Jean-Pierre Bourguignon A MATHEMATICIAN'S VISIT TO KALUZA ...

Rend. Sem. Mat. Univ. Poi. TorinoFascicolo Speciale 1989P.D.E. and Geometry (1988)Jean-Pierre BourguignonA MATHEMATICIAN'S VISIT TO KALUZA-KLEIN THEORYIn the first quarter of this century, Kaluza-Klein theory has been an attemptto unify Electromagnetism and General Relativity. It is stili of interesttoday because it has survived many changes in theoretical models in Physics.Our own interest in this theory originates in its strong geometrie structure,which undoubtedly explains its many reappearances. It also leads theway to the construction of many interesting examples of Riemannian metricshaving special curvature properties, the so-called Einstein metrics. Thesemetrics are defined on total spaces of bundles, i.e., on spaces having a locaiproduct structure, a feature which makes them interesting both on mathematica!and on physical grounds. The notion of bundle is indeed by nowaccepted as a major landmark in the quest for globality in Geometry, and wasalso recognized somewhat later as fundamental in Gauge Physics. Some ofthese examples have been obtained only recently, and, as a result, this subjectis connected with present day research.Ali along this article, we will be moving in the realm of Riemanniangeometry. More precisely, our point of view will be the following. On a givenn-dimensional manifold M (most of the time compact), we are looking formetrics which fìt this manifold as well as possible. For surfaces, Constantcurvature metrics are a perfect match. In higher dimensions (n > 3), thingscannot be so simple since space forms (i.e., manifolds admitting Constantcurvature metrics) do not by far exhaust ali possible difTerentiable types. Onefruitful idea (that mathematicians probably borrowed from physicists) comes

geometry of circle bundles as we explain in Section II. We move on in SectionIII to considering non-abelian versions of Kaluza-Klein theory, and their linksto gauge theories. Various interesting examples of Einstein metrics in theKaluza-Klein spirit will be given in Section IV. We dose this review by a sideby side comparison of concepts used by mathematicians and physicists in ourcontext.It is a pleasure to take the opportunity of this preface to thank theorgani zers of the meeting "Parfcia/ Differentìal Equations and Geometry" fortheir efficiency in setting up this conference, and for inviting us to report onthis topic for a mixed audience of analysts and geometers.145I. Kaluza-Klein theory : the early days.As was said in the introduction, Kaluza-Klein theory was an attemptto unify Electromagnetism and General Relativity. Let us begin by recallingthe most basic features of both of these theories.a) The mathematica! structure of ElectromagnetismIn 1873, James Clerk Maxwell proposed the set of equations knowntoday under his name and the basic concept of an ekctromagnetic fìeldu>. In thelanguage of exterior algebra (which Hermann Grassmann was just developingat the time of Maxwell), u; is an exterior differential 2-form over the space IR 4 ={x\x = (x° 1 x l i x 2 , a; 3 ), x* e IR} viewed as space-time. The 2-form w incorporatesthe electrical fìeld E and the magnetic fìeld B as follows(2) w = È A dx b + Y^ B^idx j dx kwhere the last summation is taken over even permutations {ij f k} of {1,2,3}.This point of view incorporates the later unification between the notions ofspace and time made by Special Relativity, and more fundamentally the introductionof a metric of signature (- + ++) over IR 4 due to Hermann Minkowskiafter earlier work by Hendrik A.Lorentz. In this setting, the Maxwell equationssay that{du = 0Su = iwhere d is the exterior differentiation operator, 6 its formai adjoint using thevolume element defined by the metric, and i denotes the current, a 4-vector

146having the charge density as time component, and the current density as spacecontent. In the vacuum, the second set of equations is of course Su = 0.Notice that the first part of the system incorporates the Faraday law,and the law expressing the absence of magnetic sources.In modem mathematica! terms, the electromagnetic field u in the vacuumis a harmonic 2-form.Moreover, if the space-time is contractible (as IR 4 is), or if we contentourselves with working locally, thanks to the Poincaré lemma one càn writeu = da where a is a differential 1-form. This result will prove of importancelater in our discussion.b) General RelativityIn 1913, A.Einstein (in [5] written jointly with Marcel Grossmann) proposeda new theory of gravitation which was very geometrie in nature. Init, he replaced the basic scalar field introduced by Newton to describe theeffect on the geometry of the distribution of mass by a Lorentzian metric l,a 10-component field, on a manifold viewed as space-time. In this paper, hefailed to find the proper field equations, but we feel that this jump was thecruciai step. As soon as one works in this context, the field equations are insome sense inevitable. The field equations, called the Einstein equations, foundby Einstein in 1915, state that(4) r t - ±« £ = Twhere again ri and st are respectively the Ricci and scalar curvatures of theLorentzian metric £, and T the stress-energy 2-tensor field which describes thePhysics of the situation (outside the effeets of gravity). (One should noticethe parallel with the Maxwell equations, where the left hand side has a stronggeometrie content whereas the right hand side describes the Physics). For thevacuum the right hand side vanishes, hence by taking the trace of equation(4) one obtains the Einstein vacuum equations(4') n=0of which equation (1) is a Riemannian generalization. For some time, Einsteinhimself considered the Lorentzian version of (1), calling A the cosmologica!Constant, but later, out of physical grounds, he viewed this modification as ablunder.The systems of Partial Differential Equations (1) and (4) are non-linear.Indeed, in a coordinate system («') in which the metric g can be expressed

nas g = ^2 g^dx^x*, the expression of the Ricci curvature r g is the given bynr g = ^ tijdx i dx i with147' ~ 2 1 4^ U**&c' ^ 0*«0s> ^d«' d^dx'J(5) *.'=i+ quadratic terms in ~ox(in this formula, (gV) denotes the inverse matrix of (gy)).The key difFerence between the Riemannian and the Lorentzian settingsis of course that in the first case the system is elliptic, and in the second itis hyperbolic. On the space of metrics, the left hand side of the Einsteinfield equatìon has a variational nature. It is indeed the difTerential of the totalscalar curvature functional S. Since this fact was discovered by David Hilbert,S is often referred to as the Einstein-Hilbert Lagrangian.e) The originai Kaluza-Klein theoryIn 1919, Theodor Kaluza proposed a 5-dimensional version of GeneralRelativity. Although at first sight the theory presented some interesting features,Kaluza's paper (cf. [7]) was only accepted for publication by Einsteinin 1921.Kaluza considered IR 5 with coordinates (j 0 ,^1,!: 2 ,^3,^) as model for anextended space-time, which can be projected by a map p onto the ordinaryspace-time IR 4 with coordinates (x° 1 x 1 tx 2 }x 3 ). The main trick, the so-calledKaluza Ansatz, is to consider a Lorentzian metric £ on IR 5 of the form(6) / = * B (pV+-*(a + «)® (a+ #))where : IR 4 —* IR is a function, a 6 IR,^ is a Lorentzian metric on IR 4 and ais a 1-form on IR 4 .Provided one takes = 1, the key fact is thenKALUZA'S DISGOVERY. - The Einstein vacuum equations for i contain theMaxwell equations for the electromagnetic field u = da, and the Einsteinequations fori with right hand side T taken to be the interaction term comingfrom the electromagnetic field w.

148This approach immediately raised the question of the physical meaningof the fìfth dimension corresponding to the coordinate £ that nobody had sofar observed. What also appeared akward was the necessity of putting = 1.Some years later, in 1926, at a time where Quantum Mechanics wasformulated, Oscar Klein (cf. [9]) took up the Kaluza model having in mindto exploit it in connection with the newly established quantum theory. He inparticular suggested that the extra-dimension be taken compact, i.e., a circle.The extended space-time M was then M = M x S 1 . Thanks to this reductionone can see the Kaluza Ansatz as the first term in a Fourier series expansionof the field with respect to the fifth variable.It is of interest to know that the 5-dimensìonal Lagrangian S(I) = / sgugJM(provided one assumes that the integrai makes sense) can be expressed as aneffective Lagrangian where, after performing an integration along the fìbresS 1 , the integrai is taken along the ordinary space-time A/, namely(7) S(Ì) =/(•*- \da\ 2 - U-'\d*?)vt ,6JM(Here, we have set a = -- as physicists usually do for dimensions reaósons). If we had brought in the fundamental constants of Physics measuringthe relative strengths of the gravitational and electromagnetic interactions,one would have seen that the size of the circle is to be taken of the order of10" 35 m, a good reason for not having yet observed the fìfth dimension.II. The Riemannian geometry of circle bundles.In this section we take up the Kaluza-Klein originai model from a mathematica!point of view. It is then convenient to consider only Riemannianmetrics. We begin with a proposition.PROPOSITION 1. - If(M t g) has a circle action by isometries with orbits ofthe same length, then ali orbits are geodesics, and the action can be made{ree by going, if necessary, to a covering of the circle acting.The proof, which is straightforward, relies on appropriately studyingthe covariant derivative of the vector fìeld V defined by the action.REMARK. - If we had inserted a non-constant function in the expression ofthe metric as Kaluza did in his first attempt, we would have forced the fìbres

not to be totally geodesie. Then, the geometrie description of the situationwould have been more involved. We will say more on that in the more generalsituation of general bundles that we address in section III.The whole geometry of the situation is then captured by the followingdata. It is first convenient to normalize g so that V has length 1. Onethen introduces the differential 1-form a dual to the vector field V via themetric g (its kernel is perpendicular to the orbits), and the space of orbitsM = M/S x which inherits a metric g since the bilinear form gjj defined on Mby g h = g - a ® a satisfìes Cv9h = 0. Of course, if p denotes the projectionfrom M to M, gn = p*g.As a result, one can view M as an S^-principal bundle over M on whicha is a connection form.One can also run the construction backwards, i.e., start from a Riemannianmanifold (M,g) y and consider an S^principal bundle p : M —* M over it.To any choice of an S 1 - connection with 1-form 5 on M, one can associate theRiemannian metric(8) g = p*g + a & .The fibres are automatically totally geodesie for the metric g by Proposition 1. Therefore, we proved the followingTHEOREM 2. - The space of S 1 -invariant Riemannian metrics on the totalspace of an S 1 -principal bundle for which the fibres are totally geodesie is in1-1 correspondance with the product of the space of Riemannian metrics onthe base by the space of S 1 -connections over the bundle.It is this fact that S. Kobayashi rediscovered in 1963 (cf. [10]). Ofcourse, what was of interest to him (and remains to us) is the description ofthe curvature of g.If we denote as above by V a unit vertical vector field, and by X and Xrespectively a tangent vector to M and its horizontal (i.e., «/-perpendicular toV) lift, we have the following basic formulas for r, the Ricci curvature of g,149(9) < f(X 1 V) = -6u 6u(X)[f(X,Y) = r(X,y)+ì W ou;(X } y)2where u is the curvature 2-form of the connection form & (i.e., p*u = rfd), 6denotes the codifferential operator on (M,g) y and the notation wou means

150the composition of u, viewed as a linear map, with itself (i.e., in a coordinatesystem, (wow),^ = J^W^ij).kìFrom (8), one easily sees that the formula for the scalar curvature of gis s = s — \\u\ 2 , hence that the formula for the Lagrangian S appears indeed asthe sum of the Lagrangian S plus the energy term coming from the connection.Hence, one can derive the precise version of Kaluza's discovery (alsothe content of Kobayashi's remark) from (8). Physicists like to see this asa superposition of energies for the extended system (often called minimalcoupling).THEOREM 3. - On the total space M of an S 1 -bundle p: M —• M, theRiemannian metric g = p*g + a® à associateci with the S 1 -connection form awith curvature u is Einstein for the Constant X if and only if:i) LJ is a harmonic 2-form with Constant length 2vÀ ;ii) r = Xg — -ut ow.CiREMARK. - Because of i), the Einstein Constant X is necessarily positiveunless the bundle is fiat in which case g is Rìcci-fiat. The equations we havebeen writing hold equally well in the Lorentzian case. We come up with thesame constraint on the sign of A provided the fìbres are space-like (as Kaluzaassumed). Hence, we see that the vacuum Einstein equations force the bundleto be triyial, and also, what makes matters worse, the electromagnetic field tovanish identically. This phenomenon is often referred to as the Kaluza-Kleininconsistency.Condition ii) does not imply that the base metric is Einstein. It isnevertheless under this special assumption that explicit éxamples were foundby Kobayashi in [10]. If r = Xg, then we must haveuow = 2(A — X)g ,so that, if A ^ 0, the curvature form must be non degenerate. This is ofcourse the case if (A/, g) is Kàhlerian with a Kàhler form being a multiple ofthe curvature form u of the bundle. A typical example of this situation isM = being then a generatorof i/ 2 (M,Z). In that case, the manifold M is a standard sphere 5 2m+1 . Theconstruction can be carried over with other Hermitian symmetric irreductiblespaces. The main draw-back of these éxamples is that the metrics obtainedturn out to be homogeneous isotropy irreducible, hence belong to a well knownfamily.

In the hands of McKenzie Wang and W. Ziller (cf. section IV and[14]), this construction provides an infinite family of new examples of Einsteinmetrics provided one does noi take the base metric to be itself Einstein.151III. Non-abeiian b un die theory.a) The Kaluza-Klein approach to gauge theoriesIn 1954, C.N. Yang and R.L. Mills proposed in [15] a variational approachas classical model for strong interactiòns (the ones responsible for thecohesion of the nucleus). Later, this model was also used to describe ci assi -cally weak interactiòns (connected to the /?-decay). This proposai was knownto be only a part of the story since, due to their ranges, these interactiònshave to be dealt with quantum mechanically.This variational theory turns out to be identical to the theory of connectionson G-bundles, as was (much) later recognized. The group G to beconsidered here is the group on invariance of the interactiòns, as of today, U\for electromagnetism, S£/ 2 for weak interactiòns, SU3 for strong interactiòns.An elementary particle subject to a given interaction being attached to anirreducible representation of the invariance group, these groups can be determinedexperimentally by letting interact different particles subject to a giveninteraction, and comparing the outcome of the interaction with the tables ofrepresentations of the presupposed symmetry group.A compact Lie group G being given, we consider p : M —• M a (7-bundle.The fìeld to be extremized in this setting is a G— connection a for the Yang-Mills functional yM defìned as(il) yM( a ) = \j |n°p^where Q a denotes the curvature of the connection a. The volume element v 9refers to a Riemannian metric g supposedly given on the space-time manifoldM. Due to the use of this model as a first perturbation term towards aQuantum theory, it is indeed a Riemannian metric that physicists pick.How does the Kaluza-Klein Ansatz interfer with this problematic? Thiswas first contemplated by B. De Witt in 1964, and later developed by A.Trautman and R. Kerner (cf. [8]). One more time, one can play the game ofusing the data to contruct a metric on the total space ofthe bundle. This canbe done by using the same recipe as before, namely by setting(12) 9=P*9 + 9

152where g is a chosen G-invariant metric on the model space of the fibers (necessarilya space where G acts) that can be extended as a bilinear form onthe whole of TM thanks to the connection. Indeed, for a tangent vector Z toM, we decompose it first into its horizontal and vertical components thanksto the connection, say Z = Z v + Zh, and set(13) RZ % Z) = 9(Tv(Z)>Tv(Z)) + HZ %ì Z m ) .The first remarkable fact is then that the G-bundles can be characterizedin terms of metrics like g that we just defined. We come to this a little later.Another fact of concern to us is of course the expression of the LagrangianS for the metric g. Anticipating a bit curvature calculations that we developlater, we have(14) S(5)= / Sv s = I (sop + s-\Q a \ 2 )v dJM JMwhere s,s and 5 denote respectively the scalar curvatures of the metrics g,gand g. In other wordsFACT 4. - The Hilbert-Einstein Lagrangian of General Relativity evaluatedon the extended space-time according to the Kaluza-Klein prescription is thesum of the Hilbert-Einstein Lagrangian of the base and fibers and of theYang-Mills functional of the connection.In the Physics literature, this fact is often referred to as minimal coupling.b) The Riemannian geometry of fibre bundlesA systematic study of the Riemannian geometry of fibre bundles wasconsidered only recently (see [13]) if one compares to the time of existence ofRiemannian geometry itself.In this setting, the basic notion is that of a Riemannian submersion,i.e., a map p : (M,g) —> (M,g) so that, at each point m of M, the restrictionof Trhp to the orthogonal space to its kernel at m is an isometry from themetric induced by g to (Tp^jAf,^^)). To stick to the discussion we startedfrom the Physics side, we will assume that p : (M,g) -*• (M,g) is a fibre bundlewhose fibre is denoted by F. Note however that completeness (a fortioricompactness) and connectedness of (M,g) ensure that property. (This is atheorem of R. Hermann). At this point, it is important to stress that it wouldbe inconsistent to assume that F is endowed with a Riemannian metric. Wecome back to this point later.

The key observation due to B. O'Neill is that two 3-tensor fields areenough to describe the geometry of the situation. They measure the difFerenceof D, the Levi-Civita connection of g, and its projection onto the verticaland horizontal subbundles respectively. If h and v denote the orthogonalprojections onto the horizontal and vertical spaces respectively, for tangentvector Z\,Z 2 to M, we have153(15) / A * lZ *= ^^1)^2)) + v &H*t) h ( z *)){T Zl Z 2 = h(D v(Zl) v(Z 2 )) + v(D v(Zl) h(Z 2 )) .The 3-tensor A measures the obstruction to integrability of this horizontaldistribution. It can be identified with the curvature of the connection.This can be done as follows. The 3-tensor A is defined over M, but one easilysees that A%Y = v([X,Y]), so that A gives rise to a 2-form on M takingits values in vector fields along the fibre. It can therefore be identified withQ°. The horizontal part of A, namely A%V for a vertical vector V, can berecovered from the previous one since g(A^Y,V) = -g(A^V f Y).The 3-tensor T measures the deviation of the fibres from being totallygeodesie. The importance of T is best captured by the following theorem.THEOREM 5 (R.HERMANN, cf. [6]). - If the fibres of a Riemannian submersionp : (M,g) —• (M,

154REMARK. - Before going on working under this assumption, let us remarkthat without it the Hilbert-Einstein Lagrangien S takes the vaine(i6) s(5) = / ( S o P + s- - MI 2 - |r| 2 -JMwhere N is the mean curvature vector of the fibre. We indeed have the reductionto (14) when the flbres are totally geodesie, Notice also that, whenone varies S among the metrics on M of the type we considered (namely,fìbered metrics with totally geodesie fibres), one does not get the Einsteinequation as Euler-Lagrange equation. This accounts for the so-called Kaluza-Klein inconsistency. The Einstein equations on M are indeed stronger thanthese equations, hence the fact that sometimes physicists drop some of theconditions that the Einstein equations imply. One then speaks of a truncatedequation.We can now come to the expression of the Ricci curvature for (M, g) interms of the geometrie data. For horìzontal vectors U, V, and horizontal liftsX>Y of tangent vectors X, Y to M, at a point m we have(17)?({/, v) = F(U, v) + £ g(K, ei ,u)m a eìte., v)f(X,V) = 5(W)x,U)nf(x,Y) = r(x,y)-2£s(n3U-«SU).8 = 1where (e,) is an orthonormal basis of the n-dimensional vector space T m M.From these formulas, we draw the following conclusions of interest tous,THEOREM 6o - The Riemannian metric induced on the total space M of aconnected G-bundle p : M —• M defined by a G-invariant Riemannian metricg on the fibre F, a Riemannian metric g on the base and a G-connection a isEinstein with Constant X if and onìy if:i) the connection a is Yang-Mills and its curvature form Q a has Constantnorm;ii) the curvature form Q Q is connected to the Ricci curvature of the fibreby the relationna») J29(^i,e i )®mi,e i ì+r = yg,8,J = 1

and the fibre (F,g) has Constant scalar curvature;iii) the curvature form fì* is connected to the Ricci curvature of the baseby the relation(19) r -2^(1)-,., 0-,.) = \g»=iand the base (M,g) has Constant scalar curvature,PROOF. - The statements contai ned in i), ii) and iii) are obvious consequencesof the system (17) except for the constancy of |O a | 2 ,s and s.To establish these facts one needs only the following clever remark.Taking the trace of (19) gives the constancy of |Q a | 2 along a fibre, and thatsince s = s-f soT- |^a| 2 > by moving along a fibre again, s must be a Constant.By taking a trace of the first formula in the system (17), one obtainsthat (n -f p)À = s + \fl a \ 2 (if p denotes the dimension of F), hence that\Cl a \ 2 is also a Constant on M. The constancy of s then follows immediately.IREMARK. - Of course, as in the case of S^bundles, this does not imply thateither the fibre, or the base are Einstein. On some occasions this implicationhas been unduly claìmed in the Physics literature.Conditions (18) and (19) seem a priori rather diffìcult to satisfy. Insection IV we give a few examples of situations where this Kaluza-Klein approachdoes give new examples of Einstein metrics. One more time, the mostconvenient case to study is when both the base and the fibres are Einstein.COROLLARY 7. - Any Einstein metrìc arising on the total space of a bundlevia a Kaluza-Klein type construction has a positive Constant A unless thebundle is fiat, and its total space is covered by the product of a covering ofthe base and of the fibre both endowed with Einstein metrics having the samenegative Constant.PROOF. - As for the case of S^-bundles the proof relies again on the formulagiving the Ricci curvature in purely vertical directions, i.e., here on formula(18). If A is negative, then necessarily f is negative since the curvature contributionis positive. But, by a well-known theorem of S. Bochner, this impliesthat (F, g) has no infinitesimal isometries. Since we know that the curvatureQ a takes its values in the vector space of infinitesimal isometries, this forcesQ a to vanish, i.e., the connection a is fiat, and (F, g) is Einstein. As a consequence,the manifold M is covered by a product of the fibre and a covering of155

156the base. By using (19), orie sees that (M t g) is also Einstein. In the absenceof the curvature term, both g and g must have the same negative Constant.•IV. Some new examples of Einstein metrics obtained by bundleconstructions.a,) The canonica/ variation of a Riemannian submersion.As explained in the last Section, solving system (17) is not an easy task.We single out one case where the construction does provide new examples ofEinstein metrics.Given a Riemannian submersion p : (M,g) —» (A7,fl0, ^ * s possible tomake a si m pi e-mi nd ed change in the metric g which turns out to be geometricallymeaningful, namely to multiply uniformly the metric g in the fibres by afactor t 2 (t > 0). (If we refer to Kaluza's originai Ansatz, this means taking $to be a Constant different from 1). We denote the resulting metric on M by g t .For any t > 0, the map p : (M,jjt) —• (M,g) remains a Riemannian submersion.(Indeed, nothing has been changed about orthogonality between horizontaland vertical directions, and the induced metric on the horizontal spaces hasbeen kept the same.) Moreover, if the fibers of p are totally geodesie for g,they remain so for g t . This construction is often referred to as the canonica/variation of the Riemannian submersion.We can then ask the following (a priori peculiar) question. Is it possiblethat more t/ian one metric in the f&mily g t be Einstein? The answer of courseis contained in comparing the expression of the system (17) for the variousEinstein metrics g t . It is reasonable to assume that g = gì is Einstein withConstant A. Since r t ig = r 9 = f the metric g t will then be Einstein with ConstantÀ< provided(20) r = ^(À, - < 2 À)j(which follows from (18)) and(21) r =r^(À,-t 2 A)0(which follows from (19)).We therefore have

THEOREM 8 (L. BÉRARD BERGERY) - Let p : (M,g) -• (M,g) be a Riemanniansubmersion with totalìy geodesie fibres and base and fibre metrics gand g Einstein for positive constants X and À. The metric g is Einstein withConstant X ifand onìy ifi) the connection a is a Yang-MMs and its curvature Q° has Constantìength e and satisfìes157(22) ì 'f ! G/H is afibration. If we choose an Ada(#) invariant complement to the Lie algebra ofH in the Lie algebra of G, and on it an AdG(/0-invariant scalar product, weobtain a G-invariant Riemannian metric g on G/H, By proceeding similarly forthe pair (H,K), we obtain also an /Mnvariant Riemannian metric g on H/K.By taking the direct sum metric on the sum of the complements of the Liesubalgebras, we obtain a Riemannian metric g on G/K which, by a theorem ofL. Bérard-Bergery, is a Riemannian submersion with totally geodesie fibres.Many important geometrie examples fall under this category, such as theusuai Hopf fibrationsof spheres over projective spaces, and also the generalizedHopf fibrations of projective spaces over other projective spaces.Then, Theorem 8 applied to this family gives "exotic" examples of Einsteinspaces on simple manifolds. In particular, from the classical Hopf fibrations5 4,+3 —• HP*, one gets two Einstein metrics on (4g + 3)-spheres; from

158the Hopf fibration S 15 -+• Ì? 8 , a third one on 5 15 ; from the generalized Hopffibration CP 2?+1 —• HP q , two Einstein metrics on odd-dimensional complexprojective spaces.This construction turns out toexhaust ali homogeneous Einstein metricson spheres and projective spaces, as was shown by W.Ziller (cf. [16]).b) Other Riemannian submersions admitting Einstein metrics.Although the topic we discuss in this paragraph takes us away from ordinaryKaluza-Klein theory, it is worth being mentioned because the examplesit gives are geometrically interesting.Most constructions in this paragraph are due to L. Bérard Bergery (cf.[1]) after initial impetus given by a physicist, D. Page, who gave the simplestmember of the family we are going to describe as a Riemannian counterpartto the Taub-NUT metric, one of the standard models in modem GeneralRelati vi ty.We start from an S 1 -bundle P classified by an indivisible 2-cohomologyclass [a] dividing the first Chern class of a compact Kàhler-Einstein manifoldM (e.g., a Hermitian symmetric space). We set c\(M) = qa.We denote the fc-foldcover of P by P k . As one easily sees, P k is classifiedby ka. We then introduce M* the 5 2 -bundle over M associated with Pk forthe usuai 5 2 =C(J{°°}- (An alternative geometrie description of Af*, more inthe spirit of complex geometry, goes as follows. Take the complex line bundleover M whose first Chern class is ka; add to it a trivial bundle, and M k isthen the projective bundle of this rank 2 vector bundle).THEOREM 9 (L. BÉRARD BERGERY). - For k satisfying 1 < k < q, themanifolds Mk admit a non homogeneous Einstein metric with positive scalarcurvature.For the proof we just refer to [2] page 273. Let us just say it involvesputting a fibered metric on M* as a bundle over a closed interval with fìbresPk degenerating at the end points of the interval with fibres M. The metric isobtained as solution of a non linear system of differential equations involvingtwo unknown functions scaling the base and the fibre metrics of the S^bundlePk. Notice that this system is heavily over determi ned since, in order to obtaina smooth metric on Mk, one has to fix the values of the two functions and atleast of their first two derivatives at the boundary.The metric obtained stili retains an isometric group action in codimension1. (One often speaks of metrics of cohomogeneity 1.) This method has

een sophisticated by N. Koiso and Y.Sakane (cf. [11]) to provide Einsteinmetrics of higher cohomogeneity, in particular on Kàhler manifolds.To dose this paragraph, we mention that the basic example Mi obtainedfor M =(^P 1 (one then necessarily has k = 1) can be identified with the nontrivialS 2 -bundle over S 2 (this is easy via the second definition we suggested ofthe manifolds Mjt), or as the manifold obtained fromCP 2 by blowing up onepoint. It therefore admits Kàhler metrics although the metric we constructedis noi Kàhler. This is an example of a Kàhler manifold admitting no Einstein-Kàhler metric.e) An infinite f&mily of interesting S l -bundles with Einstein metrics.In this paragraph we take up the subject of Section II by describingan interesting family of Einstein metrics on S^bundles obtained by W. Zillerand M. Wang in 1987. Some of these examples were discovered at about thesame time by the physicists L. Castellani, IL. D'Auria and P. Fre, as parts ofmodels of supergravity.One of the latest avatars of Kaluza-Klein theory in Physics has indeedbeen supergravity. This is an attempt to incorporate the ideas of supersymmetriesinto gravitation theory. Let us recali that in recent years physicistshave been wandering about the possible existence of transformations associating bosons, represented by tensor fields, to fermions, represented by spinorHelds that they cali supersymmetric. These transformations would allow toorganize fundamental particles into a small number of families transformedinto one another by these supersymmetric transformations. The proper geometrica!framework appropriate to deal with these objects is not yet fìrmlyestablished.Supergravity took ground in 1978 when a viable Lagrangian for it wasproposed by E. Cremmer, B. Julia and J. Scherk (cf. [4]). It mixes metric data,with spinor data. For that purpose, there was good physical reasons to look forbundles whose total spaces are 11-dimensional, he., with 7-dimensional fibres.This dimension is indeed the smallest possible one on whìch acts transitivelythe group U\ x SU2 x 5C/3 incorporating the symmetries of ali known to dayinteractions. On the other hand, it is a theorem of W. Nahm (cf. [12]) thathigher dimensionai models would lead to ill-defìned theories from a quanticpoint of vieWoPhysicists therefore put a great emphasis on constructing metrics (infact mixed with more complicated fìelds such as a differential 3-forms) on11-dimensional manifolds fìbered over a 4-dimensional manifold with a 7-159

160dimensionai fibre. Following the recipe we have been using ourselves, theylooked for Einstein metrics on the fibres. For that purpose they studied extensively7-manifolds acted upon by U\ x SU 2 x St/3, hence considered 5 1 -bundlesoverCP 1 xCP 2 , a non irreducible Hermitian symmetric space.THEOREM (M. WANG, W. ZILLER, [14]). - Let (M it gi) he Kahler Einsteinmanifolds with positive Chern classes CÌ(MÌ) = ¢,-, a,- where the ai are indivisiblecohomology classes. Any non trivial bundle M over a product of Kahler-Einstein manifolds M = n*=i Mi, whose characteristic class in H 2 (M, Z) is anintegrai linear combination ofthe a.-'s carries an Einstein metric with positiveConstant as soon as TTI(M) is Unite._The most interesting feature of the family we just described is the richnessof examples of difFerent Einstein metrics it provides on manifolds havinga given topology. For that purpose, one needs to study carefully how the topologyis affected by the integrai linear combination used to contruct M. Tobe specific, let us consider a few cases over M =(CP 1 xCP'. The S l -bundle Mis then specifìed by two integers £\ and £ 2 expressing thè characteristic classof the bundle as ^i«i + £ 2 a 2 where ai and 02 are respectively the generatingclasses inCP 1 and 3,M is difFeomorphic to S 2 x S 2q+1 when £ 2 = ±1 and £\arbitrary, difFeomorphic to S 2 x S 2q+l if ii is even and £ 2 = ±1, and tothe non-trivial 5 2?+1 -bundle over S 2 if £\ is odd and i 2 = ±1.COROLLARY. - On S 2 x S 2 « +1 and on the non-trivial S 4 < +ì bundle over S 2 ,there exist infinitely non isometric Einstein metrics with a positive Constant.In fact, this contruction gives infinitely many homotopy types of manifoldsadmitting an Einstein metric in ali but a finite number of dimensions> 7. The richness of this family goes as far as allowing some of them in dimension7 to be homeomorphic without being difFeomorphic. More precisely,as shown by M. Kreck and S. Stolz, two S^-bundles M and M' over CP 1 xCP 2corresponding to the same class overCP 2 (for example 4a 2 ) are homeomorphicif and only of £\ = t\ mod (32), and difFeomorphic if and only if £\ = £[ mod(32.28). In fact these manifolds have 28 difFerent difFerentiable structures (aliobtained by taking connected sums with the 28 distinct 7-dimensional exoticspheres), and they turn out to be homogeneous spaces under the same group,solving in the negative a long standing question in group action theory./

To come back to our Riemannian geometrie motivations, let us mentionthat if we normalize the Einstein metrics that we described, say on S 2 x S 2 * +1 ,to have volume 1, then the Einstein constants of metrics of the previous familygo to zero, together with the lengths of the geodesie fibres. This shows thatfor the moduli space of Einstein metrics on a given differentiable manifoldòne should not expect compaetness theorems to hold without making somegeometrie assumptions preventing the metric from collapsing. This also showsthat the functional S we have been considering deos not satisfy condìtion (C),emphasizing the difficulty of finding Einstein metrics by purely analyticalmeans.161V. Comparing some mathematical and physical naturai assumptions.To dose this survey article, in which we stressed the numerous occasionson which mathematicians and physicists met by considering similar objects,we would like to show side by side some of the points on which they have asyet put difFerent emphasis.Some of these discrepancies may well be the basis of some future workaiming at bringing them closer.PhysicsRole of dimensionsSpecial solutions (Ansàtze)Lorentzian metricsNon compact (even non complete)Quantum levelLocai (Tendency to global)Troncated equations acceptedMovement towards geometrization(connected to the unifìcation drive)High concentration of workon one problem at a given momentSubjects change quicklyMathematicsNo counterpartGeometrie characterizations of objectsEuclidean metricsComplete (even compact)Classica] levelGlobalEquations have to be satisfìed exactlyMovement towards geometrization(fully acknowledged)More independenceLarger time Constant

162We hope to have by now convinced the reader that there is a lessonto be learned by mathernaticians in the development of Modem Physics, andthat, at any given moment, we should pay more attention to the questionscertain physicists raise. More important certainly, we should make sure thatwe provide our students with enough exposure to the basic concepts and ideascoming from Physics that, when time comes, they will prove able to establishthe necessary contacts and understand what the physicist, most of the timeworking next door, comes and asks.REFERENCES[1] L. Bérard Bergery, Sur de nouvelles variétés riemanniennes d'Einstein, PreprintInstitut Elie Cartan, Nancy (1982).[2] A. L. Besse, Einstein manifolds, Ergeb. Math. Springer 10, Berlin-IIeidelberg-NewYork, (1987).[3] J. P. Bourguignon, Mathematica! aspects of Kaìuza-Klein theory, to appear inC.B.M.S.-N.S.F. Regional Conf. Series.[4] E. Cremmer, B. Julia, J. Scherk, Supergravity theory in 11 dimensions, Phys.Lett. 76B (1978), 394-407.[5] A. Einstein, M. Grossmann, Entwurf einer alìgemeinerten Relativitàtstheorieund einer Theorie der Gravitation, Z. Math. Phys. 62 (1913), I. PhysikalischerTeil, 225-244; idem, II. Mathematischer Teil, ibidem, 244-261.[6] R. Hermann, A suffìcient condition that a mapping of Riemannian manifoldsbe a fibre bundle, Proc. Amer. Math. Soc. 11 (1960), 236-242.[7] T. Kaluza, Zum Unitàtsproblem der Fhysik, Sitzungber. Preuss.Akad. Wiss. Phys. Math. Klasse (1921), 966-972.[8] R. Kerner, Generalization of the Kaìuza-Klein theory for an arbitrary non-Abelian gauge group, Ann. Inst. H. Poincaré 9 (1968), 143-152.[9] O. Klein, Quantentheorie und funf-dimensionale Relativitàtstheorie, Z. Phys.37 (1926), 895-906.[10] S. Kobayashi, Topology of positively pinched Kàhler manifolds, Tohòku Math.J. 15 (1963), 121-139.

[11] N. Koiso, Y. Sakane, Non homogeneous Kahier-Einstein metrics on compactcomplex manifolds, in Curvature and Topology of Riemannian manifolds, ProcKatata 1985, Lect. Notes in Math. 1201, Springer, 165-179.[12] W. Nahm, Supersymmetries and their representations, Nucl. Phys.B315 (1978), 409-413.[13] B. O'Neill, The fundamenta! equations of a submersion, Michigan Math. J. 13(1966), 459-469.[14] M. Wang, W. Ziller, New examples of Einstein metrics on principal toroidaibundles, Preprint, (1987).[15] C.N. Yang, R.L. Mills, Conservation of isotopie spin and isotopie gauge invariate,Phys Rev. 96 (1954), 191-195.[16] W. Ziller, Homogeneous Einstein metrics on spheres and projective spaces,Math. Ann. 259 (1982), 351-358.163JEAN PIERRE BOURGUIGNONCentre de MathématiquesEcole PolytechniqueU.R.A. du C.N.R.S. D 0169F-91128 Palaiseau Cedex - (France)