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HALF CONFORMALLY FLAT GRADIENT RICCI ALMOST SOLITONS M. BROZOS-VA´ZQUEZ,E. GARCÍA-RÍO,X.VALLE-REGUEIRO Abstract. The local structure of half conformally flat gradient Ricci 6 almostsolitonsisinvestigated,showingthattheyarelocallyconformally 1 0 flat in a neighborhood of any point where the gradient of the potential 2 functionisnon-null. Inopposition,ifthegradientofthepotentialfunc- tion is null, thenthesoliton isa steady traceless κ-Einstein soliton and n is realized on the cotangent bundleof an affinesurface. a J 9 1 1. Introduction ] G A triple (M,g,f), where (M,g) is a pseudo-Riemannian manifold and D f is a smooth function on M, is said to be a gradient Ricci soliton if the . following equation is satisfied: h t (1) Hes +ρ= λg a f m for some λ R, where ρ denotes the Ricci tensor and Hes denotes the f ∈ [ Hessian of f. 1 Thespecial significance of gradient Ricci solitons comes from the analysis v of the fixed points of the Ricci flow: ∂ g(t) = 2ρ(t). Ricci flat metrics 7 are genuine fixed points of the flow. M∂otreover, −if (M,g ) is Einstein with 3 0 ρ = τ0 g ,theng(t) =(1 2 τ0 )g ,whichshowsthatg isafixedpoint 8 0 dimM 0 − dimM 0 0 4 of the Ricci flow modulo homotheties. Generalizing this property, gradient 0 Ricci solitons correspond to self-similar solutions of the Ricci flow. They . 1 are ancient solutions in the shrinking case (λ > 0), eternal solutions in the 0 steady case (λ = 0), and immortal solutions in the expanding case (λ < 0). 6 Recently, a generalization of Equation (1) has been considered in [19], 1 : allowing λ to be a smooth function on M. Thus (M,g,f) is said to be a v gradientRiccialmost soliton ifEquation(1)issatisfiedforsomeλ ∞(M). i X ∈ C Since gradient Ricci almost solitons contain gradient Ricci solitons as a r particular case, we say that the gradient Ricci almost soliton is proper if a the function λ is non-constant. It is important to emphasize that, beyond being a generalization of Ricci solitons, some proper gradient Ricci almost solitons correspond to self- similar solutions of some geometric flows. The Ricci-Bourguignon flow is given by the equation ∂ g(t) = 2(ρ(t) κτ(t)g(t)), ∂t − − 2010 Mathematics Subject Classification. 53C21, 53B30, 53C24, 53C44. Key words and phrases. Gradient Riccialmost soliton, warped product,Walkermani- fold, half conformally flat. Supported by projects GRC2013-045, MTM2013-41335-P and EM2014/009 with FEDER funds (Spain). 1 2 BROZOS-VA´ZQUEZ,GARCÍA-RÍO,VALLE-REGUEIRO where κ R and τ denotes the scalar curvature. This flow can be seen ∈ as an interpolation between the Ricci flow and the Yamabe flow, which corresponds to the equation ∂ g(t) = τ(t)g(t). We refer to [10] for a ∂t − broad exposition on the Ricci-Bourguignon flow. Following [10], the self- similar solutions of this flow are called κ-Einstein solitons and correspond to the equation Hes +ρ= (κτ +µ)g f for some κ,µ R. Hence, they are a special family of gradient Ricci almost ∈ solitons with soliton function λ = κτ +µ (see [11, 12] for more information on κ-Einstein solitons). The steady case (µ = 0) will play a relevant role in our study. Gradient Ricci almost solitons exhibit some similarities but also some striking differences when comparing with usual Ricci solitons. For example, no positive-definite Ka¨hler manifold admits proper gradient Ricci almost solitons [17]incontrastwiththegradientRiccisolitoncase. Propergradient Riccialmostsolitonsareirreducibleand,moreover,theyareofconstantnon- zero curvature in the homogeneous Riemannian setting [8]. Classifying gradient Ricci almost solitons under curvature conditions is a natural problem which certainly may help in understanding the corresponding flows. Among the different curvature conditions, local conformal flatness is the most natural one, since the Ricci tensor completely deter- mines the curvature in that situation. The analysis carried out in [12] gives a classification of locally conformally flat gradient κ-Einstein solitons. The purpose of this paper is to analyze half conformally flat (i.e. self- dual or anti-self-dual) four-dimensional gradient Ricci almost solitons. The main tasks are to show the existence of proper examples and to describe their underline structure. The results are obtained as a consequence of the almost soliton equation (1), where geometric information of different nature is encoded: on the one hand information related to the curvature of (M,g) is given through the Ricci tensor; on the other hand the level hypersurfaces of the potential function f are involved by means of their second fundamental form. As a result, two cases, which are different in nature, areconsider separately. Thefirstonecorrespondstonon-degenerate level hypersurfaces ( f = 0), whereas the second one corresponds to k∇ k 6 degenerate level hypersurfaces ( f = 0) and gives rise to the socalled k∇ k isotropic solitons. Our main result shows that any half conformally flat four-dimensional gradient Ricci almost soliton is locally conformally flat if f = 0. How- k∇ k 6 ever, the isotropic case ( f = 0) allows the existence of strictly half k∇ k conformally flat proper gradient Ricci almost solitons, i.e. examples which are self-dual but not locally conformally flat. All of them are realized as the cotangent bundle of an affine surface equipped with a modified Riemannian extension. Moreover, theyaresteadytraceless κ-Einsteinsolitons, i.e. µ = 0 and κ= 1. 4 Theorem 1.1. Let (M,g,f) be a four-dimensional half conformally flat proper gradient Ricci almost soliton. HALF CONFORMALLY FLAT GRADIENT RICCI ALMOST SOLITONS 3 (1) If f = 0, then (M,g) is locally isometric to a warped product of kth∇e fkor6m I N, where I R and N is of constant sectional ϕ × ⊂ curvature. Furthermore, (M,g) is locally conformally flat. (2) If f = 0, then (M,g) is locally isometric to the cotangent bundle k∇ k T∗Σof anaffine surface (Σ,D)equipped with amodified Riemannian extension of the form g = ιT ιId+g +π∗Φ, where T is a (1,1)- D ◦ tensor field and Φ is a symmetric (0,2)-tensor field on Σ. The potential function satisfies f = π∗fˆfor some smooth function fôn Σ and is related with the soliton function λ by λ = 3Ce−f for 2 a constant C. Moreover, T is given by T = Ce−fÎd and Φ is given by Φ = 2efˆ(HesD+2ρD ). C fˆ sym The paper is organized as follows. Some basic consequences of the gradient Ricci almost soliton equation are discussed in 2.1. Half conformal § flatness is discussed at the purely algebraic level in 2.2. Walker metrics § and modifiedRiemannian extensions are introduced in 2.3 andused in Sec- § tion3toproveTheorem1.1. Theanalysisofthenon-isotropiccaseiscarried out in 3.1 to prove Theorem 1.1-(1), while the isotropic case is treated in § 3.2 to prove Theorem 1.1-(2). § 2. Preliminaries Let (M,g) be a pseudo-Riemannian manifold with Levi-Civita connection and curvature tensor (X,Y) = [ , ]. Let ρ and τ [X,Y] X Y ∇ R ∇ − ∇ ∇ denotetheRicci tensor andthescalar curvaturegiven by ρ(X,Y) = tr Z { 7→ (X,Z)Y and τ = tr Ric respectively, where Ric denotes the Ricci oper- R } { } ator defined by g(RicX,Y) = ρ(X,Y). Let Hes denote the Hessian tensor f defined by Hes (X,Y) = ( df)(Y) = XY(f) ( Y)(f). We begin by f X X ∇ − ∇ exploring the first consequences of the almost soliton equation (1). 2.1. Gradient Ricci almost solitons: consequences of the equation. Let (M,g,f) be a gradient Ricci almost soliton defined by Equation (1), whereλ is a function on M. Tracing and taking divergences on (1), one gets the following relations which extend well-known identities from the soliton case. We refer to [1, 2, 3, 19] for a detailed exposition. Lemma 2.1. Let (M,g,f) be a gradient Ricci almost soliton. Then (1) ∆f +τ = nλ , (2) ∆f + τ = λn, ∇ ∇ ∇ (3) ∆f +Ric( f)+ 1 τ = λ, ∇ ∇ 2∇ ∇ (4) τ = 2Ric( f)+2(n 1) λ, ∇ ∇ − ∇ (5) (X,Y,Z, f) = dλ(X)g(Y,Z) dλ(Y)g(X,Z) R ∇ − ( ρ)(Y,Z)+( ρ)(X,Z). X Y − ∇ ∇ Let be the Weyl conformal curvature tensor W (X,Y,Z,T) = (X,Y,Z,T) W R + τ g(X,Z)g(Y,T) g(X,T)g(Y,Z) (n−1)(n−2){ − } + 1 ρ(X,T)g(Y,Z) ρ(X,Z)g(Y,T) (n−2){ − +ρ(Y,Z)g(X,T) ρ(Y,T)g(X,Z) , − } 4 BROZOS-VA´ZQUEZ,GARCÍA-RÍO,VALLE-REGUEIRO and let C denote the Cotton tensor C(X,Y,Z) = ( ρ)(Y,Z) ( ρ)(X,Z) X Y ∇ − ∇ 1 dτ(X)g(Y,Z) dτ(Y)g(X,Z) . −2n 2{ − } − Now, an immediate application of Lemma 2.1– (4) and Lemma 2.1–(5) gives the following expression for the Weyl conformal tensor. Lemma 2.2. Let (M,g,f) be a gradient Ricci almost soliton. Then (X,Y,Z, f) = C(X,Y,Z) W ∇ − + τ g(X,Z)g(Y, f) g(X, f)g(Y,Z) (n−1)(n−2){ ∇ − ∇ } + 1 ρ(Y,Z)g(X, f) ρ(X,Z)g(Y, f) (n−2){ ∇ − ∇ } + 1 ρ(X, f)g(Y,Z) ρ(Y, f)g(X,Z) . (n−1)(n−2){ ∇ − ∇ } 2.2. Half conformally flat structure of four-dimensional manifolds. We work at the purely algebraic level. Let (V, , ) be an inner-product h· ·i vector space and let , be the induced inner product on the space ≪ · · ≫ of two-forms Λ2(V). For a given orientation vol , the Hodge star operator V : Λ2(V) Λ2(V) given by α β = α,β vol satisfies 2 = Id and V ∗ → ∧∗ ≪ ≫ ∗ induces a decomposition Λ2(V)= Λ2 Λ2, where Λ2 = α Λ2 : α = α +⊕ − + { ∈ ∗ } and Λ2 = α Λ2 : α = α . Λ2 is the space of self-dual and Λ2 is − { ∈ ∗ − } + − the space of anti-self-dual two-forms. Moreover, a change of orientation on (V, , ) reverses the roles of Λ2 and Λ2. h· ·i + − Any algebraic curvature tensor on (V, , ) (i.e., a (0,4)-tensor on V R h· ·i satisfying the symmetries of the curvature tensor) can be naturally considered as an endomorphism :Λ2(V) Λ2(V). Let :Λ2(V) Λ2(V) be R → W → the corresponding endomorphism associated to the Weyl conformal tensor. decomposes under the action of SO(V, , ) as = , where + − W h· ·i W W ⊕W = 1( + )istheself-dualand = 1( )istheanti-self-dual W+ 2 W ∗W W− 2 W−∗W Weyl curvature tensor. An algebraic curvature tensor is said to be conformally flat if = 0 R W and issaidtobehalf conformally flat ifitiseitherself-dual (i.e., = 0) − R W or anti-self-dual (i.e., = 0). + W The following algebraic characterization of self-dual algebraic curvature tensors will be used in the proof of Theorem 1.1. Lemma 2.3. Let (V, , ) be an oriented four-dimensional inner product h· ·i space of neutral signature. (i) An algebraic curvature tensor is self-dual if and only if for any R positively oriented orthonormal basis e ,e ,e ,e 1 2 3 4 { } (e ,e ,x,y) = σ ǫ ǫ (e ,e ,x,y), for any x,y V, 1 i ijk j k j k W W ∈ for i,j,k 2,3,4 and where σ denotes the signature of the cor- ijk ∈ { } responding permutation. (ii) An algebraic curvature tensor is self-dual if and only if for a pos- R itively oriented pseudo-orthonormal basis t,u,v,w (i.e., the non- { } zero inner products are t,v = u,w = 1) and for every x,y V, h i h i ∈ (t,v,x,y) = (u,w,x,y), (t,w,x,y) = 0, (u,v,x,y) = 0. W W W W HALF CONFORMALLY FLAT GRADIENT RICCI ALMOST SOLITONS 5 Proof. Let e ,e ,e ,e be an orthonormal basis of (V, , ) such that 1 2 3 4 { } h· ·i vol = e1 e2 e3 e4. Then V ∧ ∧ ∧ Λ2 = span e1 e2 ǫ ǫ e3 e4,e1 e3 ǫ ǫ e2 e4,e1 e4 ǫ ǫ e2 e3 , ± { ∧ ± 3 4 ∧ ∧ ∓ 2 4 ∧ ∧ ± 2 3 ∧ } where ǫ = e ,e . Now, = 0 if and only if i i i − h i W (e1 e2 ǫ ǫ e3 e4) = 0, (e1 e3+ǫ ǫ e2 e4)= 0, 3 4 2 4 W ∧ − ∧ W ∧ ∧ (e1 e4 ǫ ǫ e2 e3) = 0, 2 3 W ∧ − ∧ so (e ,e ) = σ ǫ ǫ (e ,e ) for i,j,k 2,3,4 and Assertion (i) fol- 1 i ijk j k j k W W ∈ { } lows. Now, if t,u,v,w is a positively oriented pseudo-orthonormal basis such { } that the only non-zero inner products are given by t,v = u,w = 1, then h i h i 1 1 1 1 e = (t v), e = (t+v), e = (w u), e = (w+u), 1 2 3 4 √2 − √2 √2 − √2 is a positively oriented orthonormal basis and Assertion (ii) follows from the previous case. (cid:3) Nowwemovetothedifferentiablesetting. Let(M,g)beafour-dimensional pseudo-Riemannian manifold. (M,g) is said to behalf conformally flat if its curvaturetensorishalfconformallyflatateachpoint. If(M,g)isLorentzian then it is half conformally flat if and only if it is locally conformally flat. Hencewerestrictourattention totheRiemannianandtheneutralsignature cases. 2.3. Modified Riemannian extensions. AWalker manifold is a pseudo- Riemannian manifold (M,g) which has a null parallel distribution D, i.e., the restriction of the metric tensor to D is totally degenerate and D is invariant by parallel transport ( D D). ∇ ⊂ The existence of a null 2-plane induces an orientation on each tangent space T M as follows. For any basis u ,u of D , all basis u ,u ,v ,v p 1 2 p 1 2 1 2 { } { } of T M determined by u ,v = δ induce the same orientation on T M p i j ij p h i [14]. Thus, in Walker coordinates (x1,x2,x1′,x2′) where the metric tensor is written as g = 2dxi dxi′ +aijdxi dxj, ◦ ◦ for aij functions on M [21], the two-form dx1′ dx2′ in the null parallel ∧ distribution is self-dual. This fixes the orientation of the manifold and we take this tobetheorientation ofany Walker manifoldthroughoutthiswork. Inthis paperwe are interested in a particular family of Walker manifolds: modifiedRiemannian extensions. Inorderto introducethesemanifoldsnext webrieflyreviewsomepropertiesofcotangentbundles(see[6]andreferences therein). Let Σ bea manifold and T∗Σ its cotangent bundle. We express any point p¯ T∗Σ as a pair p¯= (p,ω), with ω a one-form on T Σ. Thus π : T∗Σ Σ p ∈ → defined by π(p,ω) = p is the natural projection. For any vector field X on Σ, we define ιX, the evaluation map of X, to be the smooth function on T∗Σ given by ιX(p,ω) = ω(X ). Vector fields on T∗Σ are characterized by p their action on evaluation maps. For a vector field X on Σ, its complete lift XC is the vector field on T∗Σ defined by XC(ιZ) = ι[X,Z], for all vector 6 BROZOS-VA´ZQUEZ,GARCÍA-RÍO,VALLE-REGUEIRO fields Z on Σ. Tensor fields of type (0,s) on T∗Σ are also determined by their action on complete lifts of vector fields on Σ. Let (Σ,D) be a torsion-free affine manifold. Define the Riemannian extension of (Σ,D) to be the neutral signature metric g defined on T∗Σ D and determined by g (XC,YC) = ι(D Y +D X). In many situations D X Y − it is useful to consider a deformation of the Riemannian extension given by g = g +π∗Φ, where Φ is a symmetric (0,2)-tensor field on Σ. D,Φ D For any (1,1)-tensor field T on Σ, its evaluation ιT is a one-form on T∗Σ characterized by ιT(XC) = ι(T(X)). If T and S are (1,1)-tensors on Σ and Φ is a symmetric (0,2)-tensor field on Σ, then the modified Riemannian extension is the metric on T∗Σ given by (2) g = ιT ιS +g +π∗Φ. D,Φ,T,S D ◦ Local coordinates (x1,...,xn) on Σ induce local coordinates (x1,...,xn, x1′,...,xn′) on T∗Σ so that gD,Φ,T,S expresses as: (3) 1 gD,Φ,T,S = 2dxi◦dxi′+(cid:26)2xr′xs′(TirSjs+TjrSis)−2xk′DΓkij +Φij(cid:27)dxi◦dxj, where DΓk are the Christoffel symbols of the affine connection D, Φ = ij Φijdxi ⊗dxj, T = Tirdxi⊗∂xr and S = Sjsdxj ⊗∂xs. ModifiedRiemannianextensionshavebeenusedin[9]todescribeself-dual Walker manifolds as follows. Theorem 2.4. A four-dimensional Walker metric is self-dual if and only if it is locally isometric to the cotangent bundle T∗Σ of an affine surface (Σ,D), equipped with a metric tensor of the form g = ιX(ιId ιId)+ιT ιId+g +π∗Φ D ◦ ◦ where X, T, D and Φ are a vector field, a (1,1)-tensor field, a torsion-free affine connection and a symmetric (0,2)-tensor field on Σ, respectively. 3. Half conformally flat gradient Ricci almost solitons We analyze separately the isotropic and the non-isotropic cases to prove Theorem 1.1. 3.1. Non-isotropic case. First we will consider non-isotropic half conformally flat gradient Ricci almost solitons (i.e., f = 0). The fact that k∇ k 6 the level hypersurfaces of the potential function are non-degenerate hypersurfaces will be crucial to show that such a soliton is necessarily locally conformally flat. SinceanyRiemanniangradientRicci almostsoliton isnon- isotropic, the following result shows that any Riemannian half conformally flat gradient Ricci almost soliton is locally conformally flat. Theorem 3.1. Let (M,g,f) be a non-isotropic four-dimensional half conformally flat gradient Ricci almost soliton. Then (M,g) is locally isometric to a warped product of the form I N, where I R and N is of constant ϕ × ⊂ sectional curvature. Furthermore, (M,g) is locally conformally flat. HALF CONFORMALLY FLAT GRADIENT RICCI ALMOST SOLITONS 7 Proof. Since we work at the local level, let p M and orient (M,g) on ∈ a neighborhood of p so that it is self-dual. Now, since the Cotton tensor is a constant multiple of the divergence of the Weyl tensor: C(x,y,z) = 2(divW)(x,y,z), we use Lemma 2.2 to express the self-duality condition − (e ,e ,x,y) = σ ǫ ǫ (e ,e ,x,y) of Lemma 2.3-(i) applied to f as 1 i ijk j k j k W W ∇ τ g(e , f)e g(e , f)e ρ(e , f)e ρ(e , f)e i 1 1 i i 1 1 i { ∇ − ∇ }−{ ∇ − ∇ +3g(e , f)Ric(e ) 3g(e , f)Ric(e ) i 1 1 i ∇ − ∇ } (4) = σ ǫ ǫ τ g(e , f)e g(e , f)e ijk j k k j j k { ∇ − ∇ } (cid:0) ρ(e , f)e ρ(e , f)e k j j k −{ ∇ − ∇ +3g(e , f)Ric(e ) 3g(e , f)Ric(e ) k j j k ∇ − ∇ } (cid:1) for i,j,k 2,3,4 . ∈ { } Since the gradient Ricci almost soliton is non-isotropic, normalize f to ∇ be unit and complete it to an orthonormal frame E := ∇f ,E ,E ,E . 1 k∇fk 2 3 4 n o Let i,j,k 2,3,4 henceforth. Equation (4) gives rise to ∈ { } τg(E , f)g(E ,Z)+3ρ(E ,Z)g(E , f) 1 i i 1 − ∇ ∇ (5) +ρ(E , f)g(E ,Z) ρ(E , f)g(E ,Z) 1 i i 1 ∇ − ∇ = σ ǫ ǫ ρ(E , f)g(E ,Z) ρ(E , f)g(E ,Z) . ijk j k j k k j { ∇ − ∇ } Now, put Z = E in (5) to get ρ(E ,E ) = 0 for all i 2,3,4 , which 1 1 i ∈ { } shows that f is an eigenvector of the Ricci operator. Next, set Z = E in j ∇ (5) and use this fact to obtain 3ρ(E ,E )g(E , f)= σ ǫ ǫ ρ(E , f)g(E ,E )= 0, i j 1 ijk j k k j j ∇ − ∇ fromwhereitfollows thatρ(E ,E ) = 0foralli = j. Finally, settingZ = E i j i 6 in (5) one gets τg(E , f)g(E ,E ) 3ρ(E ,E )g(E , f) ρ(E , f)g(E ,E ) = 0, 1 i i i i 1 1 i i ∇ − ∇ − ∇ and thus 3ǫ ρ(E ,E )= τ ǫ ρ(E ,E ). i i i 1 1 1 − Hence the Ricci operator Ric diagonalizes on the basis E ,...,E and, 1 4 { } moreover, it has at most two distinct eigenvalues, one of multiplicity one corresponding to the eigenvector E . 1 Now, the Ricci almost soliton equation (1) shows that τ ǫ ρ(E ,E ) 1 1 1 Hes (E ,E ) = λg(E ,E ) ρ(E ,E )= λ − g(E ,E ), f i i i i i i i i − (cid:18) − 3 (cid:19) and thus the level hypersurfaces of f are totally umbilical. Since the one- dimensional distribution span E is totally geodesic one has that (M,g) 1 { } decomposes locally as a twisted product I N (see [20]). Since the Ricci ϕ × tensor is diagonal (in particular ρ(E ,E ) = 0) one has that the twisted 1 i product reduces to a warped product [16]. Finally, since I N is self- ϕ × dual, it is necessarily locally conformally flat and the fiber N is of constant sectional curvature (see [7]). (cid:3) Remark 3.2. Let I N = (I N,ǫdt2 g ) be a warped product as in ϕ N × × ⊕ Theorem 3.1, where ǫ = 1 and (N,g ) has constant sectional curvature N ± c . Since f, λ and ϕ are function of t, the almost Ricci soliton equations N 8 BROZOS-VA´ZQUEZ,GARCÍA-RÍO,VALLE-REGUEIRO (1) are obtained by direct computation (see formulas for warped products in [18]): f′′(t) ǫλ(t) 3ϕ′′(t) = 0, − − ϕ(t) ϕ(t)(f′(t)ϕ′(t) ϕ′′(t)) ǫλ(t)ϕ(t)2 2ϕ′(t)2 +2c ǫ = 0. N − − − From thefirstequation λ(t) = ǫf′′(t) 3ǫϕ′′(t) andsubstitutinginthesecond − ϕ(t) equation one gets: ϕ(t)2f′′(t)+ϕ(t)ϕ′(t)f′(t)+2ϕ(t)ϕ′′(t) 2ϕ′(t)2 +2c ǫ = 0. N − − NotethatthisisalinearODEforthepotentialfunctionf,soforanywarped productI N as abovethereexistglobally definedsolutions for f andthus ϕ × one can construct, in general, proper gradient Ricci almost solitons. Locally conformally flat Riemannian gradient Ricci almost solitons were studiedin[13],whereasexamplesofgradientRiccialmostsolitonsonwarped product manifolds of the form I N, where (N,g ) is Einstein have been ϕ N × constructed in [19, Example 2.5]. 3.2. Isotropic case. In contrast with the non-isotropic case, the level hypersurfaces of the potential function are now degenerate hypersurfaces. Lemma 3.3. Let (M,g,f) be a four-dimensional isotropic gradient Ricci almost soliton. If (M,g) is half conformally flat, then (M,g) is locally a Walker manifold and λ = 1τ. 4 Proof. We firstly determine the structure of the Ricci operator. Since f = ∇ 6 0 but g( f, f) = 0, complete it to a local pseudo-orthonormal frame ∇ ∇ = f,U,V,W , i.e. the only non-zero components of g are g( f,V) = B {∇ } ∇ g(U,W) = 1. Sinceg( f, f)= 0,onehasthat0 = g( f, f)= 2g( f, f)= X X ∇ ∇ ∇ ∇ ∇ ∇ ∇ ∇ 2g( f,X), and thus hes ( f) = 0, where hes is the Hessian opera- ∇f f f ∇ ∇ ∇ tor defined by g(hes (X),Y) = Hes (X,Y). Hence, it follows from the f f almost soliton equation (1) that f is an eigenvector of the Ricci operator: ∇ Ric( f) =λ f. ∇ ∇ Now we use the self-duality characterization given in Lemma 2.3-(ii), which with respect to the pseudo-orthonormal frame above reads as ( f,V,X,Y) = (U,W,X,Y), W ∇ W ( f,W,X,Y)= 0, (U,V,X,Y) = 0, W ∇ W for any vector fields X and Y. Setting Y = f in the first equation above ∇ and using Lemma 2.2 together with Ric( f)= λ f one has ∇ ∇ 1 0= ( f,V,X, f) (U,W,X, f) = (τ 4λ)g( f,X) W ∇ ∇ −W ∇ 6 − ∇ for any vector field X. Hence τ = 4λ. Now, setting Y = f in the third ∇ equation above: (U,V,X,Y) = 0, it follows from Lemma 2.2 that W 1 1 1 0 = (U,V,X, f) = (τ λ)g(U,X) ρ(U,X) = (λg(U,X) ρ(U,X)) W ∇ 6 − −2 2 − for all X. This shows that U is also an eigenvector of the Ricci operator associated to the eigenvalue λ: Ric(U) = λU. HALF CONFORMALLY FLAT GRADIENT RICCI ALMOST SOLITONS 9 Finally, setting X = V in the second equation above: ( f,W,X,Y) = W ∇ 0, and using Lemma 2.2 again, one has 1 0= (V,Y,W, f) = λg(Y,W) ρ(Y,W)+ρ(V,W)g(Y, f) W ∇ −2{ − ∇ } for all Y. Hence it follows that ρ(W,W) = ρ( f,W)=0. ∇ In summary, the Ricci operator expresses in the basis = f,U,V,W B {∇ } as λ 0 α β  0 λ β 0  Ric = , 0 0 λ 0    0 0 0 λ    where α and β are functions on M. Now set D= span f,U , which is a two-dimensional null distribution. {∇ } We already showed that g( f, f) = 0. Moreover, a similar argument X ∇ ∇ ∇ using that g(U,U) = 0 gives g( U,U) = 0 for all X. On the other hand, X ∇ since Ric(U) = λU, it follows from Equation (1) that hes (U) = 0. Now f g(U, f) = 0 shows that ∇ g( U, f)= g(U, f)= Hes (U,X) = 0, X X f ∇ ∇ − ∇ ∇ − for all X. Hence, since D⊥ = D and g( f, f)= 0, g( U,U) = 0, g( U, f)= 0 and g(U, f)= 0 X X X X ∇ ∇ ∇ ∇ ∇ ∇ ∇ ∇ we have that D D and (M,g) is locally a Walker manifold. (cid:3) ∇ ⊂ Thechoice of orientation did not play any role in our previous discussion. However, foraWalker manifoldtheorientation is inducedbytheorientation of the null two-dimensional distribution D. The next two lemmas show that theself-dualandtheanti-self-dual conditions arenotinterchangeable in this context. Lemma3.4. Let(M,g)beafour-dimensional Walkermanifold. If(M,g,f) is an anti-self-dual isotropic gradient Ricci almost soliton then (M,g,f) is a steady gradient Ricci soliton. Proof. It was observed in [15] that if the self-dual Weyl curvature W+ of a Walker manifold vanishes, then the scalar curvature is identically zero, and hence Lemma 3.3 shows that λ = 0. (cid:3) Lemma3.5. Let(M,g)beafour-dimensional Walkermanifold. If(M,g,f) is a self-dual isotropic gradient Ricci almost soliton, then (M,g) is locally isometric to the cotangent bundle T∗Σ of an affine surface (Σ,D) equipped with a modified Riemannian extension of the form g = ιT ιId+g + D,Φ,T,Id D ◦ π∗Φ. Moreover: (1) The potential function satisfies f = π∗fˆfor some smooth function fˆ on Σ and f is related with the soliton function by λ = 3Ce−f for a 2 constant C. (2) The (1,1)-tensor field T is given by T = Ce−fÎd and the (0,2) symmetric tensor field Φ is given by Φ = 2efˆ(HesD+2ρD ). C fˆ sym 10 BROZOS-VA´ZQUEZ,GARCÍA-RÍO,VALLE-REGUEIRO Proof. It was shown in [9] that a four dimensional Walker manifold is self- dual if and only if it is locally isometric to the cotangent bundle T∗Σ of an affine surface (Σ,D), with metric tensor (6) g = ιX(ιId ιId)+ιT ιId+g +π∗Φ D ◦ ◦ where X, T, D and Φ are a vector field, a (1,1)-tensor field, a torsion-free affineconnection andasymmetric(0,2)-tensor fieldonΣ,respectively. Now a direct calculation of the almost soliton equation (1) gives: (Hes +ρ λg)(∂ ,∂ ) = ∂2 f = 0, f − x1′ x1′ x1′x1′ (7) (Hes +ρ λg)(∂ ,∂ ) = ∂2 f = 0, f − x1′ x2′ x1′x2′ (Hes +ρ λg)(∂ ,∂ ) = ∂2 f = 0, f − x2′ x2′ x2′x2′ which shows that the potential function f of any gradient Ricci almost soliton must be of the form f = ιξ +π∗fˆfor some vector field ξ on Σ and some smooth function fôn Σ. We firstly show that no gradient Ricci almost soliton may exist if the vector field X in (6) does not vanish. Assume X = 0 at some point p Σ 6 ∈ and specialize local coordinates on Σ so that X = ∂x1. Let T = Tijdxi⊗∂xj and ξ = ξl∂ be the local expressions of the tensor field T and the vector xl field ξ. Then a straightforward calculation using that λ = 1τ shows that 4 (Hesf +ρ− 41τg)(∂x1,∂x1′)= 41 2(T11−T22+2(x1′ +∂x1ξ1) −2ξ1(3x21′ +(cid:8)2x1′T11+x2′T12−2DΓ111) +ξ2(4x1′x2′ +2x1′T21+x2′(T11+T22)−4DΓ112) , where DΓk are the Christoffel symbols of the affine connection D. Th(cid:9)is is a ij polynomial in the variables x1′, x2′, so all coefficientes must vanish and one has that ξ = 0. The previous expression reduces to 1 1 (Hesf +ρ− 4τg)(∂x1,∂x1′) = x1′ + 2(T11−T22). Since the above expression is not identically zero, we conclude that there do not exist gradient Ricci almost solitons if X does not vanish. Hence any self-dual metric (6) reduces to (8) g = ιT ιId+g +π∗Φ D,Φ,T,Id D ◦ for some(1,1)-tensor field T, an affineconnection D anda symmetric(0,2)- tensor field Φ on Σ. Moreover, Equation (7) shows that the potential function of any Ricci almost soliton must be of the form f = ιξ +π∗fˆfor some vector field ξ on Σ and fˆ ∞(Σ). In order to show that the potential ∈ C function f is the pull-back of a smooth function fôn Σ, assume the vector field ξ = 0 at some point p Σ. Let (x1,x2) be adapted local coordinates 6 ∈ so that ξ = ∂x1 and hence f = x1′ +π∗fˆ. Computing again expressions of the almost soliton equation one has (Hesf +ρ− 41τg)(∂x1,∂x1′)= 12{(1−2x1′)T11−x2′T12−T22}+DΓ111, (Hesf +ρ− 14τg)(∂x2,∂x2′)= 12{(4−2x1′)T22−x2′T12−(1+x1′)T11}+DΓ212, it follows that T1 = T2 = T2 = 0. Since the scalar curvature of any metric 1 1 2 given by (8) satisfies τ = 3(T1+T2) = 3tr T , one has that τ = 0 and thus 1 2 { }