2.5) for some smooth functionsαandβon M.
3.RICCI SOLITONS IN F-KENMOTSU MANIFOLDS
Let M be an n dimensional f-Kenmotsu manifold and let (g,V,λ)be a Ricci soliton in M. Let {ei},1≤i≤n be an orthonormal basis of TPM at P∈M. Then from (1.1), we have
S =?(λg +1 2LVg).(3.1) From (2.4), we have(Lξg)(X,Y) = f[g(X,Y)?η(X)η(Y)].(3.2) From (3.1) and (3.2), we have S(X,Y) =?λg(X,Y)?f[g(X,Y)?η(X)η(Y)].(3.3) It is easy to verify from (3.3) that S(φX,Y) =?S(X,φY)(3.4) and
S(ξ,ξ) =?λ.(3.5)
From (2.3) and (2.4), wefind that R(X,Y)ξ= f2[η(X)Y?η(Y)X] + (Y f)φ2X?(Xf)φ2Y(3.6) and S(X,ξ) =?[(n?1)f2+ξf]η(X)?(n?2)X(f).(3.7) From (3.7), we obtain S(ξ,ξ) =?(n?1)[f2+ξf].(3.8)
H.G. Nagaraja; C.R. Premalatha/Progress in Applied Mathematics Vol.3 No.2, 2012 Comparing (3.5) and (3.8), we obtainλ= (n?1)(f2+ξf)(3.9)
From (3.9), it is clear thatλis positive if f is a constant. Thus we have
Ricci soliton in a f-Kenmotsu manifold is expanding, provided f is a constant.
Suppose f is not a constant. If {ei} is an orthonormal basis of TPM at P∈M, then taking X = Y = eiin(3.3) and summing over 1≤i≤n, we get r =?λn?f(n?1),(3.10)
where r is the scalar curvature.
Differentiating (3.10) covariantly w.r.to X, we get Xr=?(n?1)Xf,(3.11) where
Xr=?Xr, Xf=?Xf.
From (3.3), we have QX =?λX?f(φ2X).(3.12) In view of (2.5), differentiation of (3.12) yields
(?YQ)X = Y f(φ2X)?f2η(X)φ2Y + fΦ(X,Y)ξ.
Contracting the above equation with respect to Y, we get(divQ)X = (φ2X) + f2(n?1)η(X).(3.13) Using (3.11) and the identity
(divQ)X =
As it is well knownthat for a Kenmotsumanifildthe curvaturer is negative. Henceλis positive forconstant r. Thus we have,
Theorem 3.3. A Ricci soliton in a Kenmotsu manifold of constant curvature is expanding.
4.RICCI SOLITONS IN 3-DIMENSIONAL TRANS-SASAKIAN MANIFOLDS
Suppose (Mn,g) is a 3-dimensional trans-Sasakian manifold and (g,V,λ) is a Ricci soliton in (Mn,g). If V is a conformal killing vectorfield, then LVg =ρg,(4.1)
2)[η(Y)η(W)?g(Y,W)] = 0. This implies r = 2ξβ?2(α2?β2)(4.8) From (4.4) and (4.8), we have 6λ=ρ?4[ξβ?(α2?β2)].(4.9)
From (4.9), we have
Theorem 4.1.In a 3-dimensionaltrans-Sasakianmanifold, a Ricci Soliton (g,V,λ),where V is conformal killing is
i) expanding forρ> 4(ξβ?(α2?β2)) ii) shrinking forρ< 4(ξβ?(α2?β2)) and iii) is steady forρ= 4(ξβ?(α2?β2))
Takingβ= 0 in (4.9), we getρ=?4α2if and only ifλ= 0. Sinceρis positive,λcannot be zero. Thus we have
Theorem 4.2.A Ricci soliton (g,V,λ) in anα-Sasakian manifold, where V is conformal killing cannot be steady.
Let (Mn,g) be a f-Kenmotsu manifold. Then from (4.2), we have
R.S = S(R(X,Y)Z,W)+ S(Z,R(X,Y)W)
= (?λ+
H.G. Nagaraja; C.R. Premalatha/Progress in Applied Mathematics Vol.3 No.2, 2012 Conversely suppose R.S = 0, i.e S(R(X,Y)Z,W)+ S(Z,R(X,Y)W) = 0.(4.10) Taking f = 1 in (3.6) and (3.7), we get R(X,Y)ξ=η(X)Y?η(Y)X,(4.11) S(X,ξ) =?(n?1)η(X).(4.12) Taking W =ξin (4.10) and using (4.11) and (4.12), we obtain S(Y,Z) =?(n?1)g(Y,Z). Substituting this in (3.1), we get
(LVg)(Y,Z) =ρg(Y,Z)
whereρ= 2((n?1)?λ). i.e V is confirmal killing. Thus we have
Theorem 4.3.Let (g,V,λ) be a Ricci soliton in a Kenmotsu manifold (Mn,g). Then (Mn,g) is Ricci-semi symmetric if and only if V is conformal killing.
REFERENCES
[1]Binh T.Q., Tamassy, L., U.C.De & M.Tarafdar (2002). Some Remarks on Almost Kenmotsu Manifolds. Math. Pannon, 13(1), 31-39.
[2]Constantin Calin & Mircea Crasmareanu (2010). From the Eisenhart Problem to Ricci Solitons in f-Kenmotsu Manifolds. Bull.Malays.Math.Sci.Soc.(2),33(3), 361-368.
[3]Cornelia Livia Bejan & Mircea Crasmareanu(2011).Ricci Solitons in Manifoldswith Quasi-Constant Curvature. Publ. Math. Debrecen, 78/1, 235-243.
[4]De U.C. & Tripathi M.M.(2003). Ricci Tensor in 3-Dimensional Trans-Sasakian Manifolds, Kyungpook Math. J, 43(2), 247-255, MR198228.
[5]De U.C. & Mondal A.K. (2009). On 3-Dimensional Normal Almost Contact Metric Manifolds Satisfying Certain Curvature Conditions. Commun, Korean Math.Soc. 24(2), 265-275.
[6]Kenmotsu K. (1972).A Class of Almost Contact RiemannianManifolds.Tohoku Math. J., 21, 93-103.
[7]Nagaraja H.G. (2010). On N(K)-Mixed Quasi Einstein Manifolds. European Journal of Pure and Applied Mathematics, 3(1), 16-25.
[8]H.G.Nagaraja (2011). Recurrent Trans-Sasakian Manifolds. Mathematicki Vesnik, 63(2), 79-86.
[9]Perelman G. (2002). The Entropy Formula for the Ricci Flow and Its Geometric Applications, arXiv: math.DG/0211159v1.
[10] Sinha B.B. & Ramesh Sharma (1983). On Para-A-Einstein manifolds, Publications De L’Institut Mathematique. Nouvelle Serie, Tome, 34(48), 211-215.
[11] Tripathi M.M. (2008). Ricci Solitons in Contact Metric Manifolds. arXiv:0801.4222v1, [math.DG], 28.