Beitr Algebra Geom DOI 10.1007/s13366-011-0038-6 ORIGINAL PAPER
K -contact metrics as Ricci solitons Amalendu Ghosh · Ramesh Sharma
Received: 18 November 2010 / Accepted: 10 January 2011 © The Managing Editors 2011
Abstract Keywords
We study η-Einstein K -contact manifold whose metric is a Ricci soliton. Ricci soliton · K -contact metric · η-Einstein
Mathematics Subject Classification (2000)
53C25 · 53C44 · 53C21
1 Introduction A Riemannian metric g on a smooth manifold is Einstein if its Ricci tensor S is a constant multiple of g. A Ricci soliton is a generalization of the Einstein metric and is defined on a Riemannian manifold (M, g) by (£V g)(X, Y ) + 2S(X, Y ) + 2λg(X, Y ) = 0
(1)
for some constant λ, a vector field V , and arbitrary vector fields X, Y on M. The Ricci soliton is said to be shrinking, steady, and expanding according as λ is negative, zero, and positive respectively. Compact Ricci solitons are the fixed points of the Ricci flow: ∂t∂ gi j = −2Ri j projected from the space of metrics onto its quotient modulo diffeomorphisms and scalings, and often arise as blow-up limits for the Ricci flow on
A. Ghosh (B) Department of Mathematics, Krishnagar Government College, Krishnanagar, West Bengal, 741101, India e-mail:
[email protected] R. Sharma Department of Mathematics, University of New Haven, West Haven, CT 06516, USA e-mail:
[email protected]
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compact manifolds. If the vector field V is a gradient of a potential function − f , then g is called a gradient Ricci soliton. On a compact Riemannian manifold a Ricci soliton is always a gradient Ricci soliton (see Perelman). For details we refer to Chow et al. (2004). Gradient Ricci solitons have been studied by Sharma (2008) for a K -contact manifold, and by Ghosh et al. (2008) for (κ, μ)-space [a generalization of Sasakian manifold, introduced by Blair, Koufogiorgos and Papantoniou Blair et al. (1995)]. A (2n + 1)-dimensional K -contact metric manifold is said to be η-Einstein if the Ricci tensor can be written as S(X, Y ) = αg(X, Y ) + βη(X )η(Y )
(2)
for some smooth functions α and β. For n > 1, we know [see Yano et al. (1984)] that α and β are constants. Zhang (2009) has shown that a compact Sasakian manifold with constant scalar curvature and quasi-positive holomorphic bisectional transverse curvature is an η-Einstein Sasakian manifold. Recently, Sharma and Ghosh proved that, if a 3-dimensional Sasakian metric is a non-trivial Ricci soliton, then it is homothetic to the standard Sasakian structure on the Heisenberg group nil 3 . It is well known that in dimension 3, a K -contact metric reduces to a Sasakian metric but in higher dimensions, this is not true. Thus, we are motivated to study η-Einstein K -contact metrics as Ricci solitons in higher dimensions. Precisely, we prove Theorem If the metric g of a K -contact η-Einstein manifold M(η, ϕ, ξ, g) is a nontrivial (non-Einstein) Ricci soliton with potential vector field V , then (i) V is Jacobi along the geodesics determined by ξ , (ii) V is a non-strict infinitesimal contact transformation and is equal to − 21 ϕ D f + f ξ for a smooth function f on M, (iii) V preserves the fundamental collineation ϕ, and (iv) the Ricci soliton is expanding. Corollary If the metric g of a complete and simply connected Sasakian space form M(c) (with constant ϕ-sectional curvature c) is a non-trivial Ricci soliton, then, in addition to the conclusions (i) through (iv), c = −3, and M(c) is R 2n+1 (−3) which can be indentified with the (2n + 1)-dimensional Heisenberg group. 2 Preliminaries A (2n +1)-dimensional smooth manifold M is said to be a contact manifold if it carries a global 1-form η such that η ∧ (dη)n = 0 everywhere on M. For a given contact 1-form η there exists a unique vector field ξ such that dη(ξ, X )= 0 and η(ξ ) = 1. Polarizing dη on the contact subbundle η = 0, one obtains a Riemannian metric g and a (1,1)-tensor field ϕ such that dη(X, Y ) = g(X, ϕY ), η(X ) = g(X, ξ ), ϕ 2 = −I + η ⊗ ξ
(3)
g is called an associated metric of η and (ϕ, η, ξ, g) a contact metric structure. We denote the Levi-Civita connection, curvature tensor and Ricci operator of g by ∇ R and Q respectively. Following Blair (2002) we recall two self-adjoint operators h = 21 £ξ ϕ
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and l = R(., ξ )ξ . We know that h and hϕ are trace-free and hϕ = −ϕh. We also have the following formulas for a contact metric manifold: ∇ X ξ = −ϕ X − ϕh X l − ϕlϕ = −2(h 2 + ϕ 2 ) ∇ξ h = ϕ − ϕl − ϕh 2
(4) (5) (6)
T r.l = S(ξ, ξ ).
(7)
A contact metric structure is said to be K -contact if ξ is Killing with respect to g, equivalently, h = 0, or T r.l = 2n. The contact structure on M is said to be normal if the almost complex structure on M × R defined by J (X, f d/dt) = (ϕ X − f ξ, η(X )d/dt), where f is a real function on M × R, is integrable. A normal contact metric manifold is called a Sasakian manifold. Sasakian manifolds are K -contact and 3-dimensional K -contact manifolds are Sasakian. For a K -contact manifold, ∇ X ξ = −ϕ X
(8)
R(X, ξ )ξ = X − η(X )ξ Qξ = 2nξ
(9) (10)
For details we refer to Blair (2002). A plane section of the tangent space T p M 2n+1 to a Sasakian manifold M 2n+1 is called a ϕ-section if there exists a vector X ∈ T p M 2n+1 orthogonal to ξ such that {X, ϕ X } span the section. The sectional curvature K (X, ϕ X ) of M is called ϕ-sectional curvature. If this sectional curvature is independent of the choice of X at any point, then it is constant (say, c) on M and then M is called a Sasakian space form (denoted M(c)) which is η-Einstein with and β = (n+1)(1−c) . In particular, a complete simply connected α = n(c+3)+c−1 2 2 2n+1 (−3) which has the standard normal contact metric strucM(c) with c = −3, is R n n y i d x i ), ξ = 2∂/∂z and g = η⊗η+ 41 i=1 ((d x i )2 +(dy i )2 ), ture: η = 21 (dz − i=1 and can be identified with a (2n + 1)-dimensional Heisenberg group having g as a left invariant metric. For details we refer to Blair (2002) and also Tanno (1996). 3 Proof of the theorem Proof of the theorem Since M is η-Einstein, Eq. 2 shows that the scalar curvature r is constant and r = (2n + 1)α + β
(11)
Now using (2), Eq. 1 takes the form (£V g)(Y, Z ) = −2(λ + α)g(Y, Z ) − 2βη(Y )η(Z )
(12)
Differentiating (12) along an arbitrary vector field X and using (4) provides (∇ X £V g)(Y, Z ) = 2β{g(Y, ϕ X )η(Z ) + g(Z , ϕ X )η(Y )}
(13)
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Making use of the commutation formula [see Yano (1970), p. 22]: (£V ∇ X g − ∇ X £V g − ∇[V,X ] g)(Y, Z ) = −g((£V ∇)(X, Y ), Z ) − g((£V ∇)(X, Z ), Y ) we obtain (∇ X £V g)(Y, Z ) = g((£V ∇)(X, Y ), Z ) + g((£V ∇)(X, Z ), Y )
(14)
Now, use of equation (13) in (14) and a straightforward combinatorial computation shows (£V ∇)(Y, Z ) = 2β{η(Z )ϕY + η(Y )ϕ Z }
(15)
Substituting Y = Z = ξ in the above equation we have (£V ∇)(ξ, ξ ) = 0. Using this and that ξ is geodesic [in view of Eq. 8] in the identity [Duggal and Sharma (1999), p.39]: (£V ∇)(X, Y ) = ∇ X ∇Y V − ∇∇ X Y V + R(V, X )Y provides ∇ξ ∇ξ V + R(V, ξ )ξ = 0, i.e. V is Jacobi along ξ , which proves part (i). Next, differentiating Eq. 15 along an arbitrary vector field X and using (4) we get (∇ X £V ∇)(Y, Z ) = 2β{−g(Z , ϕ X )ϕY − g(Y, ϕ X )ϕ Z + η(Z )(∇ X ϕ)Y + η(Y )(∇ X ϕ)Z }
(16)
Making use of Eq. 16 and the identity: (£V R)(X, Y )Z = (∇ X £V ∇)(Y, Z ) − (∇Y £V ∇)(X, Z ) one obtains (£V R)(X, Y )Z = 2β[−g(Z , ϕ X )ϕY + g(Z , ϕY )ϕ X + 2g(X, ϕY )ϕ Z + η(Z ){(∇ X ϕ)Y − (∇Y ϕ)X }+η(Y )(∇ X ϕ)Z −η(X )(∇Y ϕ)Z ] (17) Setting Y = Z = ξ in (17) shows that (£V R)(X, ξ )ξ = 4β{η(X )ξ − X }
(18)
Next, Lie differentiating (9) along V and recalling (18) and (12) we find that 4β{η(X )ξ − X } + R(X, £V ξ )ξ + R(X, ξ )£V ξ = −η(X )£V ξ + 2(λ + α + β)η(X )ξ − g(£V ξ, X )ξ.
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(19)
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Contracting (19) over X , recalling (10) and g(£V ξ, ξ ) = (λ + α + β) (follows from (12) by taking Y = Z = ξ ) provides α−β +λ=0
(20)
We now recall the Ricci soliton integrability condition [see p. 6 of Chow et al. (2004) and p. 139 of Sharma (2008)] £V r = − r + 2λr + 2|S|2
(21)
where r = −div.Dr . Computing the value of |S|2 from Eq. 2 and using (10) and (11) we find that β(α + 2) = 0. Since β = 0, otherwise M will be Einstein, we find that α = −2 and β = 2(n + 1). Thus, it follows that λ = 2(n + 2) > 0, i.e. the Ricci soliton is expanding, which proves part (iv). Contracting Eq. 17 at X , and using the formula: (divϕ)X = −2nη(X ) for a contact metric one gets (£V S)(Y, Z ) = 4β[g(Y, Z ) − (2n + 1)η(Y )η(Z )] Next, taking the Lie-derivative of (2) along V and making use of (12) provides (£V S)(Y, Z ) = −2(α 2 + αλ)g(Y, Z ) + β[(£V η)(Y )η(Z ) +η(Y )(£V η)Z ] − 2αβη(Y )η(Z ) Comparing the last two equations, substituting Z = ξ , and using the values of α, β and λ obtained earlier, we obtain £V η = −4(n +1)η, i.e. V is a non-strict infinitesimal contact transformation, which proves part (ii). Also, by a straightforward calculation, we find that £V ξ = 4(n + 1)ξ . Now, Lie-differentiating dη(X, Y ) = g(X, ϕY ) along V (noting that Liedifferentiation commutes with d), and using the equation £V η = − 4(n + 1)η (obtained earlier) we conclude that £V ϕ = 0, which proves part (iii). Furthermore, as V is an infinitesimal contact transformation, it follows that [see Blair (2002)] V = − 21 ϕ D f + f ξ for a smooth function f on M. This completes the proof. Proof of the Corollary A Sasakian space form M(c) is K -contact and η-Einstein, and hence all the conclusions of Theorem 1 are valid. As α = n(c+3)+c−1 and β = 2 (n+1)(1−c) for M(c), comparing these with the values of α and β found in the proof 2 2n+1 of Theorem 1, we find that c = −3. Hence M is R (−3) identifiable with the (2n + 1)-dimensional Heisenberg group. This proves the corollary. Acknowledgments The authors are thankful to the referee for numerous suggestions for the improvement of this paper. A.G. has been supported by the University Grants Commission (India) under the scheme: Minor Research Project in Science, Sanction No. F.PSW-106/09-10, Dated 8th Oct. 2009. R.S. has been supported by University Research Scholarship of the University of New Haven.
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