Bach-flat isotropic gradient Ricci solitons
Esteban Calviño-Louzao, Eduardo García-Río, Ixchel Gutiérrez-Rodríguez, Ramón Vázquez-Lorenzo
aa r X i v : . [ m a t h . DG ] F e b BACH-FLAT ISOTROPIC GRADIENT RICCI SOLITONS
E. CALVI ˜NO-LOUZAO, E. GARC´IA-R´IO, I. GUTI´ERREZ-RODR´IGUEZ,R. V ´AZQUEZ-LORENZO
Abstract.
We construct examples of Bach-flat gradient Ricci solitonswhich are neither half conformally flat nor conformally Einstein. Introduction
Let (
M, g ) be a pseudo-Riemannian manifold. Let f ∈ C ∞ ( M ). We saythat ( M, g, f ) is a gradient Ricci soliton if the following equation is satisfied:(1) Hes f + ρ = λ g , for some λ ∈ R , where ρ is the Ricci tensor , and Hes f = ∇ df is the Hessiantensor acting on f .A gradient Ricci soliton is said to be trivial if the potential function f is constant, since equation (1) reduces to the Einstein equation ρ = λg .Besides being a generalization of Einstein manifolds, the main interest ofgradient Ricci solitons comes from the fact that they correspond to self-similar solutions of the Ricci flow ∂ t g ( t ) = − ρ g ( t ) . Ricci solitons are ancientsolutions of the flow in the shrinking case ( λ > λ = 0), and immortal solutions in the expanding case( λ < M, g ) in terms of the Ricci curvature and the second fundamental formof the level sets of the potential function f . The fact that the Ricci ten-sor completely determines the curvature tensor in the locally conformallyflat case has yielded some results in this situation [14, 27, 30]. Any locallyconformally flat gradient Ricci soliton is locally a warped product in theRiemannian setting [21]. The higher signature case, however, allows otherpossibilities when the level sets of the potential function are degeneratehypersurfaces [7]. Generalizing the locally conformally flat condition, four-dimensional half conformally flat (i.e., self-dual or anti-self-dual) gradientRicci solitons have been investigated in the Riemannian and neutral signa-ture cases [6, 16]. While they are locally conformally flat in the Riemanniansituation, neutral signature allows other examples given by Riemannian ex-tensions of affine gradient Ricci solitons. Mathematics Subject Classification.
Key words and phrases.
Gradient Ricci soliton, Bach tensor, Riemannian extension,Affine surface.
Let W be the Weyl conformal curvature tensor of ( M, g ). The Bach ten-sor, B ij = ∇ k ∇ ℓ W kijℓ + ρ kℓ W kijℓ , is conformally invariant in dimensionfour. Bach-flat metrics contain half conformally flat and conformally Ein-stein metrics as special cases [4]. Hence, a natural problem is to classifyBach-flat gradient Ricci solitons. The Riemannian case was investigated in[12, 13] both in the shrinking and steady cases. In all situations the Bach-flat condition reduces to the locally conformally flat one under some naturalconditions.Our main purpose in this paper is to construct new examples of Bach-flat gradient Ricci solitons. The corresponding potential functions havedegenerate level set hypersurfaces and their underlying structure is neverlocally conformally flat, in sharp contrast with the Riemannian situation.These metrics are realized on the cotangent bundle T ∗ Σ of an affine surface(Σ , D ), and they may be viewed as perturbations of the classical Riemannianextensions introduced by Patterson and Walker in [29].Here is a brief guide to some of the most important results of this pa-per. In Theorem 3.1 we show that, for any affine surface (Σ , D ) admittinga parallel nilpotent (1 , T , the modified Riemannian extension( T ∗ Σ , g D,T, Φ ) is Bach-flat. Moreover we show that Bach-flatness is indepen-dent of the deformation tensor field Φ, thus providing an infinite family ofBach-flat metrics for any initial data (Σ , D, T ). Affine surfaces admitting aparallel nilpotent (1 , T are characterized in Proposition 3.3by the recurrence of the symmetric part of the Ricci tensor, being ker T a parallel one-dimensional distribution whose integral curves are geodesics.This class of surfaces generalizes those considered in [28].The previous construction is used in Theorem 4.3 to show that, for anysmooth function h ∈ C ∞ (Σ), there exist appropriate deformation tensorfields Φ so that ( T ∗ Σ , g D,T, Φ , f = h ◦ π ) is a steady gradient Ricci solitonif and only if dh (ker T ) = 0. This provides infinitely many examples ofBach-flat gradient Ricci solitons in neutral signature.Theorem 5.1 and Theorem 6.1 show that ( T ∗ Σ , g D,T, Φ ) is genericallystrictly Bach-flat, i.e., neither half conformally flat nor conformally Ein-stein. Moreover, Theorem 5.1 is used in Proposition 5.2 to construct newexamples of anti-self-dual metrics. Turning to gradient Ricci solitons, weshow in Theorem 5.3 the existence of anti-self-dual steady gradient Riccisolitons which are not locally conformally flat.The paper is organized as follows. Some basic results on the Bach tensorand gradient Ricci solitons are introduced in Section 2, as well as a sketchof the construction of modified Riemannian extensions g D, Φ ,T . We use thesemetrics in Section 3 to show that, for any parallel tensor field T on (Σ , D ), g D, Φ ,T is Bach-flat if and only if T is either a multiple of the identity ornilpotent (cf. Theorem 3.1). In Section 4 we show that for each initial data(Σ , D, T ) there are an infinite number of Bach-flat steady gradient Riccisolitons (cf. Theorem 4.3). Non-triviality of the examples is obtained afteran examination of the half conformally flat condition (cf. Section 5) and theconformally Einstein property (cf. Section 6) of the modified Riemannianextensions introduced in Section 2. As a consequence, new anti-self-dual ACH-FLAT ISOTROPIC GRADIENT RICCI SOLITONS 3 gradient Ricci solitons are exhibited in Theorem 5.3. Finally, we specializethis construction in Section 7.1 to provide some illustrative examples.2.
Preliminaries
Let ( M n , g ) be a pseudo-Riemannian manifold with Ricci curvature ρ andscalar curvature τ . Let W denote the Weyl conformal curvature tensor anddefine W [ ρ ]( X, Y ) = P ij ε i ε j W ( E i , X, Y, E j ) ρ ( E i , E j ), where { E i } is a localorthonormal frame and ε i = g ( E i , E i ). Then the Bach tensor is defined by(see [2])(2) B = div div W + n − n − W [ ρ ] , where div is the divergence operator.Let S = ρ − τ n − g denote the Schouten tensor of ( M, g ). Let the
Cotton tensor , C ijk = ( ∇ i S ) jk − ( ∇ j S ) ik ; it provides a measure of thelack of symmetry on the covariant derivative of the Schouten tensor. Sincediv W = − n − n − C , the Bach and the Cotton tensors of any four-dimensionalmanifold are related by B = ( − div C + W [ ρ ]) .The Bach tensor, which is trace-free and conformally invariant in dimen-sion n = 4, has been broadly investigated in the literature, both from thegeometrical and physical viewpoints (see for example [15, 18, 20] and refer-ences therein). It is the gradient of the L functional of the Weyl curvatureon compact manifolds. The field equations of conformal gravity are equiva-lent to setting the Bach tensor equal to zero and it is also central in the studyof the Bach flow, a geometric flow which is quadratic on the curvature andwhose fixed points are the vacuum solutions of conformal Weyl gravity [3].Besides the half conformally flat metrics and the conformally Einsteinones, there are few known examples of strictly Bach-flat manifolds, meaningthe ones which are neither half conformally flat nor conformally Einstein(see, for example, [1, 23, 26]). Motivated by this lack of examples, wefirst construct new explicit four-dimensional Bach-flat manifolds of neutralsignature.2.1. Riemannian extensions.
In order to introduce the family of metricsunder consideration, we recall that a pseudo-Riemannian manifold (
M, g )is a
Walker manifold if there exists a parallel null distribution D on M .Walker metrics, also called Brinkmann waves in the literature, have beenwidely investigated in the Lorentzian setting (pp-waves being a special classamong them). They appear in many geometrical situations showing a spe-cific behaviour without Riemannian counterpart (see [8]).Let ( M, g, D ) be a four-dimensional Walker manifold of neutral signatureand D of maximal rank. Then there are local coordinates ( x , x , x ′ , x ′ )so that the metric g is given by (see [31])(3) g = 2 dx i ◦ dx i ′ + g ij dx i ◦ dx j , where “ ◦ ” denotes the symmetric product ω ◦ ω := ( ω ⊗ ω + ω ⊗ ω )and ( g ij ) is a 2 × D = span { ∂ x ′ , ∂ x ′ } . CALVI ˜NO-LOUZAO, GARC´IA-R´IO, GUTI´ERREZ-RODR´IGUEZ, V ´AZQUEZ-LORENZO
A special family of four-dimensional Walker metrics is provided by theRiemannian extensions of affine connections to the cotangent bundle of anaffine surface. Next we briefly sketch their construction. Let T ∗ Σ be thecotangent bundle of a surface Σ and let π : T ∗ Σ → Σ be the projection. Let˜ p = ( p, ω ) denote a point of T ∗ Σ, where p ∈ Σ and ω ∈ T ∗ p Σ. Local coordi-nates ( x i ) in an open set U of Σ induce local coordinates ( x i , x i ′ ) in π − ( U ),where one sets ω = P x i ′ dx i . The evaluation functions on T ∗ Σ play a cen-tral role in the construction. They are defined as follows. For each vectorfield X on Σ, the evaluation of X is the real valued function ιX : T ∗ Σ → R given by ιX ( p, ω ) = ω ( X p ). Vector fields on T ∗ Σ are characterized by theiraction on evaluations ιX and one defines the complete lift to T ∗ Σ of a vectorfield X on Σ by X C ( ιZ ) = ι [ X, Z ], for all vector fields Z on Σ. Moreover, a(0 , s )-tensor field on T ∗ Σ is characterized by its action on complete lifts ofvector fields on Σ.Next, let D be a torsion free affine connection on Σ. The Riemannianextension g D is the neutral signature metric g D on T ∗ Σ characterized by theidentity g D ( X C , Y C ) = − ι ( D X Y + D Y X ) (see [29]). They are expressed inthe induced local coordinates ( x i , x i ′ ) as follows(4) g D = 2 dx i ◦ dx i ′ − x k ′ D Γ ij k dx i ◦ dx j , where D Γ ijk denote the Christoffel symbols of D . The geometry of ( T ∗ Σ , g D )is strongly related to that of (Σ , D ). Recall that the curvature of any affinesurface is completely determined by its Ricci tensor ρ D . Moreover, thesymmetric and skew-symmetric parts given by ρ Dsym ( X, Y ) = (cid:8) ρ D ( X, Y ) + ρ D ( Y, X ) (cid:9) , and ρ Dsk ( X, Y ) = (cid:8) ρ D ( X, Y ) − ρ D ( Y, X ) (cid:9) play a distinguishedrole.Let Φ be a symmetric (0 , deformed Rie-mannian extension , g D, Φ = g D + π ∗ Φ, is a first perturbation of the Rie-mannian extension. A second one is as follows. Let T = T ki dx i ⊗ ∂ x k be a(1 , ιT defines a one-form on T ∗ Σ char-acterized by ιT ( X C ) = ι ( T X ). The modified Riemannian extension g D, Φ ,T is the neutral signature metric on T ∗ Σ defined by (see [10])(5) g D, Φ ,T = ιT ◦ ιT + g D + π ∗ Φ , where Φ is a symmetric (0 , g D, Φ ,T = 2 dx i ◦ dx i ′ + { x r ′ x s ′ ( T ri T sj + T rj T si ) − x k ′ D Γ ijk +Φ ij } dx i ◦ dx j . The case when T is a multiple of the identity ( T = c Id , c = 0) is of specialinterest. It was shown in [10] that for any affine surface (Σ , D ), the modifiedRiemannian extension g D, Φ ,c Id is an Einstein metric on T ∗ Σ if and only ifthe deformation tensor Φ is the symmetric part of the Ricci tensor of (Σ , D ).Moreover, a slight generalization of the modified Riemannian extension al-lows a complete description of self-dual Walker metrics as follows.
Theorem 2.1. [10, 19]
A four-dimensional Walker metric is self-dual ifand only if it is locally isometric to the cotangent bundle T ∗ Σ of an affinesurface (Σ , D ) , with metric tensor g = ιX ( ι id ◦ ι id) + ι id ◦ ιT + g D + π ∗ Φ ACH-FLAT ISOTROPIC GRADIENT RICCI SOLITONS 5 where X , T , D and Φ are a vector field, a (1 , -tensor field, a torsion freeaffine connection and a symmetric (0 , -tensor field on Σ , respectively. As a matter of notation, we will write ∂ k = ∂∂x k and ∂ k ′ = ∂∂x k ′ , unlesswe want to emphasize some special coordinates. We will let φ k = ∂∂x k φ and φ k ′ = ∂∂x k ′ φ to denote the corresponding first derivatives of a smoothfunction φ .2.2. Gradient Ricci solitons and affine gradient Ricci solitons.
Let(
M, g, f ) be a gradient Ricci soliton, i.e., (
M, g ) is a pseudo-Riemannianmanifold and f ∈ C ∞ ( M ) is a solution of equation (1) for some λ ∈ R . Thelevel set hypersurfaces of the potential function play a distinguished role inanalyzing the geometry of gradient Ricci solitons. Hence we say that thesoliton is non isotropic if ∇ f is nowhere lightlike (i.e., k∇ f k = 0), and thatthe soliton is isotropic if k∇ f k = 0, but ∇ f = 0.Non isotropic gradient Ricci solitons lead to local warped product de-compositions in the locally conformally flat and half conformally flat cases,and their geometry resembles the Riemannian situation [6, 7]. The isotropiccase is, however, in sharp contrast with the positive definite setting since ∇ f gives rise to a Walker structure. Self-dual gradient Ricci solitons which arenot locally conformally flat are isotropic and, moreover, they are describedin terms of Riemannian extensions as follows. Theorem 2.2. [6]
Let ( M, g, f ) be a four-dimensional self-dual gradientRicci soliton of neutral signature which is not locally conformally flat. Then ( M, g ) is locally isometric to the cotangent bundle T ∗ Σ of an affine surface (Σ , D ) equipped with a modified Riemannian extension g D, Φ , .Moreover any such gradient Ricci soliton is steady and the potential func-tion is given by f = h ◦ π for some h ∈ C ∞ (Σ) satisfying the affine gradientRicci soliton equation (6) Hes Dh +2 ρ Dsym = 0 , for any symmetric (0 , -tensor field Φ on Σ . The previous result relates the affine geometry of (Σ , D ) and the pseudo-Riemannian geometry of ( T ∗ Σ , g D, Φ , ), allowing the construction of an in-finite family of steady gradient Ricci solitons on T ∗ Σ for any initial data(Σ , D, h ) satisfying (6). It is important to remark here that the existence ofaffine gradient Ricci solitons imposes some restrictions on (Σ , D ), as shownin [9] in the locally homogeneous case.3.
Bach-flat modified Riemannian extensions
The use of modified Riemannian extensions with T = c Id allowed theconstruction of many examples of self-dual Einstein metrics [10]. One ofthe crucial facts in understanding the metrics g D, Φ ,c Id is that the (1 , T = c Id is parallel with respect to the connection D . Hence, anatural generalization arises by considering arbitrary tensor fields T whichare parallel with respect to the affine connection D . CALVI ˜NO-LOUZAO, GARC´IA-R´IO, GUTI´ERREZ-RODR´IGUEZ, V ´AZQUEZ-LORENZO
Let (Σ , D, T ) be a torsion free affine surface equipped with a parallel(1 , T . Parallelizability of T guaranties the existence of localcoordinates ( x , x ) on Σ so that T ∂ = T ∂ + T ∂ , T ∂ = T ∂ + T ∂ , for some real constants T ji . Let ( T ∗ Σ , g D, Φ ,T ) be the modified Riemannianextension given by (5). Further note that D and Φ are taken with full gen-erality. Thus, the corresponding Christoffel symbols D Γ kij and the coefficientfunctions Φ ij are arbitrary smooth functions of the coordinates ( x , x ).Our first main result concerns the construction of Bach-flat metrics: Theorem 3.1.
Let (Σ , D, T ) be a torsion free affine surface equipped with aparallel (1 , -tensor field T . Let Φ be an arbitrary symmetric (0 , -tensorfield on Σ . Then the Bach tensor of ( T ∗ Σ , g D, Φ ,T ) vanishes if and only if T is either a multiple of the identity or nilpotent.Proof. In order to compute the Bach tensor of ( T ∗ Σ , g D, Φ ,T ), first of allobserve that being T parallel imposes some restrictions on the components T ji as well as on the Christoffel symbols of the connection D :(7) DT = 0 : T D Γ − T D Γ = 0 ,T D Γ − T D Γ = 0 ,T D Γ + ( T − T ) D Γ − T D Γ = 0 ,T D Γ + ( T − T ) D Γ − T D Γ = 0 ,T D Γ + ( T − T ) D Γ − T D Γ = 0 ,T D Γ + ( T − T ) D Γ − T D Γ = 0 . Then, expressing the Bach tensor B ij = B ( ∂ i , ∂ j ) in induced coordinates( x i , x i ′ ), a long but straightforward calculation shows that(8) ( B ij ) = B B B B ˜ B ˜ B , where˜ B = 16 (( T − T ) + 4 T T ) · ( T + T ) · (cid:18) T − T T T T − T (cid:19) and where the coefficients B , B and B can be written in terms of d = det( T ) and t = tr( T ) as follows: ACH-FLAT ISOTROPIC GRADIENT RICCI SOLITONS 7 B = − (cid:8) d − t + 13 T t − T ) ) d +(5 t − T )( t − T ) t d − ( t − T ) t (cid:9) x ′ − (cid:8) ( T ) (30 d + t d − t ) (cid:9) x ′ − (cid:8) (13 t − T ) d + (3 t − T ) t d − ( t − T ) t (cid:9) T x ′ x ′ − (cid:8) ( D Γ + 2 D Γ )( t − T ) + 2 T D Γ (cid:9) ( t − d ) t x ′ − (cid:8) D Γ ( t − T ) + 2 T D Γ (cid:9) ( t − d ) t x ′ − (cid:8) d + (3 t − T t + 14( T ) ) d − ( t − T t + 2( T ) ) t (cid:9) Φ − (cid:8) (11 t − T ) d − t − T ) t (cid:9) T Φ + (cid:8) t − d (cid:9) ( T ) Φ − ( ∂ D Γ − ∂ D Γ )(4 d − t ) , B = − (cid:8) (13 t − T ) d + (3 t − T ) t d − ( t − T ) t (cid:9) T x ′ + (cid:8) (17 t − T ) d − (2 t + T ) t d + T t (cid:9) T x ′ + (cid:8) d + 4(4 t − T t + 15( T ) ) d − (3 t + 2 T t − T ) ) t d + 2( t − T ) T t (cid:9) x ′ x ′ − (cid:8) D Γ ( t − T ) + 2 T D Γ (cid:9) ( t − d ) t x ′ − (cid:8) D Γ ( t − T ) + 2 T D Γ (cid:9) ( t − d ) t x ′ − (cid:8) (11 t − T ) d − t − T ) t (cid:9) T Φ + (cid:8) d + (6 t − T t + 28( T ) ) d − ( t − T ) t (cid:9) Φ + (cid:8) (3 t − T ) d + 2 T t (cid:9) T Φ − (cid:8) ( ∂ D Γ − ∂ D Γ − ∂ D Γ + ∂ D Γ )( t − d ) (cid:9) , B = − (cid:8) d − t + t d (cid:9) ( T ) x ′ − (cid:8) d + 2( t − T t + 15( T ) ) d +(4 t + T ) T t d − ( T ) t (cid:9) x ′ + (cid:8) (17 t − T ) d − (2 t + T ) t d + T t (cid:9) T x ′ x ′ − (cid:8) D Γ ( t − T ) + 2 T D Γ (cid:9) ( t − d ) t x ′ + (cid:8) D Γ ( t − T ) − T D Γ (cid:9) ( t − d ) t x ′ − (7 d − t )( T ) Φ + (cid:8) (3 t − T ) T d + 2 T T t (cid:9) Φ − (cid:8) d − (5 t + 6 T t − T ) ) d + t − T ) t (cid:9) Φ − ( ∂ D Γ − ∂ D Γ )( t − d ) . CALVI ˜NO-LOUZAO, GARC´IA-R´IO, GUTI´ERREZ-RODR´IGUEZ, V ´AZQUEZ-LORENZO
Suppose first that the Bach tensor of ( T ∗ Σ , g D, Φ ,T ) vanishes. We startanalyzing the case T = 0. In this case, the expression of ˜ B in Equation (8)reduces to(9) ˜ B = 16 ( T − T ) · ( T + T ) · (cid:18) T − T T T − T (cid:19) . If T = T , we differentiate the component B in Equation (8) twice withrespect to x ′ to obtain T T = 0. Thus, either T = 0 and T is a multipleof the identity, or T = 0 and, in such a case, T is determined by T ∂ = T ∂ and therefore it is nilpotent. If T = T , then Equation (9) implies that T = − T . In this case, we differentiate the component B in Equation (8)twice with respect to x ′ and obtain T = 0. Thus, as before, T is nilpotent.Next we analyze the case T = 0. We use Equation (7) to express D Γ = T − T T D Γ + T T D Γ , D Γ = T T D Γ , D Γ = T T D Γ , D Γ = D Γ − T − T T D Γ . Considering the component ˜ B in Equation (8),˜ B = 16 ( T − T ) · ( T + T ) · (( T − T ) + 4 T T ) , we analyze separately the vanishing of each one of the three factors in ˜ B .Assume that T = T . In this case, component ˜ B in Equation (8)reduces to ˜ B = T ( T ) T ; since we are assuming that T = 0, theneither T = 0 or T = 0 and T = 0. If T = 0, the only non-zero componentof the Bach tensor is given by B = − ( T ) ( T ) (3( T ) x ′ + Φ ), fromwhere it follows that T = 0 and hence T is determined by T ∂ = T ∂ andis nilpotent. If T = 0 and T = 0, then we differentiate the component B in Equation (8) with respect to x ′ and x ′ to get T T = 0, which is notpossible since both T and T are non-null.Suppose now that T = − T . In this case, we differentiate the component B in Equation (8) twice with respect to x ′ and as a consequence weobtain T ( T T + ( T ) ) = 0; since we are assuming T = 0, it follows that T = − ( T ) T . Thus, the (1,1)-tensor field T is given by T ∂ = T ∂ − ( T ) T ∂ and T ∂ = T ∂ − T ∂ , and therefore it is nilpotent as well.Finally, suppose that ( T − T ) + 4 T T = 0; since T = 0, this isequivalent to T = − ( T − T ) T . Now, we differentiate the component B inEquation (8) twice with respect to x ′ to obtain T ( T + T ) = 0. Thus,we have that T = − T and T is given by T ∂ = T ∂ − ( T ) T ∂ and T ∂ = T ∂ − T ∂ , which again implies that T is nilpotent.To conclude the proof we show the “only if” part. If T is a multiple ofthe identity, then ( T ∗ Σ , g D, Φ ,T ) is self-dual by Theorem 2.1 and thereforeit has vanishing Bach tensor. Thus, we suppose T is parallel and nilpotentand, in this case, we can choose a system of coordinates ( x , x ) such that T is determined by T ∂ = ∂ and T ∂ = 0. Hence, examining Equation (8), ACH-FLAT ISOTROPIC GRADIENT RICCI SOLITONS 9 clearly ˜ B = 0 and, since d = t = 0, one easily checks that B = B = B = 0, showing that the Bach tensor of ( T ∗ Σ , g D, Φ ,T ) vanishes. (cid:3) Remark 3.2.
We emphasize that even though the Bach tensor of the met-rics g D, Φ ,T depends on the choice of Φ (as shown in the proof of Theorem 3.1),the existence of Bach-flat metrics in Theorem 3.1 is independent of the sym-metric (0 , , D, T ). Moreover, note that the metrics g D, Φ ,T aregenerically non-isometric for different deformation tensor fields Φ.The Bach-flat modified Riemannian extensions in Theorem 3.1 obtainedfrom a (1 , T = c Id are not of interest for ourpurposes since they all are half conformally flat (cf. Theorem 2.1). Hence,in what follows we focus on the case when T is a parallel nilpotent (1 , g D, Φ ,T as a nilpotent Riemannian extension .3.1. Affine connections supporting parallel nilpotent tensors.
Theproof of Theorem 3.1 shows that the existence of a parallel nilpotent tensorfield T on a torsion free affine surface (Σ , D ) imposes some restrictions on D . Proposition 3.3.
Let (Σ , D, T ) be a torsion free affine surface equippedwith a nilpotent (1 , -tensor field T . If T is parallel, then (i) ker T is a parallel one-dimensional distribution whose integral curvesare geodesics of (Σ , D ) . (ii) The symmetric part of the Ricci tensor, ρ Dsym , is zero or of rank oneand recurrent, i.e., Dρ Dsym = η ⊗ ρ Dsym , for some one-form η .Proof. Let (Σ , D ) be a torsion free affine surface admitting a parallel nilpo-tent (1 , T . Then there exist suitable coordinates ( x , x )where T ∂ = ∂ , T ∂ = 0 and it follows from (7) that the Christoffel sym-bols of D satisfy(10) D Γ = 0 , D Γ = D Γ , D Γ = 0 , D Γ = 0 . In such a case the one-dimensional distribution ker T (= span { ∂ } ) is paralleland ∂ is a geodesic vector field, thus showing Assertion (i). Moreover, theRicci tensor of any affine connection given by (10) satisfies ρ D = ∂ D Γ − ∂ D Γ ∂ D Γ − ∂ D Γ ! , from where it follows that the symmetric and the skew-symmetric parts ofthe Ricci tensor are given by ρ Dsym = ( ∂ D Γ − ∂ D Γ ) dx ◦ dx , ρ Dsk = ∂ D Γ dx ∧ dx . Hence ρ Dsym is either zero or of rank one. Moreover, a straightforward calcu-lation of the covariant derivative of the symmetric part of the Ricci tensorgives( D ∂ ρ Dsym )( ∂ , ∂ ) = ∂ D Γ − ∂ D Γ − D Γ ( ∂ D Γ − ∂ D Γ ) , ( D ∂ ρ Dsym )( ∂ , ∂ ) = ∂ D Γ − ∂ D Γ , the other components being zero. This shows that ρ Dsym is recurrent, i.e., Dρ Dsym = η ⊗ ρ Dsym , with recurrence one-form(11) η = { ∂ ln ρ Dsym ( ∂ , ∂ ) − D Γ } dx + ∂ ln ρ Dsym ( ∂ , ∂ ) dx , which proves (ii). (cid:3) Remark 3.4.
It follows from the expression of ρ Dsk in the proof of Propo-sition 3.3 that any connection given by (10) has symmetric Ricci tensor ifand only if ∂ D Γ = 0, in which case ρ D is recurrent. Now, it follows fromthe work of Wong [32] that any such connection can be described in suit-able coordinates (¯ u , ¯ u ) by D∂ ¯ u ∂ ¯ u = ¯ u Γ (¯ u , ¯ u ) ∂ ¯ u , where ¯ u Γ (¯ u , ¯ u )is an arbitrary function satisfying ∂ ¯ u ¯ u Γ (¯ u , ¯ u ) = 0. Moreover, the onlynon-zero component of the Ricci tensor is ρ D ( ∂ ¯ u , ∂ ¯ u ) = ∂ ¯ u ¯ u Γ , and therecurrence one-form ω is given by(12) ω = ∂ ¯ u (ln ∂ ¯ u ¯ u Γ ) d ¯ u + ∂ ¯ u (ln ∂ ¯ u ¯ u Γ ) d ¯ u . Further assume that T is a parallel nilpotent (1 , , D ).Then a straightforward calculation shows that its expression in the co-ordinates (¯ u , ¯ u ) is given by T ∂ ¯ u = T ∂ ¯ u and T ∂ ¯ u = 0, for some T ∈ R , T = 0. Hence, considering the modified coordinates ( u , u ) =(¯ u , ( T ) − ¯ u ) one has that T ∂ u = ∂ u and T ∂ u = 0, and the connectionis determined by the only non-zero Christoffel symbol u Γ . Moreover itfollows from the expression of the recurrence one-form ω that ω (ker T ) = 0if and only if ∂ u Γ = 0.4. Bach-flat gradient Ricci solitons
Let Φ be a symmetric (0 , , D, T ). One uses the nilpo-tent structure T to construct an associated symmetric (0 , b Φ given by b Φ( X, Y ) = Φ(
T X, T Y ), for all vector fields
X, Y on Σ. Fur-ther, let ( x , x ) be local coordinates where T ∂ = ∂ , T ∂ = 0 and letΦ = Φ ij dx i ⊗ dx j . Then b Φ expresses as b Φ = b Φ ij dx i ⊗ dx j = Φ dx ⊗ dx .4.1. Einstein nilpotent Riemannian extensions.Theorem 4.1.
Let (Σ , D, T ) be an affine surface equipped with a parallelnilpotent (1 , -tensor field T and let Φ be a symmetric (0 , -tensor fieldon Σ . Then ( T ∗ Σ , g D, Φ ,T ) is Einstein (indeed, Ricci-flat) if and only if b Φ = − ρ Dsym .Proof.
Let ( x , x ) be local coordinates on Σ so that T ∂ = ∂ , T ∂ = 0,and consider the induced coordinates ( x , x , x ′ , x ′ ) on T ∗ Σ. A straightfor-ward calculation shows that the Ricci tensor of any nilpotent Riemannnianextension g D, Φ ,T is determined by ρ ( ∂ , ∂ ) = Φ( ∂ , ∂ ) + 2 ρ Dsym ( ∂ , ∂ ) , the other components being zero. Hence the Ricci operator is nilpotent and g D, Φ ,T has zero scalar curvature. Moreover, the Ricci tensor vanishes if andonly if Φ( ∂ , ∂ ) + 2 ρ Dsym ( ∂ , ∂ ) = 0. The result now follows. (cid:3) ACH-FLAT ISOTROPIC GRADIENT RICCI SOLITONS 11
Remark 4.2.
The Weyl tensor of a pseudo-Riemannian manifold is har-monic if and only if the Cotton tensor vanishes. Let (Σ , D, T ) be an affinesurface equipped with a parallel nilpotent (1 , T and let Φbe a symmetric (0 , x , x ) be local coordinateson Σ so that T ∂ = ∂ , T ∂ = 0, and consider the induced coordinates( x , x , x ′ , x ′ ) on T ∗ Σ. A straightforward calculation shows that the Cot-ton tensor of ( T ∗ Σ , g D, Φ ,T ) is given by C ( ∂ , ∂ , ∂ ) = −{ ∂ Φ( ∂ , ∂ ) + 2 ∂ ρ Dsym ( ∂ , ∂ ) } , the other components being zero. Hence ( T ∗ Σ , g D, Φ ,T ) has harmonic Weyltensor if and only if d D Φ = − b η ⊗ ρ Dsym , where b η ( X ) = η ( T X ) , η being therecurrence one-form in (11) , and d D Φ( X, Y ; Z ) = D Φ( T X, T Y ; T Z ) . Gradient Ricci solitons on nilpotent Riemannian extensions.
Recall from Theorem 2.2 that the affine gradient Ricci soliton equationHes Dh +2 ρ Dsym = 0 determines the potential function of any self-dual gra-dient Ricci soliton which is not locally conformally flat, independently ofthe deformation tensor Φ. The next theorem shows that, in contrast withthe previous situation, for any h ∈ C ∞ (Σ) with dh (ker T ) = 0, one mayuse the symmetric (0 , Dh +2 ρ Dsym to determine a deforma-tion tensor field Φ so that the resulting nilpotent Riemannian extension isa Bach-flat steady gradient Ricci soliton with potential function f = h ◦ π .Let ( T ∗ Σ , g D, Φ ,T , f ) be a gradient Ricci soliton with potential function f ∈ C ∞ ( T ∗ Σ). Let ( x , x ) be local coordinates on Σ so that T ∂ = ∂ , T ∂ = 0, and consider the induced coordinates ( x , x , x ′ , x ′ ) on T ∗ Σ.Since Hes f ( ∂ i ′ , ∂ j ′ ) = ∂ i ′ j ′ f ( x , x , x ′ , x ′ ), it follows from the expressionof the Ricci tensor in Theorem 4.1 and the metric tensor (5) that f = ιX + h ◦ π for some h ∈ C ∞ (Σ) and some vector field X on Σ. Set X = A ( x , x ) ∂ + B ( x , x ) ∂ in the local coordinates ( x , x ), for some A, B ∈ C ∞ (Σ). Then Hes f ( ∂ , ∂ ′ ) = ∂ A ( x , x ), from where it follows that X = A ( x ) ∂ + B ( x , x ) ∂ . Considering the component Hes f ( ∂ , ∂ ′ ) = − A ′′ ( x ) + ∂ B ( x , x ), one has that X = A ( x ) ∂ + ( P ( x ) + x A ′ ( x )) ∂ for some smooth function P ( x ). Next the componentHes f ( ∂ , ∂ ′ ) = A ( x ) D Γ − x ′ A ( x )+ D Γ ( P ( x ) + x A ′ ( x )) + P ′ ( x ) + x A ′′ ( x ) , shows that A = 0 and it reduces to Hes f ( ∂ , ∂ ′ ) = P ′ ( x ) + P ( x ) D Γ .A solution P ( x ) of the equation P ′ ( x ) + P ( x ) D Γ = 0 either vanishesidentically (and hence X = 0) or it is nowhere zero, in which case ∂ D Γ = 0(see the proof of Theorem 6.1). In the later case Proposition 3.3 shows thatthe Ricci tensor of (Σ , D ) is symmetric and thus recurrent of rank one.Since we are mainly interested in the case when ρ Dsk is non-zero, the nexttheorem examines the simpler situation when X = 0 and f = h ◦ π . Theorem 4.3.
Let (Σ , D, T ) be an affine surface equipped with a parallelnilpotent (1 , -tensor field T and let Φ be a symmetric (0 , -tensor field on Σ . Let h ∈ C ∞ (Σ) be a smooth function. Then ( T ∗ Σ , g D, Φ ,T , f = h ◦ π ) is a Bach-flat gradient Ricci soliton if and only if dh (ker T ) = 0 and (13) b Φ = − Hes Dh − ρ Dsym . Moreover the soliton is steady and isotropic.Proof.
Taking local coordinates on T ∗ Σ as above and setting f = h ◦ π ,one has that Hes f ( ∂ , ∂ ′ ) + ρ ( ∂ , ∂ ′ ) = λg ( ∂ , ∂ ′ ) leads to λ = 0, whichshows that the soliton is steady. A straightforward calculation shows thatthe remaining non-zero terms in the gradient Ricci soliton equation are givenby Hes f ( ∂ , ∂ ) + ρ ( ∂ , ∂ ) = ∂ h , Hes f ( ∂ , ∂ ) + ρ ( ∂ , ∂ ) = ∂ h − D Γ ∂ h , Hes f ( ∂ , ∂ ) + ρ ( ∂ , ∂ ) = x ′ ∂ h − D Γ ∂ h + ∂ h − D Γ ∂ h +Φ + 2 ∂ D Γ − ∂ D Γ . It immediately follows from the equation (Hes f + ρ )( ∂ , ∂ ) = 0 that ∂ h = 0,which shows that dh (ker T ) = 0. The only remaining equation now becomesHes f ( ∂ , ∂ ) + ρ ( ∂ , ∂ ) = ∂ h − D Γ ∂ h + Φ + 2 ∂ D Γ − ∂ D Γ = Φ( ∂ , ∂ ) + Hes Dh ( ∂ , ∂ ) + 2 ρ Dsym ( ∂ , ∂ ) , from which Equation (13) follows. Moreover, it also follows from the formof the potential function that ∇ f = h ′ ( x ) ∂ ′ , and thus k∇ f k = 0 (equiv-alently the level hypersurfaces of the potential function are degenerate sub-manifolds of T ∗ Σ), which shows that the soliton is isotropic. (cid:3)
Remark 4.4.
The tensor field D ijk = C ijk + W ijkℓ ∇ ℓ f introduced in [13]plays an essential role in analyzing the geometry of Bach-flat gradient Riccisolitons. Local conformal flatness in [12, 13] follows from D = 0, which isobtained under some natural assumptions.Gradient Ricci solitons in Theorem 4.3 satisfy ∇ f = h ′ ( x ) ∂ ′ . Then, astraightforward calculation shows that D is completely determined by D = − h ′ ( x ) ∂ D Γ ( x , x ), the other components being zero. Hence it followsfrom the proof of Proposition 3.3 that the tensor field D vanishes if andonly if the Ricci tensor ρ D is symmetric. However Theorem 5.1 shows that( T ∗ Σ , g D, Φ ,T ) is never locally conformally flat. Further observe that, sincethe soliton is isotropic, ∇ f does not give rise to a local warped productdecomposition unlike the Riemannian case.A special situation of Theorem 4.3 occurs if h ∈ C ∞ (Σ) is a solution ofthe affine gradient Ricci soliton equation (6). The following result will beused in Section 5.1 to construct examples of anti-self-dual steady gradientRicci solitons (cf. Theorem 5.3). Proposition 4.5.
Let (Σ , D, T ) be an affine surface equipped with a parallelnilpotent (1 , -tensor field T . Then (Σ , D, T, h ) is an affine gradient Riccisoliton with dh (ker T ) = 0 if and only if ( T ∗ Σ , g D, b Φ ,T , f = h ◦ π ) is a Bach-flat gradient Ricci soliton for any symmetric (0 , -tensor field Φ .Moreover, (Σ , D, T, h ) is a non-flat affine gradient Ricci soliton with po-tential function h satisfying dh (ker T ) = 0 if and only if the recurrenceone-form η in (11) satisfies b η = 0 . ACH-FLAT ISOTROPIC GRADIENT RICCI SOLITONS 13
Proof.
Let Φ be an arbitrary symmetric (0 , b Φ( X, Y ) = Φ(
T X, T Y ) be the associated tensor field induced by T . Since b Φ = 0, (13) shows that ( T ∗ Σ , g D, b Φ ,T , f = h ◦ π ) is a gradient Ricci soliton ifand only if (Σ , D, T, h ) is an affine gradient Ricci soliton with dh (ker T ) = 0.Next take local coordinates ( x , x ) on Σ so that T ∂ = ∂ , T ∂ = 0. Sincethe Christoffel symbols D Γ kij are given by (10), using the expression of ρ Dsym in Proposition 3.3, one has (Hes Dh +2 ρ Dsym )( ∂ , ∂ ) = ∂ h . Thus h ( x , x ) = x P ( x ) + Q ( x ) for some P, Q ∈ C ∞ (Σ). Hence dh (ker T ) = 0 holds if andonly if P = 0. Since h ( x , x ) = Q ( x ) one has that (Hes Dh +2 ρ Dsym )( ∂ , ∂ ) =0, and the only remaining equation is0 = (Hes Dh +2 ρ Dsym )( ∂ , ∂ ) = Q ′′ + 2( ∂ D Γ − ∂ D Γ ) = Q ′′ + 2 ρ D ( ∂ , ∂ ) . Therefore, the integrability condition becomes ∂ ρ D ( ∂ , ∂ ) = 0. Hence, itfollows from (11) that (Σ , D, T, h ) is an affine gradient Ricci soliton with dh (ker T ) = 0 if and only if the symmetric part of the Ricci tensor ρ Dsym isrecurrent with recurrence one-form η satisfying η (ker T ) = 0. (cid:3) Half conformally flat nilpotent Riemannian extensions
The existence of a null distribution D on a four-dimensional manifold( M, g ) of neutral signature defines a natural orientation on M : the onewhich, for any basis u, v of D , makes the bivector u ∧ v self-dual (see [17]). Weconsider on T ∗ Σ the orientation which agrees with D = ker π ∗ , and thus self-duality and anti-self-duality are not interchangeable. The following resultshows that they are essentially different for nilpotent Riemannian extensions. Theorem 5.1.
Let (Σ , D, T ) be an affine surface equipped with a parallelnilpotent (1 , -tensor field T . Then (i) ( T ∗ Σ , g D, Φ ,T ) is never self-dual for any deformation tensor field Φ . (ii) If ( T ∗ Σ , g D, Φ ,T ) is anti-self-dual, then D is either a flat connectionor (Σ , D ) is recurrent with symmetric Ricci tensor of rank one.In the later case there exist local coordinates ( u , u ) where theonly non-zero Christoffel symbol is u Γ and the tensor field T isgiven by T ∂ u = ∂ u , T ∂ u = 0 . Moreover, ( T ∗ Σ , g D, Φ ,T ) is anti-self-dual if and only if the symmetric (0 , -tensor field Φ satisfiesthe equations: (14) d D Φ = − b ω ⊗ ρ D , b Φ ⊗ b Φ( ∂ , ∂ , ∂ , ∂ ) + 2( b Φ ⊗ ρ D )( ∂ , ∂ , ∂ , ∂ )+ D Φ( ∂ , ∂ ; T ∂ , T ∂ ) + D Φ( T ∂ , T ∂ ; ∂ , ∂ ) − D Φ( ∂ , T ∂ ; T ∂ , ∂ ) , where d D Φ( X, Y, Z ) = D Φ( T X, T Y ; T Z ) , ω is the recurrence one-form given by Dρ D = ω ⊗ ρ D , and b ω ( X ) = ω ( T X ) .Proof. A direct computation using the expression of the anti-self-dual curva-ture operator of any four-dimensional Walker metric obtained in [19] shows that, for any nilpotent Riemannian extension g D, Φ ,T , W − takes the form(15) W − = 12 − − , thus showing that the anti-self-dual Weyl curvature opertor W − is nilpotentand hence ( T ∗ Σ , g D, Φ ,T ) is never self-dual, which proves (i).Next we show (ii). Let ( M, g ) be a four-dimensional Walker metric (3)and set the metric components g = a , g = c and g = b , where g ij are functions of the Walker coordinates ( x , x , x ′ , x ′ ). Then the self-dualWeyl curvature operator takes the form (see [19])(16) W + = W +11 W +12 W +11 + τ − W +12 τ − W +12 − W +11 − τ − W +12 − W +11 − τ , where(17) W +11 = (6 ca b − a b ′ − ba c + 12 a c ′ − ca b + 6 a b ′ + 6 ba c + 6 a ′ b − a ′ b − a ′ c + 6 ab c − ab c + 12 b c ′ − b ′ c − a − c a − bca + 24 ca ′ − b a + 12 ba ′ − a ′ ′ − a b + 12 ab ′ − b − b ′ ′ + 12 acc − c + 6 abc − cc ′ − ac ′ − bc ′ + 24 c ′ ′ ) , and(18) W +12 = ( − ca − ba + 2 a ′ + ab − b ′ + ac − cc − c ′ − bc + 2 c ′ ) . Since any anti-self-dual metric is Bach-flat, we proceed as in the proofof Theorem 3.1 considering local coordinates ( x , x ) on the surface Σ suchthat T is determined by T ∂ = ∂ and T ∂ = 0. Since T is parallel, theChristoffel symbols must satisfy (10), i.e., D Γ = 0 , D Γ = D Γ , D Γ = 0 , D Γ = 0 . Next, we analyze the self-dual Weyl curvature operator, which is completelydetermined by the scalar curvature and its components W +11 and W +12 alreadydescribed in equations (17) and (18). The scalar curvature is zero by Theo-rem 4.1, and W +12 = − ∂ D Γ , from where it follows that the Ricci tensor ρ D is symmetric of rank one and recurrent (see Remark 3.4). Take localcoordinates ( u , u ) as in Remark 3.4 so that the only non-zero Christof-fel symbol is u Γ and T ∂ u = ∂ u , T ∂ u = 0. Finally, we compute thecomponent W +11 given by (17) in the coordinates ( u , u , u ′ , u ′ ) of T ∗ Σ,obtaining W +11 = ( ∂ Φ + 2 ∂ u Γ ) u ′ − (Φ ) − ∂ u Γ − ∂ Φ u Γ +2 ∂ Φ − ∂ Φ − ∂ Φ . ACH-FLAT ISOTROPIC GRADIENT RICCI SOLITONS 15
Thus ( T ∗ Σ , g D, Φ ,T ) is anti-self-dual if and only if ∂ Φ + 2 ∂ u Γ = 0 , (Φ ) + 2Φ ∂ u Γ + ∂ Φ u Γ = 2 ∂ Φ − ∂ Φ − ∂ Φ , from where (14) follows. (cid:3) Anti-self-dual gradient Ricci solitons.
Half conformally flat gradi-ent Ricci solitons are locally conformally flat in the Riemannian setting. Thesame result holds true in neutral signature for non-isotropic ( k∇ f k = 0)gradient Ricci solitons (see [6, 16]). Isotropic half conformally flat gradientRicci solitons are not necessarily locally conformally flat, and they are re-alized on Walker manifolds [6]. However, although the self-dual ones werealready described in Theorem 2.2, no explicit examples of strictly anti-self-dual gradient Ricci solitons were previously reported. In order to constructthe desired examples we firstly specialize Theorem 5.1 to get the followinganti-self-dual nilpotent Riemannian extensions. Proposition 5.2.
Let (Σ , D, T, Φ) be an affine surface equipped with a par-allel nilpotent (1 , -tensor field T and a parallel symmetric (0 , -tensorfield Φ . If ρ D is symmetric then ( T ∗ Σ , g D, Φ ,T ) is anti-self-dual if and onlyif b ω = 0 and b Φ = 0 , where ω is the recurrence one-form given by (12) .Proof. If the Ricci tensor ρ D is symmetric of rank one and Φ is parallel,then the equations in Theorem 5.1 reduce to b ω = 0 and b Φ = 0, whichproves the result. Moreover if (Σ , D ) is a flat surface, a straightforwardcalculation shows that anti-self-duality is equivalent to b Φ = 0, being Φ aparallel tensor. (cid:3)
The condition b Φ = 0 in previous proposition restricts the consideration ofRicci solitons on T ∗ Σ to those originated by an affine gradient Ricci solitonon Σ (see Theorem 4.3). An application of Propositions 4.5 now gives thedesired examples of strictly anti-self-dual gradient Ricci solitons.
Theorem 5.3.
Let (Σ , D, T, Φ) be an affine surface equipped with a parallelnilpotent (1 , -tensor field T and a parallel symmetric (0 , -tensor field Φ .If the Ricci tensor ρ D is symmetric, then any h ∈ C ∞ (Σ) is an affine gradientRicci soliton with dh (ker T ) if and only if the nilpotent Riemannianextension ( T ∗ Σ , g D, b Φ ,T , f = h ◦ π ) is an anti-self-dual gradient steady Riccisoliton which is not locally conformally flat.Moreover, there exist local coordinates ( u , u ) on Σ so that the only non-zero Christoffel symbol is given by u Γ = α ( u ) + u β ( u ) and the potentialfunction h ( u ) is determined by h ′′ ( u ) = − β ( u ) , for any α, β ∈ C ∞ (Σ) .Proof. First of all, note that ( T ∗ Σ , g D, b Φ ,T , f = h ◦ π ) is a gradient Riccisoliton by Proposition 4.5. Anti-self-duality follows from Proposition 5.2.Next, since the Ricci tensor is symmetric, Remark 3.4 shows that it is recur-rent. Take local coordinates ( u , u ) on Σ as in Remark 3.4. Then (Σ , D, T )admits an affine gradient Ricci soliton with potential function h such that dh (ker T ) = 0 if and only if h ( u , u ) = S ( u ) for some S ∈ C ∞ (Σ) and(Hes Dh +2 ρ Dsym )( ∂ , ∂ ) = ∂ h + 2 ∂ u Γ = 0 becomes S ′′ ( x ) = − ∂ u Γ . Hence the integrability condition ∂ u Γ = 0 is equivalent to u Γ = α ( u ) + u β ( u ) for some α, β ∈ C ∞ (Σ) and h ′′ ( u ) = − β ( u ). (cid:3) Conformally Einstein nilpotent Riemannian extensions
A pseudo-Riemannian manifold ( M n , g ) is said to be (locally) confor-mally Einstein if every point p ∈ M has an open neighborhood U and apositive smooth function ϕ defined on U such that ( U , ¯ g = ϕ − g ) is Ein-stein. Brinkmann [5] showed that a manifold is conformally Einstein if andonly if the equation(19) ( n −
2) Hes ϕ + ϕ ρ − n { ( n − ϕ + ϕ τ } g = 0has a positive solution. Besides its apparent simplicity, the integration ofthe conformally Einstein equation is surprisingly difficult (see [25] and ref-erences therein for more information). It was shown in [22, 24] that anyfour-dimensional conformally Einstein manifold satisfies(20) ( i ) C + W ( · , · , · , ∇ σ ) = 0 , ( ii ) B = 0 . where the conformal metric is given by ¯ g = e σ g .Conditions (i)-(ii) above are also sufficient to be conformally Einsteinif ( M, g ) is weakly-generic (i.e., the Weyl tensor viewed as a map
T M → N T M is injective). Since nilpotent Riemannian extensions are not weakly-generic (see the expression of W − in the proof of Theorem 5.1), we willanalyze the conformally Einstein equation (19), seeking for solutions onnilpotent Riemannian extensions ( T ∗ Σ , g D, Φ ,T ). Theorem 6.1.
Let (Σ , D, T ) be a torsion free affine surface equipped witha parallel nilpotent (1 , -tensor field T . Then any solution of (19) is of theform ϕ = ιX + φ ◦ π for some vector field X on Σ such that X ∈ ker T and tr( DX ) = 0 .Moreover ( T ∗ Σ , g D, Φ ,T ) is conformally Einstein if and only if one of thefollowing holds: (i) The conformally Einstein equation (19) admits a solution ϕ = φ ◦ π for some φ ∈ C ∞ (Σ) with dφ (ker T ) = 0 , and the deformation tensor Φ is determined by φ b Φ + 2(Hes Dφ + φ ρ Dsym ) = 0 . (ii) The conformally Einstein equation (19) admits a solution ϕ = ιX + φ ◦ π for some φ ∈ C ∞ (Σ) and some non-zero vector field X on Σ such that X ∈ ker T and tr( DX ) = 0 .In this case, the Ricci tensor ρ D is symmetric of rank one andrecurrent. Moreover there are local coordinates ( u , u ) on Σ so that ϕ ( u , u , u ′ , u ′ ) = κu ′ + φ ( u , u ) is a solution of (19) if andonly if dφ ( T ∂ ) = κ Φ( T ∂ , T ∂ ) , Hes Dφ ( ∂ , ∂ ) + φ ρ D ( ∂ , ∂ ) = − ( φ + 2 κ u Γ )Φ( T ∂ , T ∂ )+ κ { D ∂ Φ)(
T ∂ , ∂ ) − ( D T ∂ Φ)( ∂ , ∂ ) } . Proof.
Let ( x , x ) be local coordinates on Σ so that T ∂ = ∂ , T ∂ = 0,and consider the induced coordinates ( x , x , x ′ , x ′ ) on T ∗ Σ. Since T is ACH-FLAT ISOTROPIC GRADIENT RICCI SOLITONS 17 parallel, and we obtain directly from Equation (7) that D Γ = 0 , D Γ = D Γ , D Γ = 0 , D Γ = 0 . In order to analyze the conformally Einstein equation (19) consider thesymmetric (0 , E = 2 Hes ϕ + ϕ ρ − { ϕ + ϕ τ } g and set E = 0. Let E ij = E ( ∂ i , ∂ j ) and let ϕ ∈ C ∞ ( T ∗ Σ) be a solution of (19). Thenone computes E = 2 ∂ ′ ′ ϕ, E = 2 ∂ ′ ′ ϕ, E = 2 ∂ ′ ′ ϕ , to show that any solution of (19) must be of the form(21) ϕ ( x , x , x ′ , x ′ ) = A ( x , x ) x ′ + B ( x , x ) x ′ + ψ ( x , x ) , for some smooth functions A , B and ψ depending only on the coordinates( x , x ). This shows that any solution of the conformally Einstein equationon ( T ∗ Σ , g D, Φ ,T ) is of the form ϕ = ιX + ψ ◦ π , where ιX is the evaluationof a vector field X = A∂ + B∂ on Σ, ψ ∈ C ∞ (Σ) and π : T ∗ Σ → Σ is theprojection.Now, the conformally Einstein condition given in Equation (19) can beexpressed in matrix form as follows:(22) ( E ij ) = E E ∂ A − ∂ B D Γ A + D Γ B + ∂ B − Ax ′ ) ∗ E ∂ A − ∂ A + ∂ B ∗ ∗ ∗ ∗ ∗ where positions with ∗ are not written since the matrix is symmetric, andwhere E = − ( ∂ A − ∂ B − D Γ A ) x ′ + { A Φ + 2( ∂ A − D Γ ∂ A + D Γ ∂ B + A∂ D Γ − B∂ D Γ ) } x ′ − { B Φ + 2 A Φ − ∂ B + D Γ ∂ A − D Γ ∂ B +( ∂ D Γ − D Γ D Γ ) A + ( ∂ D Γ − D Γ ) ) B + ∂ ψ ) } x ′ + 2 ∂ Ax ′ x ′ − ( ∂ A + ∂ B )Φ + 2( D Γ A + D Γ B )Φ + (2 D Γ B + ψ )Φ − A∂ Φ + B∂ Φ − B∂ Φ + 2 ∂ ψ − D Γ ∂ ψ − D Γ ∂ ψ − ∂ D Γ − ∂ D Γ ) ψ , E = 2( ∂ A − D Γ ∂ A + A∂ D Γ ) x ′ + 2( ∂ B + D Γ ∂ A + A∂ D Γ ) x ′ − ( ∂ A + ∂ B )Φ + 2 D Γ B Φ − A∂ Φ − B∂ Φ + 2 ∂ ψ − D Γ ∂ ψ , E = 2 ∂ Ax ′ + 2( ∂ B + 2 A∂ D Γ ) x ′ − ( ∂ A + ∂ B + 2 D Γ A )Φ − A∂ Φ + A∂ Φ − B∂ Φ + 2 ∂ ψ . First, we use component E = 2( D Γ A + D Γ B + ∂ B − Ax ′ ) in Equa-tion (22); note that ∂ ′ E = − A , and therefore A ( x , x ) = 0, which showsthat X ∈ ker T . Now component E in Equation (22) gives ∂ B = 0, whichimplies B ( x , x ) = P ( x ) for some smooth function P depending only onthe coordinate x , i.e., the vector field X = B∂ satisfies tr( DX ) = 0.At this point, the conformal function ϕ has the coordinate expression ϕ ( x , x , x ′ , x ′ ) = P ( x ) x ′ + ψ ( x , x )and the possible non-zero components in Equation (22) are E , E , E and E . Considering the component E = 2( P ′ ( x ) + D Γ ( x , x ) P ( x )), wedistinguish two cases depending on whether the function P vanishes iden-tically or not. Indeed, if P ( x ) is a solution of the equation E = 0, then ∂ (cid:16) P ( x ) e R D Γ ( x ,x ) dx (cid:17) = e R D Γ ( x ,x ) dx (cid:8) P ′ ( x ) + P ( x ) D Γ ( x , x ) (cid:9) = 0, which shows that P ( x ) e R D Γ ( x ,x ) dx = Q ( x ) for some smoothfunction Q ( x ). Now, if the function Q ( x ) vanishes at some point, then P ( x ) = 0 at each point. Otherwise, if Q ( x ) = 0 at each point, so is P ( x ).First, suppose that P ( x ) ≡
0, and hence ϕ = ψ ◦ π . In this case,component E in Equation (22) yields ∂ ψ = 0, which implies ψ ( x , x ) = Q ( x ) x + φ ( x ) for some smooth functions Q and φ depending only on thecoordinate x . Now, the only components in Equation (22) which could benon-null are E = 2 Qx ′ + ( Q Φ + 2 Q ′′ − D Γ Q ′ − ∂ D Γ − ∂ D Γ ) Q ) x + φ Φ + 2 φ ′′ − D Γ φ ′ − ∂ D Γ − ∂ D Γ ) φ − D Γ Q , E = 2( Q ′ − D Γ Q ) . Now, ∂ ′ E = 2 Q , implies Q = 0, thus showing that dϕ (ker T ) = 0. Then E = 0 and the component E reduces to E = φ Φ + 2 φ ′′ − D Γ φ ′ − ∂ D Γ − ∂ D Γ ) φ . Since ϕ ( x , x , x ′ , x ′ ) = φ ( x ), φ must be non-null and we obtain that E = 0 is equivalent toΦ = − φ (cid:8) φ ′′ − D Γ φ ′ − ( ∂ D Γ − ∂ D Γ ) φ (cid:9) , = − φ (cid:8) Hes Dφ ( ∂ , ∂ ) + φ ρ Dsym ( ∂ , ∂ ) (cid:9) , from where (i) is obtained.Finally, we analyze the case in which the function P ( x ) does not vanishidentically. Since E = 2( P ′ ( x ) + D Γ ( x , x ) P ( x )), we have ∂ D Γ = 0.Now it follows from Remark 3.4 that the Ricci tensor ρ D is symmetric ofrank one and recurrent. Specialize the local coordinates ( u , u ) on Σ so thatthe only non-zero Christoffel symbol of D is u Γ ( u , u ) and T ∂ u = ∂ u , T ∂ u = 0. Then any solution of the conformally Einstein equation takes theform ϕ ( u , u , u ′ , u ′ ) = A ( u ) u ′ + φ ( u , u ) . Now, considering the component E of the conformally Einstein equationin the new coordinates ( u , u ), one has E = 2 A ′ ( u ), which shows that ACH-FLAT ISOTROPIC GRADIENT RICCI SOLITONS 19 ϕ ( u , u , u ′ , u ′ ) = κu ′ + φ ( u , u ) for some κ = 0. Considering now thecomponent E = (2 ∂ φ − κ Φ ) u ′ + 2 ∂ φ − ∂ φ u Γ + 2 φ∂ u Γ + φ Φ + 2 κ Φ u Γ + κ∂ Φ − κ∂ Φ it follows that the conformally Einstein equation reduces to κ Φ = 2 ∂ φ, ( φ + 2 κ u Γ )Φ = − Dφ ( ∂ u , ∂ u ) + φρ D ( ∂ u , ∂ u ))+ κ (2 ∂ Φ − ∂ Φ ) , from where (ii) is obtained. (cid:3) Examples
Nilpotent Riemannian extensions with flat base.
Let (Σ , D ) bea flat torsion free affine surface. Take local coordinates on Σ so that allChristoffel symbols vanish. Let T be a parallel nilpotent (1 , T is parallel, its components T ji are necessarily constant on the givencoordinates. Hence one may further specialize the local coordinates ( x , x ),by using a linear transformation, so that T ∂ = ∂ , T ∂ = 0 and all theChristoffel symbols D Γ kij remain identically zero. Now Theorem 3.1 showsthat ( T ∗ Σ , g D, Φ ,T ) is Bach-flat for any symmetric (0 , -tensor field Φ on Σ. Moreover it follows from Theorem 4.3 that ( T ∗ Σ , g D, Φ ,T , f = h ◦ π ) is asteady gradient Ricci soliton for any h ∈ C ∞ (Σ) with dh ◦ T = 0 and anysymmetric (0 , -tensor field Φ such that Φ ( x , x ) = − h ′′ ( x ).Further note from Remark 4.4 that the steady gradient Ricci soliton( T ∗ Σ , g D, Φ ,T , f = h ◦ π ) satisfies D = 0. Moreover, since Φ = − h ′′ ( x ), onehas that ( T ∗ Σ , g D, Φ ,T ) is in the conformal class of an Einstein metric (justconsidering the conformal metric ¯ g = φ − g D, Φ ,T determined by the equation φ ′′ ( x ) − φ ( x ) h ′′ ( x ) = 0). Remark 7.1.
Set Σ = R with usual coordinates ( x , x ) and put T ∂ = ∂ , T ∂ = 0. For any any smooth function h ( x ) consider the deformation tensorΦ given by Φ ( x , x ) = − h ′′ ( x ) (the other components being zero). Then,the non zero Christoffel symbols of g D, Φ ,T are given byΓ = − x ′ = − Γ ′ ′ , Γ ′ = − h ′′ ( x ) x ′ , Γ ′ = − h (3) ( x ) = − Γ ′ . Hence a curve γ ( t ) = ( x ( t ) , x ( t ) , x ′ ( t ) , x ′ ( t )) is a geodesic if and only if¨ x ( t ) = 0 , ¨ x ( t ) − x ′ ( t ) ˙ x ( t ) = 0 , ¨ x ′ ( t ) + 2 x ′ ( t ) ˙ x ( t ) ˙ x ′ ( t ) + h (3) ( x ( t )) ˙ x ( t ) = 0 , ¨ x ′ ( t ) − h ′′ ( x ( t )) x ′ ( t ) ˙ x ( t ) − h (3) ( x ( t )) ˙ x ( t ) ˙ x ( t ) = 0 . Thus x ( t ) = at + b for some a, b ∈ R and¨ x ( t ) − a x ′ ( t ) = 0 , ¨ x ′ ( t ) − h ′′ ( at + b ) a x ′ ( t ) − h (3) ( at + b ) a ˙ x ( t ) = 0 . ¨ x ′ ( t ) + 2 a x ′ ( t ) ˙ x ′ ( t ) + h (3) ( at + b ) ˙ x ( t ) = 0 , Now the first two equations above are linear and thus x ( t ) and x ′ ( t ) areglobally defined. Finally, since ¨ x ′ ( t )+2 a x ′ ( t ) ˙ x ′ ( t )+ h (3) ( at + b ) ˙ x ( t ) =0 is also linear on x ′ ( t ), one has that geodesics are globally defined.Then it follows from Theorem 4.3 that ( T ∗ R , g D, Φ ,T , f = h ◦ π ) is ageodesically complete steady gradient Ricci soliton, which is conformallyEinstein by Theorem 6.1.7.2. Nilpotent Riemannian extensions with non recurrent base.
Let( T ∗ Σ , g D, Φ ,T , f = h ◦ π ) be a non-trivial Bach-flat steady gradient Riccisoliton as in Theorem 4.3. Further assume that the Ricci tensor ρ D is nonsymmetric, i.e., ρ Dsk = 0 (equivalently ∂ D Γ = 0 as shown in the proof ofProposition 3.3). Then it follows from Theorem 5.1 that ( T ∗ Σ , g D, Φ ,T ) isnot half conformally flat.Theorem 6.1 shows that ( T ∗ Σ , g D, Φ ,T ) is conformally Einstein if and onlyif there exists a positive φ ∈ C ∞ (Σ) with dφ ◦ T = 0 such that φ b Φ +2(Hes Dφ + φ ρ Dsym ) = 0. Hence it follows from Theorem 4.3 that Hes Dh = φ Hes Dφ , which means (2 φ ′ φ − h ′ ) D Γ = 2 φ ′′ φ − h ′′ . Taking derivatives withrespect to x and, since ∂ D Γ = 0, the equation above splits into2 φ ′ φ − h ′ = 0 , and 2 φ ′′ φ − h ′′ = 0 , which only admits constant solutions. Summarizing the above one has thefollowing: Let (Σ , D, T ) be an affine surface with non-symmetric Ricci tensor(i.e., ρ Dsk = 0 ). Then any Bach-flat gradient Ricci soliton ( T ∗ Σ , g D, Φ ,T , f = h ◦ π ) is neither half conformally flat nor conformally Einstein. Acknoledgments.
It is a pleasure to acknowledge useful conversations onthis subject with Professor P. Gilkey.Research partially supported by projects GRC2013-045, MTM2013-41335-P, MTM2016-75897-P and EM2014/009 with FEDER funds (Spain).
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Monatsh. Math. (1964), 175–184. (E. C.-L.) Conseller´ıa de Cultura, Educaci´on e Ordenaci´on Universitaria,Edificio Administrativo San Caetano, 15781 Santiago de Compostela, Spain(E. G.-R.) Faculty of Mathematics, University of Santiago de Compostela,15782 Santiago de Compostela, Spain(I. G.-R.) Faculty of Mathematics, University of Santiago de Compostela,15782 Santiago de Compostela, Spain (R. V.-L.) Department of Mathematics, IES de Ribadeo Dionisio Gamallo,Ribadeo, Spain E-mail address ::