bach tensor in ew geometry

j. reine angew. Math. 441 (1993), 99—113 Journal für die reine undangewandte Mathematik© Walter de GruyterBerlin - New York 1993

Einstein-Weyl geometry, the Bach tensor andconformal scalar curvature

By Henrik Pedersen and Andrew Swann at Odense

1. Introduction

A conformal manifold with compatible torsion-free connection is said to be Einstein-Weyl if the symmetrised Ricci tensor of the conformal connection is proportional to arepresentative metric. This is a conförmally invariant generalisation of the Einsteinequations of Riemannian geometry. The Einstein-Weyl equations have mainly been studiedin three-dimensions (see [16] and the references therein) but recently attention has movedto higher dimensions [7], [8], [12], [13]. Here we study the conformal scalar curvature ofEinstein-Weyl manifolds. It turns out that this behaves in a special way on compact Einstein-Weyl four-manifolds which enables us to calculate a conformal invariant, the Bach tensor,in these cases. We end with new examples of compact Einstein-Weyl manifolds the per-tinence of which will be explained below.

The conformal scalar curvature is essentially the function of proportionality in theEinstein-Weyl equations; a röle which is played by the scalar curvature on an Einsteinmanifold. In the Einstein case it is well-known that the scalar curvature is constant. Such aStatement cannot apply to the conformal scalar curvature äs this curvature may be multi-plied by an arbitrary positive function by changing the choice of representative metric.However, at each point, the conformal scalar curvature has a well-defined sign (positive,negative or zero). Compact three-dimensional Einstein-Weyl manifolds have been classifiedby Tod [16] and for each of the manifolds one sees that the conformal scalar curvature is ofconstant sign. We give an abstract argument which shows that the conformal scalarcurvature is also of constant sign on compact Einstein-Weyl four-manifolds. However, inhigher dimensions this need not be the case. Amongst the Einstein-Weyl structures con-structed in the last part of the paper will be found compact examples for which the con-formal scalar curvature changes sign.

Although the conformal scalar curvature need not be constant it can not be anarbitrary smooth function, since a result of Pedersen and Tod [l 5] shows that the conformalscalar curvature is an analytic function for a suitable choice of metric and local coordinates.This combined with a Bochner argument is used to show that a compact Einstein-Weyl

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1 00 Pedersen and S wann, Einstein- Weyl geometry

manifold with non-negative conformal scalar curvature has first Betti number at most lunless the manifold is Einstein. (For an Einstein manifold of non-negative scalar curvaturethe upper bound is the dimension of the manifold, see Besse [3].)

In the third part of the paper we study the Bach tensor on four-manifolds. It is knownthat this vanishes on Einstein manifolds and on self-dual manifolds. We show that forcompact Einstein -Weyl manifolds the Bach tensor vanishes exactly when M is locally con-formal to Einstein. For example, this implies that no self-dual conformal structure on theconnected sum of k copies of the complex projective plane CP(2) can be Einstein -Weyl

The final section is devoted to new examples of Einstein -Weyl manifolds. Until nowno examples of Einstein -Weyl four-manifolds were known which were not locally conformalto Einstein. We extend a construction of Berard Bergery [2] for Einstein metrics to con-struct compact Einstein -Weyl structures not locally conformal to Einstein on two-spherebundles over certain almost Kahler manifolds. An essential feature of the Einstein -Weylconstruction is that the analyticity result of Pedersen and Tod [15] implies that the con-formal class of Solutions with only continuous second order derivatives contains smoothstructures. The new examples in four-dimensions are conformal structures on the productof two-spheres S2 x S2 and the blow-up of CP(2) at one point. In higher dimensions weobtain Einstein -Weyl structures whose conformal scalar curvature is not single-signed onfor example S2 x S2 x S2, CP(2) χ S2 and the bundles P(0(l) Θ &) and P (0(2) Θ 0)over CP(2). The new examples are also of interest in complex Einstein -Weyl geometry.

Acknowledgement* We would like to thank C. R. LeBrun for useful comments andsuggestions.

2. Conformal scalar curvature and topology

Let M be an «-dimensional manifold (n ΐ> 3) with conformal structure [g]. A Weylconnection on M is a torsion-free connection D preserving the conformal class [g]. Thisimplies the existence of a l-form ω such that Dg = ω ® g. Let rD denote the Ricci curva-ture of D, then using g this has trace sD

9 the conformal scalar curvature of D. Note thatάω is conformally invariant and that SD has conformal weight — 2, that is if we replace gby /2g for some non-vanishing function / then SD is replaced by f~2sD.

A Weyl manifold M is said to be Einstein- Weyl if the Symmetrie part of the Ricci cur-vature rD is proportional to the metric g at each point of M. Unlike the Einstein case thefunction of proportionality is not in general constant. In terms of the Ricci curvature rv ofg the Einstein-Weyl equations are

v n — lrv+ —

where 9ω(Χ9Υ) = (Vxa>)Y+(VYQ))X+co(X)co(Y) and 2A = -SD+ d*ω+?—^\\co\\2

n 2(see Pedersen and Swann [13]). If M is compact then Gauduchon [6] showed that, up to



Pedersen and S wann, Einstein-Weyl geometry 101

homothety, there is a unique choice of metric g in the conformal class such that the cor-responding l -form ω is co-closed. For this choice of metric on a compact manifold thevector field ω* dual to ω is Killing (Tod [16]) and we may rewrite the Einstein-Weyl equa-tions in the form

— 2 „ „9 l n\ n-1-— ω ® ω .

Proposition 2.1. IfMisa compact Einstein - Weyl manifold ofdimension n and g is therepresentative of the conformal class for which ω is co-closed then the conformal scalarcurvature satisfies

Inparticular, on a compact Einstein-Weyl four-manifold the conformal scalar curvature, withrespect to any choice of compatible metric, is of constant sign.

Proof. Let sv denote the scalar curvature of g. Taking the trace of the differentialBianchi identity and using the Einstein-Weyl equations gives

dsv = -2V*rv

Since ω* is Killing we have 2ί/*νω,ω = — Δ||ω||2, so taking the co-differential yields

However, the Einstein-Weyl equations also imply that

Substituting this into the previous equation gives the claimed formula.

In dimension four we now see that SD is harmonic and hence constant. Changing themetric by a conformal factor multiplies SD by a positive function, so the sign of SD doesnot change. D

Note that SD is also of constant sign for compact Einstein-Weyl three-manifolds, butthe proof of this is a case by case examination after the classification of Tod [16].

An examination of the proof in Pedersen and Tod [15] that Einstein-Weyl three-manifolds are analytic yields the following analogue of a result of DeTurck and Kazdan[5] for Einstein manifolds.



102 Pedersen and Swann, Einstein-Weyl geometry

Theorem 2.2. Lei M" be cm Einstein-Weyl manifold (n 3) andsuppose locally thereis a choice of representative meine g with i»form ω and local coordinates {xt} such that g isofdass C2 and ω is ofdass Cl. Then locally there is another choice ofcompatible metric gwith corresponding i-form ώ and local coordinates {xj for which g and ω are analytic.Moreover, this is the case when the local coordinates are conformally-harmonic (that isD*Dxt = 0) and d®& = 0. Such coordinates always exist in a sufficiently small neigh-bourhood of any given point. α

Recall the following result which was proved by Pedersen and Tod in dimension threeand extended to all dimensions by Gauduchon.

Proposition 2.3 (Pedersen and Tod [15], Gauduchon [7]). Lei M" be a connected,compact Einstein-Weylmanifold. Ifthe conformalscalar curvature SD isnon-positive (SD g 0)but not identically zero then M is Einstein (with negative scalar curvature). If SD is iden-tically zero then M is locally conformally Einstein.

Note that if we choose the metric for whibh d*co = 0 then ω Ξ 0 implies M is Einsteinwhereas άω ^ 0 implies M is locally conformal to Einstein. In dimensions other than fourwe may give an alternative proof of the above Proposition by using analyticity. Gauduchon

2[7] shows that Δω = -s^co, so

and if SD is non-positive and not identically zero, analyticity implies SD is non-zero on adense open set and the result now follows by Integration over M.

Theorem 2,4. Lei M be a compact, connected Einstein - Weyl manifold and let g be therepresentative metric with i- form ω co-closed. Suppose ω is not identically zero. If SD isnon-negative (SD ^ 0) but not identically zero then thefirst Betti number bl(M) vanishes andifsD>0 everywhere then n^M) isfinite. IfsD^0 then b^M) = 1.

Proof. Rearrange the Einstein-Weyl equations so that

When ω is non-zero, the last bracket is positive semi-definite with kerael spanned by ω*.If SD is strictly positive then rv has only positive eigenvalues and Myers' Theorem impliesn^M) is inite. For the remaining cases, let α be a harmonic l -form. Integrating the innerproduct of the Weitzenb ck formula Δ« «= ?*?a + r?a with α implies Va =* 0 and hencerva »Β 0. Now, SD is analytic, so if it is positive somewhere then it is non-zero on a dense




open set U of M9 hence α is zero on U and by continuity on the whole of M. Thus b± (M)vanishes. When SD s Ο, α is forced to be a multiple of ω, but α and ω are parallel so thismultiple is a constant and b±(M) is 1. D

From the Euler characteristic of M and the Poincare-Hopf Theorem one now deduces:

CoroUary 2.5. IfM is a compact, connected Einstein - Weylfour-manifold with positiveconformal scalar curvature, then the \-form ω given by any choice of compatible meine gmust vanish somewhere. o

In Pedersen and Swann [13] various examples of Einstein-Weyl manifolds were con-structed. The only four-dimensional examples there arise s 2-torus bundles over CP(1),They have the property that ||ω|| is constant so the above Corollary gives an easy proofthat they are locally conformal to Einstein. The problem of constructing four-dimensionalexamples not locally conformal to Einstein will be addressed in the last section of thispaper.

The Situation for compact Einstein-Weyl four-manifolds may be summarised sfollows. Either the conformal scalar is negative SD < 0, in which case M is Einstein ofnegative scalar curvature; or SD Ξ 0 and M is locally conformal to Einstein and has

= l unless M is Einstein; or SD > 0 and M has finite fundamental group.

3. The Bach tensor

Let M be a four-dimensional Riemannian manifold. The Bach tensor B of (M, g) isdefined to be the gradient of the conformally invariant functional

M

where Wv is the Weyl curvature tensor of g. The Weyl curvature acts on a Symmetrie 2-tensor h via

(Wvh)(X, Υ) = Σ Wv(e^ X, ej9 Y)h(ei9 *,) ,u

where {ej is a local orthonormal basis for g. There is also a differential operator dv whichacts on h s

(d*h)(x, Y, z) = (v, A)(y, z) - (vr h)(x, z) ,with adjoint δν given by

, K) = -£(?„*)(«„ r, f)·i

One may now write the Bach tensor s



104 Pedersen and S wann, Einstein-Weyl geometry

(see Bach [1] and Besse [3]). Direct computation shows that B vanishes for Einstein metricsand hence for metrics locally confonnal to Einstein. One may also show that if g is seif-dual then it is an absolute minimum of the functional, so B is also zero in this case (seeDerdzinski [4]).

Writing the Weyl curvature in terms of the Riemann curvature, the Ricci curvatureand the scalar curvature gives the Bach tensor in the form

B = dvd V - V* Vrv - Vdsv

- 2C(rVv) + (3||r||2 -

(see Derdzinski [4]) where C(·, ·) denotes contraction with the metric over the first indicesand

(V*Vr)(JST, Y) = -Vei(Vxr)(ei, Y) .

Proposition 3.1. IfM is a compact Einstein- Weyl four-manifold and g is the represen-tative metric with ω co-closed then the Bach tensor is given by

Proof. Write / for the function - (SD + 2 ||co||2), so that rv becomes

Fix a point χ of M and choose a local orthonormal basis of tangent vectors {ej suchthat Vtfj vanishes at x. Let X = ej and F= ek for some j and k and write Vf for Ver Thenthe first two terms of the Bach tensor are

0* </vrv _ v* Vrv)(Z, 7) = -V, (V, rv)(7, X) + V^r*)^ X) + Vt(Vxrv)(ei9 7) .

Now, ω* is Killing, so άω = - Vca,

Vi(Virv)^Viω®Viω+-ei^(Vidω)®ω + -

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Pedersen and Swann, Einstein-Weyl geometry 105

and iν,(νζΓν)(^ 7) = - -

+ (eiXf)g(ei9Y)

= ~ {ω* ((ν*

Combining these formulae with Δω = -^ω, Δ||ω||2 = -sD||ci)||2 — \\άω\\2 and the fact

that SD is constant gives the required result. D

Corollary 3.2. The Bach tensor of a compact Einstein-Weyl four-manifold M is zeroif and only if M is locally conformally Einstein. Furthermore, when the latter is the case, ifM is connected and the conformal scalar curvature of M is not zero then M is Einstein.

Proof. If M is locally conformally Einstein then B is automatically zero. Assume Bvanishes. Since Λ2Τ*Μ = so (3)+ 0 so(3)_ and SO (3) acts transitively on the unit spherein so (3), we may choose a local orthonormal basis {e{} of 1-forms such that

ω = e l Λ £?2 + έ?3 Λ e ^ + e i Λ e2-e3/\ e4

for some functions f±. Direct computation then gives

C(da>9 άώ) = (/+ +/-)2(ef + φ + (Γ ~D2(el + ei) ·

There is still enough freedom left in the choice of basis for us to write ω = w+eifor functions w±. The vanishing of B now implies SD w+ w_ =0 and

-

^κ + ̂ -)- \*Dw2- -r r = o = ΐ^^κ +w2_)-/v- .Hence, by analyticity, either ω is zero and M is Einstein or SD = 0. In the latter case, άωis self-dual or anti-self-dual, so by Stokes' Theorem άω is zero and hence M is locallyconformally Einstein. D

Corollary 3.3. A compact, simply-connected, connected Einstein-Weyl four-manifoldwhose conformal class is self-dual admits an Einstein meine in the same conformal class.Thus, ifk>3no self-dual conformal class on kCP(2) is Einstein-Weyl. Also the conformal



106 Pedersen and S wann, Einstein- Weyl geometry

classes of self-dual structures on kCP(2), h> l, constructed by LeBrun [11] can not beEinstein-WeyL

Proof. If the conformal class is self-dual then the Bach tensor vanishes and anEinstein-Weyl structure gives an Einstein metric, since the manifold is simply connected. Theresult now follows from Pedersen and Tod [14] in the case of LeBrun's structures and bythe signature-Euler characteristic inequality of Hitchin [9] in the other cases. α

4. New examples

We now give a construction of some compact four-dimensional Einstein -Weyl mani-folds with positive conformal scalar curvature. Our method is an extension of work ofBerard Bergery [2] on Einstein metrics on S2-bundles (see also Besse [3]) and producesnon-trivial examples of Einstein -Weyl manifolds in all even dimensions greater than or equalto four. Note that since the conformal scalar curvature is positive the Ricci curvature ofthe representative metric with ω co-closed is positive. We start by recalling Berard Bergery'ssetting.

Let (B, gB) be a compact almost-K hler Einstein manifold of positive scalar curvatureand real dimension (n — 2) normalised so that

where r* is the Ricci curvature. Let ΩΒ be the Kahler form and assume there exists apositive constant q such that

where a*e H2 (B, ff$) is the image of an indivisible class α e H2 (B, Z). For any integer s,let P(s) denote the S1-bundle over B with characteristic class SOL*. Then P(s) carries ametric g (a, b) which gives a Riemannian submersion with base metric bgB and totallygeodesic fibres of length 2na. If S1 acts on S2 in the usual way fixing 0 and oo, then wemay define M (s) to be the S2-bundle over B associated to P(s), that is

where {0} χ (b, x) ~ {0} x (b, y) and {ί } x (b, w) ~ {(} x (b, z). Let t be the coordinate on[0, t ] and consider the metric

g = dt2 + g ( f ( t ) , h 2 ) - dt2 + AV+/«202 ,

where θ is the connection l -form on P (s) -+ B and pull-back signs have been omitted. Themetric g is of class C2 if /is C2 with /> 0 on (O,/),

/(O) -/"(O) -/(/)- /" (0 = 0 and .f(0)-l.-/'(O

(s^ Kazdan and Warner [10] for example). Note that unlike Berard Bergery [2] we take hto be constant rather than a function of t. This will cause no further loss of generality




provided the l -form defining the Weyl structure has a component proportional to dt, sinceif h is a function we may conformally rescale g and change { and the parameter t.

We obtain a Weyl connection on M (s) by defining

for some functions α (ί), β (t). This is of class C1 if α and /? are C1 and

<x(0) = j»(0) = /T (0) = ocGO = (f) = /»'(O = 0 .

Lemma 4.1. 7%e Einstein-Weyl equations on M (s) are equivalent to

f" n -2 , n -2 , ,-J + — α + — «2 = Λ ,

/" n -2 /' , n-2 2 , s2K2(n-V

2q

'-2^- +a = 0,

where A is the Einstein-Weyl function.

Proof. The Ricci tensor has been calculated by Berard Bergery [2] and gives theterms above not involving α or . It only remains to calculate 3>ω.

Let {71( . . . , Y„_ 2} be an orthonormal basis for gs. Let 'be the tangent vector whichis vertical for P(s) -» B and for which Θ(Χ) = l and let H = d / d t . Then

is an orthonormal basis for g. First we have

(Va(o)(H) = Η(ω(Η)) - ωφΗΗ) = α',

(νβω)(ΑΓ) = Η(ω(Χ)) - (o(VHX) = p-L t

since θ(νΗΧ) = g(VHX, X)lf2 = (Hg(X, JST))/(2/2) =/'//. Also, X and H commute so

(V,a>)(Jr) = -ωφχΧ) = α//',

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108 Pedersen and S wann, Einstein-Weyl geometry

If Υ is Yt for some i, then (VHco)(7) = 0, (Vr ω) (H) = 0 and

= -βθ(νχΥ+νγΧ) = -

s dB is a multiple of the pull-back of ΩΒ. Finally,

since g(VY. YJ9 H) = 0 and A" is a Killing vector field on P (s) so

g(VYt Yj + VYj Yt, X) = -g([Yt, XI, Yj) - g(Y(, [Yj, X])

vanishes. Thus

αΛ7χ'7χ)^7 + Τ>and all other components of 2ω vanish. D

Theorem 4.2. The manifold M (s) admits an Einstein-Weyl structure not locallyconformal to Einstein whenever \s\< q.

Proof. The last of the Einstein-Weyl equations suggests that we introduce thefunction γ = /f2 which then satisfies yf + ay = 0. If γ is zero then άω vanishes and weobtain the conformal class of the Einstein metrics constructed by Berard Bergery [2]. Wetherefore assume γ is non-zero and solve the last equation for a. Substituting this solutioninto the first three equations and introducing the constant k = s2n2/(q2h4) gives

(A,. Γ n~2y" . 3 ( K - 2 ) y ' 2 _ A(?*·*) 7 Λ ' ~ A 2~ ~~ '/ 2 y 4 y*

(4.2)

(4.3)

/" n-ly'f n-2f 2 jf 4

n k

' 2 f 2 , n 2 f2j _ A} J 1 4 f ί k Λ,

2 = A .

Subtracting (4.1) from (4.2) and dividing by (n — 2)/4 we obtain



Pedersen and Swann, Einstein-Weyl geometry 109

which implies

(4-4) /2 = -Τ77Γ "'2

for some constant ci. Note that if we assume cv ^ 0 then y must lie between

l l- νΊ — l/c2 + 4k) and - (q -f 1/c2 + 4fc) .L L

Define Γ to be y / 2 , then substituting (4.4) into twice (4.1) gives

(4.5) ^ = -f-

2 *

3(5 - «)*72 + 3 (n -

where dots indicate derivatives with respect to y. The homogeneous Version of (4.5) hasa solution

and substituting an arbitrary function t; times this expression into (4.5) gives

Hence, assuming c± ^ 0 and y > 0, we have

2n y("-2)/2 c2ynl2

'h2 (k 4- c ty - y2)3/2 " (fc + clT - y2)3/2 '

for some constant c2.

lNow we introduce φ defined by γ = A sin φ 4- 5, where ^4 = - |/c2 + 4k and

= c1/2. Then we may write

for some polynomial P2r and some constant fc2

Γ = ( T? Pn-2(S\"8 Journal f r Mathematik. Band 441

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110 Pedersen and Swann, Einstein· Weyl geometry

Note that P2r has degree (r — 1) and the coefficients are A~2 times non-zero homogeneouspolynomials in A and B of degree r.

In satisfying the boundary conditions, note that (4.1) implies that we need only show/(O) =/(/) = 0, /(O) = /(/) = 0 and /'(O) = l = -/'(/)· We have

(4.7) y'2~y<4~»v2(k + Cly-y2)3l2v and f2 = y~nl2(k + c,y -y2)i/2v .

Without loss of generality, we may assume y is increasing on [0, f~]. We will takey 00 = A + B and suppose y(0) = X. Our previous assumptions imply 0 < X < A + B andthat v is increasing on (0, ( ).

The condition f ( £ ) = 0 implies -7jPn_2(l) 4- c2Pn(l) = 0. However, the function

COS0JJ y (w~2) /2sec3<£äfy vanishes at (, so P„(l) = (A + B)Pn_2(i) and

2n l 4«2 h2 A + B

The derivative of / is given by

-y>+^(^^^

and the condition /'(/) = — l gives

c3 = »(^ + 5)<-

Now /(O) = 0 implies 0(0) = 0 so /'(O) = l gives

Note that this is always less than A + B. The condition that X be positive reduces tos2 < q2, hence the topological restriction on P (s). Now i?(0) = 0 implies

(48) l = (tan o g)(|/c?+4j2

where ß(x) is a non-zero rational function whose polynomial part has degree (n — 2). (Infour dimensions the right hand side is tan (A2c t/(2n)).) For fixed h this gives variousvalues of CA which in turn determine c2 and c3. Note that v has no zero in (0, /], so / ispositive on (0, /), and that y' is automatically zero at 0 and f.

We thus have a solution of the Einstein-Weyl equations with a metric of class C2 andl-form of class C1. Now Theorem 2.2 implies that this defines the conformal class of asmooth solution.



Pedersen and S wann, Einstein- Weyl geometry 111

A table of values of q for compact Hermitian Symmetrie spaces may be found in Besse[3]. In particular, for B = CP(n/2 — 1) one has g = w/2, so in four dimensions there areonly two possibilities.

Corollary 4.3. The quadric S2 x S2 and the blow-up CP(2) # CP(2) of CP(2) atonepoint admit Einstein -Weyl structures of positive conformal scalar curvature which are notlocally conformal t o Einstein. o

The positivity of the conformal scalar curvature in these two cases is guaranteed byProposition 2.1, however in higher dimensions we have:

Corollary 4.4. In even dimensions strictly bigger thanfour, there exist Einstein-Weylmanifolds whose conformal scalar curvature does not have constant sign.

Proof. In general the conformal scalar curvature is given by

3- = Λ-\ά*ω-η-=1\\ω\\*.n 2 4

In the examples constructed in the Theorem we have

For simplicity we only consider the case n = 6 and k = 0, but the argument may be easilyadapted to cover all the above examples with « ̂ 6. From (4.7) and (4.6) we see

i6 n cin l

So at γ = ct we have sD(f ) = 36/A2 which is positive.

Integrating (4.6) gives



112 Pedersen and S wann, Einstein- Weyl geometry

12where c3 = 4ct — -^ s"* *(!)· Now γ lies between X and c1? whereΛ

x-

Note that for fixed A, (4.8) has Solutions for arbitrarily large c1% For ci > 6/A2, we haveX< Cj/2, so we may put y = c t/2 in the expression for ^ to obtain

which for large ct is negative. (In general one obtains a polynomial in c± of degree(w — 4)/2 whose leading coefficient is negative if n ^ 6.) α

In all cases the manifold M (s) carries an almost complex structure / defined usingthe Standard structure on the fibre S2 together with that from the base manifold B. Thisalmost complex structure is compatible with the constructed conformal class and is inte-grable when B is Kahler. However this is not a Hermitian- Einstein -Weyl structure in thesense of Pedersen et al. [12] since / is not parallel with respect to the Weyl connection. Tosee this let Ω be the 2-form defined by / and g. On a Hermitian -Weyl manifold one hasάΩ == ω Λ Ω, but this is not satisfied in our case.

Proposition 4.5. IfBisa Kahler manifold and M (s) is s in the Theorem above, thenM (s) is complex and Einstein-Weyl. The complex structure is compatible with the conformalclass of the meine but not with the Weyl connection. o

References

[I] R. Bach, Zur Weylschen Relativit tstheorie und der Weylschen Erweiterung des Kr mmungstensorbegriffs,Math. Z. 9 (1921), 110-135.

[2] L. Βέτατά Bergery, Sur des nouvelles vari£t£s riemanniennes d'Einstein, Publications de PInstitut E. Cartan 4(Nancy) (1982), 1-60.

[3] A. L. Besse, Einstein Manifolds, Erg. Math. Grenzgeb. 10, Berlin-Heidelberg-New York 1987.[4] A. Derdzinski, Self-dual Kahler manifolds and Einstein manifolds of dimension four, Comp. Math. 49 (l983),

405-433.[5] D. M. DeTurck and L. Kazdan, Some Regularity Theorems in Riemannian Geometry, Ann. Scient. Ec. Norm.

Sup. 14 (1981), 249-260.[6] P. Gauduchon, La 1-forme de torsion d'une variete hermitienne compacte, Math. Ann. 267 (1984), 495-518.[7] P. Gauduchon, Structures de Weyl-Einstein, Espaces de Twisteurs et Varietes de Type S1 χ S3, Preprint.[83 P. Gauduchon, Structures de Weyi et theoremes d'annulation sur une variete conforme autoduale, Ann. Sc.

Norm. Sup. Pisa XVffl (1991), 563-629.[9] N.J. ffitchin, Compact four-dimensional Einstein manifolds, J. Diff. Geom. 9 (1974), 435-441.[10] J.L. Kazdan and F. W. Warner, Curvature functions for open 2-manifolds, Ann. Math. 99 (1974), 203-219.[II] C.R. LeBrw, Explidt self-dual metrics on €P(2) # ··· * CP(2), J. Diff. Geom. 34 (1991), 223-253.[12] H. Pedersen^ Y.S. Poon and A. F. Swann, The Einstein-Weyl Equations in Complex and Quaternionic Geo-

metry, Diff. Geom. Appl., to appear.[l 3] H. Pedersen and A. F. Swann, Riemannian Submersions, Four-Manifolds and Einstein-Weyl Geometry, Proc.

London Math. Soc. (3) €6 (1993), 381-399.[14] H. Pedersen and K.P. Tod, Einstein metrics and hyperbolic monopoles, Class. Quantum Grav. 8 (1991),

751*760.Brought to you by | University of California


Pedersen and S wann, Einstein- Weyl geometry 113

[15] H. Pedersen and K.P. Tod, Three-dimensional Einstein-Weyl geometry, Adv. Math. (1) 97 (1993), 74-109.[16] K.P. Tod, Compact 3-dimensional Einstein-Weyl structures, J. London Math. Soc. (2) 45 (1992), 341-351.

Department of Mathematics and Computer Science, Odense University, Campusvej 55,5230 Odense M, Denmark

Eingegangen 16. November 1992



bach tensor in ew geometry

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