aa r X i v : . [ m a t h . DG ] J a n NEUTRAL BI-HERMITIAN GRAY SURFACES.
Wlodzimierz Jelonek
Abstract.
The aim of this paper is to give examples of compact neutral 4-manifolds(
M, g ) whose Ricci tensor ρ satisfies the relation ∇ X ρ ( X, X ) = Xτg ( X, X ). Wepresent also a family of new Einstein bi-Hermitian neutral metrics on ruled surfacesof genus g > This paper I dedicate to Professor Kouei Sekigawa on his sixtieth birthday.
0. Introduction.
In the present paper we are concerned with the classof neutral semi-Riemannian 4-manifolds (
M, g ) whose Ricci tensor ρ satisfies thecondition(*) ∇ X ρ ( X, X ) = 13
Xτ g ( X, X )where τ is the scalar curvature of ( M, g ). These class of manifolds was introducedby A. Gray ( see [G],[Be]). For general facts and some results concerning neutral 4-manifolds we refer to [K], [Pe],[A], [D],[M-L]. In [M] the condition for the existenceof two opposite almost complex structures on 4-manifolds are studied. Note that in[A] (p.187) it is stated that there are no known neutral Hermitian-Einstein metricson ruled surfaces except trivial ones. In this note we give among others manynew, non-trivial examples of such manifolds. The main subject of the paper isthe construction of new Einstein and Gray neutral metrics on ruled surfaces. Themethods however are very similar to those used by me in [J-1]-[J-4].In our papers [J-2]-[J-4] we have described Riemannian metrics g on compactcomplex surfaces ( M, J ) such that (
M, g ) satisfies the condition ∗ and has J -invariant Ricci tensor. In particular we have given a complete classification ofbi-Hermitian Gray surfaces on ruled surfaces of genus g ≥ g = 0. We denote by g also theRiemannian metric but it should not cause any misunderstandings. The technics MS classification:53C05,53C20,53C25. The paper was partially supported by KBN grant 2P03A 02324. Typeset by
AMS -TEX WLODZIMIERZ JELONEK used in [J-3] can be applied to describe a large class of neutral bi-Hermitian Graysurfaces. The equations, describing such surfaces whose Ricci tensor has exactlytwo real eigenvalues such that the corresponding two dimensional eigendistributionsare space-like and time-like, are described by the same equations as in [J-3],[J-4].In that way we present a large class of explicit examples of neutral bi-HermitianGray surfaces. In particular we get a large class of Einstein neutral bi-Hermitiansurfaces on ruled surfaces of genus g >
1. Neutral A C ⊥ -surfaces. By an A C ⊥ - manifold (see [Be],[G]) we mean asemi-Riemannian manifold ( M, g ) satisfying the condition(*) C XY Z ∇ X ρ ( Y, Z ) = 2(dim M + 2) C XY Z
Xτ g ( Y, Z ) , where ρ and τ are the Ricci tensor and the scalar curvature of ( M, g ) respectivelyand C means the cyclic sum. In this paper, an A C ⊥ -manifold with neutral metricis also called a neutral Gray manifold. A Riemannian manifold ( M, g ) is an A C ⊥ manifold if and only if the Ricci endomorphism Ric of (
M, g ) is of the form
Ric = S + n +2 τ Id where S is a Killing tensor and n =dim M . Let us recall that asymmetric (1,1) tensor S on a semi-Riemannian manifold ( M, g ) is called a Killingtensor if g ( ∇ S ( X, X ) , X ) = 0 for all X ∈ T M and that a semi-Riemannian manifoldwhose Ricci tensor is a Killing tensor is called an A -manifold.On a semi-Riemannian manifold ( M, g ) a distribution
D ⊂
T M is called umbil-ical (see [J-3]) if D is non-degenerate (i.e. a metric g is non-degenerate on D ) and ∇ X X |D ⊥ = g ( X, X ) ξ for every X ∈ Γ( D ), where X |D ⊥ is the D ⊥ component of X with respect to the orthogonal decomposition T M = D ⊕ D ⊥ . The vector field ξ is called the mean curvature normal of D . The foliation tangent to involutivedistribution D is called totally geodesic if its every leaf is a totally geodesic ( i.e ∇ X X ∈ D for any X ∈ D ) non-degenerate submanifold of ( M, g ). In the sequelwe shall not distinguish between D and a foliation tangent to D and we shall alsosay that D is totally geodesic in such a case.It is not difficult to prove exactly as in the Riemannian case (see [J-3]) thefollowing lemma: Lemma 1.
Let S ∈ End ( T M ) be a (1,1) tensor on a neutral semi-Riemannian4-manifold ( M, g ) . Let us assume that S has exactly two everywhere different eigen-values λ, µ of the same multiplicity 2, i.e. dim D λ = dim D µ = 2 , where D λ , D µ are non-degenerate eigendistributions of S corresponding to λ, µ respectively. Then S is a Killing tensor if and only if both distributions D λ and D µ are umbilical withmean curvature normal equal respectively ξ µ = ∇ µ λ − µ ) , ξ λ = ∇ λ µ − λ ) . We shall call a bi-Hermitian surface with bi-Hermitian Ricci tensor simply asbi-Hermitian surface If (
M, g ) is also an A C ⊥ manifold, then we call it a neutralbi-Hermitian Gray surface. Proposition 1.
Let us assume that ( M, g ) is a simply connected neutralGray 4-manifold and the Ricci tensor S ( ρ ( X, Y ) = g ( SX, Y ) ) has exactly two real EUTRAL BI-HERMITIAN GRAY SURFACES. 3 eigenvalues λ, µ and no null eigenvectors. Then there exist two Hermitian complexstructures J, ¯ J commuting with S and ( M, g ) is a bi-Hermitian neutral Gray surface.Proof. Analogous to [J-1]. ♦ We shall prove
Proposition 2.
Let ( M, g ) be a 4-dimensional neutral semi-Riemannian mani-fold. Let D be a two dimensional totally geodesic bundle-like and space-like foliationon M . Then M admits (up to 4-fold covering) two opposite Hermitian structures J, ¯ J such that J |D = − ¯ J |D , J |D ⊥ = ¯ J |D ⊥ . The nullity distributions of both J, ¯ J contain D .Proof. To prove the first part of the Proposition it is enough to show that on M there exists a Killing tensor with eigendistributions D , D ⊥ . Since the foliation D istotally geodesic and bundle-like it follows that ∇ X X ∈ Γ( D ) (resp.) ∇ X X ∈ Γ( D ⊥ )if X ∈ Γ( D ) (resp.) X ∈ Γ( D ⊥ ). Consequently a tensor S defined by SX = λX if X ∈ D SX = µX if X ∈ D ⊥ where λ = µ are two different real numbers is a smooth Killing tensor. We canassume (up to 4-fold covering) that the distributions D , D ⊥ are orientable. Letus denote by J the only almost Hermitian structure which preserves D , D ⊥ andagrees with their orientations. We define ¯ J by J |D = − ¯ J |D , J |D ⊥ = ¯ J |D ⊥ . FromProposition 1 it follows that both J, ¯ J are Hermitian. Now let { E , E } be a localorthonormal frame on D . Then ∇ J ( E , E ) + J ( ∇ E E ) = ∇ E E . It follows that ∇ J ( E , E ) ∈ D and consequently ∇ E J = 0. Analogously ∇ E J =0 . ♦ Now we give a theorem, whose proof is analogous as in the Riemannian case (see [M-S],[B],[S]).
Theorem 1.
Let us consider the manifold U = ( a, b ) × P , where ( P, g ) is a3-dimensional semi-Riemannian A -manifold (a principle S bundle p : P → Σ )over a Riemannian surface (Σ , g can ) of constant sectional curvature K , with an A C ⊥ -metric (1.5) g = dt + f ( t ) θ − g ( t ) p ∗ g can , where g = θ − p ∗ g can , f, g ∈ C ∞ ( a, b ) and θ is the connection form of P . Thenthe metric g on U extends to a smooth A C ⊥ metric on the ruled surface M whichis a C P -bundle over Σ and such that U is an open and dense subset of M if andonly if the functions f, g ∈ C ∞ ( a, b ) satisfy the conditions:(a) f ( a ) = f ( b ) = 0 , f ′ ( a ) = 1 , f ′ ( b ) = − , (b) g ( a ) = 0 = g ( b ) , g ′ ( a ) = g ′ ( b ) = 0 . Remark.
Let us note that the metric (1.5) induces a semi-Riemannian metric on M if the functions f, g satisfy: WLODZIMIERZ JELONEK (i) f is positive on ( a, b ) , and f is odd at a and b , i.e. f is the restriction of afunction f on R satisfying f ( a + t ) = − f ( a − t ) , f ( b + t ) = − f ( b − t );(ii) g is positive on [ a, b ] and even at a and b which means that g is the restrictionof a function g such that g ( a + t ) = g ( a − t ) , g ( b + t ) = g ( b − t );The proof is similar to the description of the metric in polar coordinates (see[Be] Lemma 9.114 and Theorem 9.125.)Note that it is an easy exercise to prove that functions f, g satisfying the ODEcharacterizing A C ⊥ -metrics together with the initial conditions ( a ) , ( b ) extend torespectively odd and even (with respect to the points a, b ) real analytic functionsaround a, b . The fact that the conditions ( a ) , ( b ) of Theorem 1 imply (i) and (ii) isbased on the following elementary Lemma. Lemma 2.
Let us assume that the function Q is real analytic in a closed interval [ x , x ] , Q is positive in ( x , x ) and Q ( x ) = Q ( x ) = 0 , Q ′ ( x ) = 0 , Q ′ ( x ) = 0 .Let a real function φ ( t ) satisfy an equation (1.6) φ ′′ = 12 Q ′ ( φ ) , φ (0) = x , φ ′ (0) = 0 , Then ( φ ′ ) = Q ( φ ) and φ is a real analytic and periodic function φ ∈ C ω ( R ) satisfying conditions φ (0) = x , φ ( l ) = x where l is the first positive real numbersuch that φ ( l ) = x . Also φ is an even function at t = 0 and t = l .Proof. The first part of Lemma is well known (see [B],p.445, Lemma 16.37). If φ satisfies equation (1.6) then ( φ ′ ) = Q ( φ ) where Q is a primitive function of Q ′ . From the boundary conditions Q = Q . It is enough to show that φ is even at t = 0 (for t = l the proof is similar) which is equivalent to φ (2 k − (0) = 0 for every k ∈ N . We prove by induction with respect to k that for every F ∈ C ω ( R ) thefunction ψ ( t ) = F ( φ ( t )) is even at t = 0. If k = 1 then ψ ′ (0) = F ′ ( φ (0)) φ ′ (0) = 0.Let us assume that the result holds for 2 k −
1. Let us write Q ′ = Q . We get ψ (2 k +1) = ( ψ ′′ ) (2 k − = ( F ′′ ( φ )( φ ′ ) + F ′ ( φ ) φ ′′ ) (2 k − = ( F ′′ ( φ ) Q ( φ ) + F ′ ( φ ) Q ( φ )) (2 k − = X (2 k − p !(2 k − − p )! (( F ′′ ( φ )) ( p ) Q ( φ ) (2 k − − p ) + ( F ′ ( φ )) ( p ) ( Q ( φ )) (2 k − − p ) ) . Thus ψ (2 k +1 (0) = 0. It follows that F ( φ ) is even at 0 for every F . If F = id thenwe get that φ is even at 0. ♦ Using this lemma one can show that in every case considered below in the paperfunction g is even and f is odd. Both functions are non-negative and g is positivewhich follows from the formula on g .
2. Neutral bi-Hermitian Gray surfaces.
In this section we shall constructbi-Hermitian neutral metrics on ruled surfaces M k,g of genus g . The ruled surface( M k,g , g ) is locally of co-homogeneity 1 with respect to the group of all local isome-tries of ( M k,g , g ) and an open, dense submanifold ( U k,g , g ) ⊂ ( M k,g , g ) is isometricto the manifold ( a, b ) × P k where ( P k , g k ) is a 3-dimensional semi-Riemannian A -manifold (a principal circle bundle p : P k → Σ g ) over a Riemannian surface(Σ g , g can ) of constant sectional curvature K ∈ {− , , } with a metric(2.1) g f,g = dt + f ( t ) θ − g ( t ) p ∗ g can , EUTRAL BI-HERMITIAN GRAY SURFACES. 5 where g k = θ − p ∗ g can and θ is the connection form of P k such that dθ = 2 πk p ∗ ω ,where the de Rham cohomology class [ ω ] ∈ H (Σ g , R ) defined by the form ω is anintegral class corresponding to the class 1 ∈ H (Σ g , Z ) = Z . Let θ ♯ be a vector fielddual to θ with respect to g k . Let us consider a local orthonormal frame { X, Y } on(Σ g , g can ) and let X h , Y h be horizontal lifts of X, Y with respect to p : M k,g → Σ g (i.e. dt ( X h ) = θ ( X h ) = 0 and p ( X h ) = X ) and let H = ∂∂t . Let us define twoalmost Hermitian structures J, ¯ J on M as follows JH = 1 f θ ♯ , JX h = Y h , ¯ JH = − f θ ♯ , ¯ JX h = Y h . Proposition 3.
Let D be a distribution spanned by the fields { θ ♯ , H } . Then D is a non-degenerate totally geodesic foliation with respect to the metric g f,g where g is a non constant function. Both structures J, ¯ J are Hermitian and D is containedin the nullity of J and ¯ J . The distribution D ⊥ is umbilical with the mean curvaturenormal ξ = −∇ ln g . Let λ, µ be eigenvalues of the Ricci tensor S of g f,g corre-sponding to eigendistributions D , D ⊥ respectively. Then the following conditionsare equivalent:(a) There exists D ∈ R such that λ − µ = Dg ,(b) There exist C, D ∈ R such that µ = Dg − C ,(c) λ − µ is constant,(d) ( U k,g , g f,g ) is a neutral bi-Hermitian Gray surface.Proof. The first assertion of Proposition 3 is a consequence similar to Propo-sition 3 in [J-2]. Note that ∇ λ = HλH, ∇ µ = HµH . Consequently tr g ∇ S = ∇ τ = ( Hλ + Hµ ) H . On the other hand one can easily check that tr g ∇ S =2( µ − λ ) ξ + HλH . Thus ∇ µ = 2( λ − µ ) ∇ ln g. Now we prove that (a) ⇒ (b). If (a) holds then ∇ µ = 2 Dg ∇ gg = D ∇ g . Thus ∇ ( µ − Dg ) = 0 which implies (b).(b) ⇒ (a). We have − ∇ gg µ − λ ) = ∇ µ = 2 Dg ∇ g, and consequently ∇ g ( ( µ − λ ) g − Dg ) = 0 which is equivalent to (a).(a) ⇒ (c). We have λ − µ = Dg and consequently ∇ µ = 2 Dg ∇ g = D ∇ g . Thus ∇ λ = ∇ ( µ + Dg ) = 2 D ∇ g and ∇ λ − ∇ µ = 0 which gives (c).(c) ⇒ (a). If ∇ λ = 2 ∇ µ then ∇ λ = 4( λ − µ ) ∇ gg . Consequently ∇ λ − ∇ µ =2( λ − µ ) ∇ gg and ∇ ln | λ − µ | = 2 ∇ gg = 2 ∇ ln g , which means that ∇ ln | λ − µ | g − = 0or λ = µ on the whole of M . In the last case we obtain an Einstein metric. It followsthat ln | λ − µ | g = ln D for some D ∈ R + or λ − µ = 0 (D=0), which is equivalent to(a).(d) ⇔ (c). This equivalence follows from [J-3]. ♦ Theorem 2.
On any ruled surface M k,g of genus g with k > there exista one-parameter family of neutral bi-Hermitian A C ⊥ -metrics { g x : x ∈ (0 , ǫ s ) } , WLODZIMIERZ JELONEK where ǫ s > depends only on g and k , which consists of neutral bi-Hermitian Graymetrics on M k,g .Proof. Note that for the first Chern class c (Σ g ) ∈ H (Σ g , Z ) of the complexcurve Σ g we have the relation c (Σ g ) = χα , where α ∈ H (Σ g , Z ) is an indivisibleintegral class and χ = 2 − g is the Euler characteristic of Σ g . Let us write s = k | χ | if g = 1 and s = k if g = 1. Then it is easy to show using O’Neill formulas fora semi-Riemannian submersion (see [ON],[B],[S]) that the manifold ( M k,g , g ) withthe metric g given by (2 .
1) has the Ricci tensor with the following eigenvalues : λ = − g ′′ g − f ′′ f , (2.2a) λ = − f ′′ f − f ′ g ′ f g + 2 s f g , (2.2b) λ = − g ′′ g − f ′ g ′ f g − ( g ′ g ) − s f g − Kg , (2.2c)where λ , λ , correspond to eigenfields T = ddt , θ ♯ and λ corresponds to a two-dimensional eigendistribution orthogonal to T and θ ♯ . Note that the equationswe have got are practically the same as equations for Gray manifolds obtainedin the Riemannian case. The only difference is that we now have − K instead of K , so the cases K >
K <
M, g ) ∈ A C ⊥ is a bi-Hermitian Gray surface then λ = λ = λ , and if wedenote µ = λ , Proposition 3 b implies an equation(2.3) µ = Dg − C for some D, C ∈ R . Since λ = λ we get(2.4) f = ± gg ′ p s + Ag . Using a homothety of the metric we can assume that A ∈ {− , , } . In the case A = 0 we get a neutral K¨ahler metric and these metrics on compact complexsurfaces we shall describe in section 4. So we restrict our considerations to the case A ∈ {− , } . Now we introduce a function h such that h = s + Ag . Note thatim h ⊂ ( − s, s ) if A = −
1, and im h ⊂ ( s, ∞ ) if A = 1. Then g = p | s − h | . Let usintroduce a function z such that h ′ = p z ( h ). Note that(2.5) f = h ′ and f ′ = 12 z ′ ( h ) . It follows that equation (2.3) is equivalent to(2.6) z ′ ( h ) − z ( h ) s + h h ( s − h ) = 4 ǫh + D ( s − h ) h − C ( s − h ) h , EUTRAL BI-HERMITIAN GRAY SURFACES. 7 where ǫ = − sgn KA ∈ {− , , } . It follows that z ( h ) = (1 − ( hs ) ) − ( − ǫ ( hs ) − Ds hs ) + ( Ds − Cs hs ) +(2.7) +(2 Cs − Ds )( hs ) − ǫ + Cs − Ds + Es hs ) . Let us denote again C = Cs , D = Ds , E = Es and let(2.8) z ( t ) = (1 − t ) − ( − ǫ (1+ t )+ D ( − t + t − t − C ( − t +2 t +1)+ Et ) . Write(2.9) P ( t ) = − ǫt − D t + ( D − C t + (2 C − D ) t + Et − ǫ + C − D. Then z ( t ) = P ( t )1 − t . Note that z ( h ) = z ( hs ) and z ′ ( h ) = s z ′ ( hs ). In view of Th. 1.we are looking for real numbers x, y ∈ R , x > y such that z ( x ) = 0 , z ′ ( x ) = − s, (2.10a) z ( y ) = 0 , z ′ ( y ) = 2 s, (2.10b)and z ( t ) > t ∈ ( y, x ). Note that equations (2.10a) are equivalent to − ǫx − D x + ( D − C x + (2 C − D ) x − ǫ + C − D + Ex = 0(2.11a) − ǫx − D x + 4( D − C x + 2(2 C − D ) x + E = − s (1 − x ) . (2.11b)Equations (2 .
11) yield D = 5( − E − s − ǫx + 3 Ex − sx − ǫx + 2 sx )2( − x ) x (1 + x )(15 + 10 x − x ) , (2.12a) C = 3(5 E + 10 s + 80 ǫx + 30 sx − Ex + 5 Ex − sx − ǫx + 2 sx )2( − x ) x (1 + x )( − − x + x )(2.12b)Solving in a similar way equations (2 . b ) one can see that there exists a function z satisfying the equations (2.10) if( x + y )( − ǫ ( − x + x + 5 y + 2 x y − xy − y )(2.13) + s (5 + 2 x y + 2 xy + 3 y + 3 x + x y − xy )) = 0 , where x > y , x, y ∈ ( − ,
1) in the case A = − x, y ∈ (1 , ∞ ) in the case A = 1.Using standard methods one can check that in the case of the genus g ≤ K = 4 or K = 0) the only solutions of (2.13) giving a positive function z are x = − y ∈ (0 , g ≥ K = − x = − y (see [J-3]) there is an additional family of solutions with ǫ = − WLODZIMIERZ JELONEK s ∈ (0 , η ) on M k,g where η = (15 + 4 √ q (13(10 √ − ≃ . .. . Onecan check that there exist families of solutions with − < y < x < k ∈ N suchthat k < ( g − η . In fact it is not difficult to check that if g > F = { ( x, y ) : − < y < x < } and H = { ( x, y ) : 1 < y < x } the function G ( x, y ) = − ǫ ( − x + x + 5 y + 2 x y − xy − y )+ s (5 + 2 x y + 2 xy + 3 y + 3 x + x y − xy )is negative and positive somewhere on the boundaries of both of F and H for s < g ≤ G is positive on both F and H . We have G x ( x, y ) = − ǫ ( − x + 4 xy − y ) + 2 s (3 x y + y x + 3 x − y + y ) ,G y ( x, y ) = − ǫ (5 + 2 x − xy − y ) + 2 s ( x + 3 y + x y − x + 3 x y ) . The equations G x = 0 , G y = 0 are equivalent to − ǫ ( − x + 4 xy − y ) + 2 s (3 x y + y x + 3 x − y + y ) = 0 , ( x + y )(10 ǫ ( y − x ) + s ( − x + 3 xy + y )) = 0 . Consequently for ǫ = − s ∈ [2 , η ) G attains its negative minimum in F on theline x = − y . It follows that if g ≤ x = − y ∈ (0 , , E = 0 and ǫ = 1 or ǫ = 0.These solutions are also valid for g ≥ x = − y both for ǫ = − s ∈ (0 , η ) (then − < y < x <
1) and for ǫ = 1 and s ∈ (0 ,
2) (in this case there exist solutions with 1 < y < x ). Now we give explicitformulas for the case x = − y ∈ (0 , P ( t )(2.14)= 1 x (15 − x − x + x ) (( t − x )( s ( −
15 + 10 x − x + t (10 + 12 x − x )+ t ( − − x + x )) + 4 ǫx ( x ( − x ) − t (3 + x ) + t (5 + 2 x + x )))) . Thus P (0) = − ǫx ( x −
5) + sx (15 − x + 3 x )15 − x − x + x and, since lim x → + P (0) x = s >
0, there exists ǫ s > P ( t ) > t ∈ (0 , x )for all x ∈ (0 , ǫ s ). In fact in the case ǫ = − ǫ s is the first positiveroot of the polynomial − x ( x −
5) + s ( −
15 + 10 x − x ) and ǫ s = 1 if ǫ = 1.Note also that in both cases ǫ s = 1 if s ≥
2. Now the function z ( t ) = − t ) P ( t )is positive on ( − x, x ) , x ∈ (0 , ǫ s ). If x ∈ (0 , ǫ s ) then there exists a solution h :( − a, a ) → ( − sx, sx ), where a = lim t → sx − Z t dh q z ( hs ) , of an equation h ′ = r z ( hs ) , EUTRAL BI-HERMITIAN GRAY SURFACES. 9 such that h ( − a ) = − sx, h ( a ) = sx, h ′ ( − a ) = h ′ ( a ) = 0 , h ′′ ( − a ) = 1 , h ′′ ( a ) = −
1. Itfollows that functions f = h ′ , g = √ s − h are smooth on ( − a, a ) and satisfy theboundary conditions described in Th.2. Consequently the metric g x = dt + f ( t ) θ − g ( t ) p ∗ g can , on the manifold ( − a, a ) × P k extends to the smooth metric on the compact ruledsurface M = P k × S S which is a 2 − sphere bundle over Riemannian surface Σ g .Note that g ( − a ) = g ( a ) = s √ − x . ♦ Theorem 3.
On the surfaces M ,g = CP × Σ g where g ≥ there exists aone-parameter family { g α : α > } of bi-Hermitian A C ⊥ -metrics. The Ricci tensor ρ = ρ α of (Σ , g α ) is bi-Hermitian and has two eigenvalues, which are everywheredifferent.Proof. Analogous to [J-3]. Let us write g ′ = p P ( g ). Then(2.15) g P ′′ ( g ) − P ′ ( g ) g − P ( g ) + 16 + 6 Cg = 0 . Consequently(2.16) P ( g ) = Ag + Bg + Cg + 4 , where A, B ∈ R are arbitrary. Now let D = 1 and let us consider the equations (weare looking for unknown real numbers A, B, C and ( x, y ) where 0 < y < x ) P ( y ) = 0 , P ( x ) = 0 , (2.17) P ′ ( y ) = 2 , P ′ ( x ) = − . Then y = α − α +3 α +1) α (2 α + α +2) , x = αy = α − α +3 α +1)(2 α + α +2) where α >
1. Note that x, y >
C > , A, B <
0. Let us consider an equation ( P = P α , h = h α dependon the parameter α > d hdt = 12 P ′ ( h ) , h ′ (0) = 0 , h (0) = y = 4( α − α + 3 α + 1) α (2 α + α + 2) . This equation is equivalent to (if t ∈ D = { t ≥ h ′ ( t ) ≥ } )(2.19) dhdt = p P ( h ) , h (0) = y = 4( α − α + 3 α + 1) α (2 α + α + 2) . It follows that P = P α , where α >
1, has exactly two positive roots { x, y } and P ( t ) > t ∈ ( x, y ). Note that equation (2 .
18) admits a smooth periodic solution h defined on the whole of R and such that im h = [ x, y ]. Now it is easy to checkthat λ = − Bh − C and µ = − Bh − C . The tensor ρ − τ g is a Killing tensorwith eigenvalues C, Bh + C corresponding to D , D ⊥ respectively. Then we obtaina one parameter family of bi-Hermitian A C ⊥ -metrics { g α : α > } on M ,g . ♦
3. Einstein neutral bi-Hermitian surfaces.
Let us note that the solutions( M k,g , g x ) with D ( x ) = 0 correspond to Einstein neutral surfaces. Let us recallthat(3.1) D = 5( − s − ǫx − sx − ǫx + 2 sx )2( − x ) x (1 + x )(15 + 10 x − x )Consequently there exist Einstein metrics in the family of Gray metrics ( M k,g , g x )if and only if the equation(3.2) − s − ǫx − sx − ǫx + 2 sx = 0has a real root x ∈ (0 , g = 0 wehave ǫ = 1 and s ∈ N and that equation (3.2) does not have any solution in (0 , g = 1. In the case g > s = kg − > ǫ = −
1. Inthat case equation (3 .
2) has a real root in (0 ,
1) if and only if s ∈ (0 , Q ( x ) = − s − ǫx − sx − ǫx + 2 sx . Then Q ′′ ( x ) = − s + 2 ǫx − sx ) . Consequently Q ′′ ( x ) < ,
1) for ǫ ≥
0. In the case where ǫ = −
1, since Q ′′ (0) = − s < Q ′′ (1) = 48 >
0, the equation Q ′′ ( x ) = 0 has exactlyreal one root, say α s . (In fact, α s = s ( √ s + 1) − .) Then, we have Q ′ ( α s )(=16(2 − (1 + s )( √ s + 1) − > < ) s < ( √ / ( √ < s < < ( √ / ( √ Q ′ ( x ) ≥ Q ′ ( α s ) >
0, and hence Q ( x )is monotone increasing, in (0 , Q (0) = − s < Q (1) = 32 − s , itfollows that Q ( x ) has exactly one root in (0 ,
1) if s ∈ (0 , s/ge
2, then, since 3 + 6 x − x > x < x in (0 , Q ( x ) = 24 x + 8 x − s (3 + 6 x − x ) ≤ x + 8 x − x − x ) < x + 8 x − − x + 4 x = 12( x − x + 2 x −
1) =12( x − x + 2 x −
1) = 12( x − x − x + 1) < . Thus the equation Q ( x ) = 0 has exactly one real root x ∈ (0 ,
1) if and only if s ∈ (0 , s = kg − it follows that the real root x ∈ (0 ,
1) exists if and onlyif k ∈ { , , , .., g − } . Note that for D = 0 the polynomial P is even and ofdegree 4 with exactly one positive root. Consequently we get an Einstein metricon M k,g for all k ∈ { , , .., g − } . Thus we obtain on the ruled surfaces M k,g ofgenus g > g − M k,g for k ∈ { , , ..., g − } .Note that for g = 2 we obtain only one metric corresponding to the RiemannianEinstein Bergery-Page metric on the first Hirzebruch surface F . Consequently wehave proved Theorem 4.
On a ruled surface M k,g of genus g ≥ for k ∈ { , , ..., g − } there exists an Einstein bi-Hermitian non-K¨ahler neutral metric. EUTRAL BI-HERMITIAN GRAY SURFACES. 11
Note that in the Riemannian case we have the Page metric which is only oneEinstein co-homogeneity 1 Hermitian, non-K¨ahler metric. (see [P],[B],[LeB],[S]) .
4. Neutral K¨ahler Gray surfaces.
The solutions with A = 0 analogously asin the Riemannian case give neutral K¨ahler Gray surfaces. We shall prove Theorem 5.
On a ruled surface M k,g of genus g ≥ for k ∈ { , , ..., g − } there exists bi-Hermitian K¨ahler neutral metric.Proof. Since A = 0 we get f = gg ′ s . Consequently from (2.3) we get an equation(4.1) − g ′′ g − g ′ g ) − Kg = Dg + C. Let us write g ′ = p P ( g ). Then we can rewrite (4.1) as(4.2) P ′ ( g ) + 4 g P ( g ) + Kg + Dg + Cg = 0 , with the boundary conditions(4.3) 12 yP ′ ( y ) = s, xP ′ ( x ) = − s, where g ( a ) = y < g ( b ) = x , P ( x ) = P ( y ) = 0 and P ( t ) > t ∈ ( y, x ).Consequently P ( g ) = − D g − C g + Eg − K . Now exactly as in [J-2] we seethat the solution satisfying the boundary conditions exists if K = − s +2) y + ( s − x = 0. This condition implies that s < , C = 0, D > y = q − s ) D , x = q s ) D and E = s − D <
0. In fact it is not difficult tocheck that a positive solution satisfying the boundary condition exists if and onlyif s ∈ (0 ,
2) hence for k < g −
1) and that the metrics corresponding to the same s with different D are homothetic. ♦ Note that the situation in the case of neutral 4-manifolds is again quite differentfrom Riemannian case where we have only one irreducible non-Einstein compactK¨ahler Gray surface (see [A-C-G],[J-2]).The author is very grateful to the referee for the valuable comments and pointingout a mistake in the first version of the paper.
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