Nilpotent Structures and Invariant Metrics on Collapsed Manifolds Author(s): Jeff Cheeger, Kenji Fukaya, Mikhael Gromov Source: Journal of the American Mathematical Society, Vol. 5, No. 2 (Apr., 1992), pp. 327-372 Published by: American Mathematical Society Stable URL: http://www.jstor.org/stable/2152771 Accessed: 27/09/2010 11:22
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http://www.jstor.org JOURNALOF THE AMERICANMATHEMATICAL SOCIETY Volume5, Number2, April1992
NILPOTENT STRUCTURES AND INVARIANT METRICS ON COLLAPSED MANIFOLDS
JEFF CHEEGER, KENJI FUKAYA, AND MIKHAEL GROMOV
CONTENTS
I. Introduction 0. Background 1. Statementof mainresults and outlineof theirproof II. Preliminaries 2. SmoothingHausdorff approximations 3. Equivariantand parameterizedversion of thetheorem on al- mostflat manifolds 4. NilpotentKilling structures on fibrations III. The nilpotentKilling structure and invariantround metric 5. Local fibrationof theframe bundle 6. Makingthe local fibrationscompatible 7. Makingthe local groupactions compatible 8. The inducedstructure and metricon thebase Appendix1. Local structureof Riemannianmanifolds of bounded curvature Appendix2. Fibrationisotopy
I. INTRODUCTION
0. BACKGROUND
Let Mn be a completeRiemannian manifold of bounded curvature,say IKI < 1. Given a smallnumber, c > 0, we put Mn = Wn(c) U Fn (c), where ?1n (e) consistsof thosepoints at whichthe injectivity radius of theexponential map is > e. The complementaryset, &"n (c) is called the e-collapsedpart of Mn. If x E 4 (n), r < E, thenthe metricball Bx(r) is quasi-isometric,with smalldistortion, to theflat ball B0(r) in theEuclidean space, Rn. Afterslightly Receivedby the editors May 15, 1991. 1991 MathematicsSubject Classification. Primary 53C20. The firstauthor was partiallysupported by NSF GrantDMS 840 596.
(? 1992American Mathematical Society 0894-0347/92$1.00 + $.25 perpage
327 328 JEFF CHEEGER, KENJI FUKAYA, AND MIKHAEL GROMOV adjustingthe boundaryof Wn(e), we obtaina set whosequasi-isometry type is determinedup to a finitenumber of possibilitiesby theratio, dia(7n (e))/c, where dia(7n (e)) denotes the diameter of f(V)* (Compare [C, GLP, GW, PI). In thispaper, we are concernedwith what can be said aboutthe E-collapsed part, Wn(,), for e = e(n) a suitablysmall constantdepending only on n. Roughlyspeaking, our main resultsshow that the essential features of thelocal geometryare encoded in the symmetrystructure of a nearbymetric. More precisely,any metric of boundedcurvature on Mn can be closelyapproximated byone thatadmits a sheafof nilpotentLie algebrasof local Killingvector fields thatpoint in all sufficientlycollapsed directions of Cn(E). This sheafis called thenilpotent Killing structure. A second sheafof nilpotentLie algebrasof vectorfields, called the nilpo- tentcollapsing structure will be discussedelsewhere. It playsa crucialrole in constructions,which collapse away all sufficientlycollapsed directions in the manifold(while keepingits curvaturebounded). The factthat twodifferent sheavesarise simply reflects the distinction between right and leftinvariant vec- tor fieldson a nilpotentLie group(compare Example 1.6 and the discussion precedingit). The firstnontrivial example of a collapsingsequence of Riemannianman- ifoldswas pointedout by Marcel Bergerin about 1962. Bergerstarted with the Hopf fibration,S -4 S3 - S2, whereS3 carriesits standardmetric. He observedthat if one multipliesthe lengthsof the fibresby c, while leaving themetric in theorthogonal directions unchanged, then the sectional curvature staysbounded independent of e, as E -E 0. Butas E -E 0, S3 moreand more closelyresembles S2 (equippedwith a metricof constantcurvature 4). In the process,the injectivity radius converges to zero everywhere. The firsttheorem on collapse characterizes"almost flatmanifolds" [G1]. These manifolds,XnA, have bounded curvature, say IKI < 1 , and are collapsed in thestrongest sense possible. Namely, the diameter satisfies, dia (Xn) < e(n). The theoremasserts that a finitenormal covering space, Xn, is diffeomorphic to a nilmanifold,A\N. Subsequently,by employingadditional analytic arguments, Ruh provedthat Xn itselfis infranil[R]. This meansthat the covering group of X"n-A X actsby affinetransformations with respect to thecanonical flat affine connection on the tangentbundle of "n'.Otherwise, put Xn is diffeomorphicto A\N, wherethe coveringgroup, A, actsby affine transformations, with respect to thecanonical connectionon N and theimage of theholonomy homomorphism is finite.By the canonicalconnection, we mean the one forwhich all leftinvariant vector fieldsare parallel.An importantby-product of Ruh'sproof is thestatement that the diffeomorphismbetween Xn and A\ N can be chosencanonically, given the geometryof Xn and a choiceof base point, X E Xn. Moreover,in this case, a canonicalleft invariant metric on N thatis actuallyinvariant under A can also be chosen. It is easy to see thatalthough most infranil manifolds admit no flatmetric, any such manifoldadmits a sequenceof metricswith IKI < 1, forwhich the NILPOTENT STRUCTURES AND INVARIANTMETRICS ON COLLAPSED MANIFOLDS 329 diameterbecomes arbitrarily small; see [G 1]. Thus,the above-mentioned results implythe existence of a criticaldiameter, if X' admitsa metricwith IKI < 1 and dia(Xn) < e(n), thenit admitsa sequenceof metricswith IKI < 1 and dia( Xn) ___0. The case of infranilmanifolds illustrates a basic point. Collapse can take place simultaneouslyon severaldifferent length scales and notjust on the scale of theinjectivity radius. Indeed, the simplest nonflat nilmanifolds (with almost flatmetrics) can be viewedas the total spaces of a nontrivialcircle bundles, whosebase spaces are isometricproducts of twocircles of lengthE and whose 2 fibreshave lengthe . This kindof inhomogeneousscaling is actuallyessential, in orderfor the curvature to remainbounded as E -- 0. The ideas on almostflat manifolds were extended along two rather different lines,in orderto studythe collapsingphenomenon in greatergenerality. The goal of thepresent paper is to combinethese two approaches. In [CG3, CG4], generalizingthe conceptof a groupaction, the notionof an actionof a sheafof groups was introduced.An F-structureis an actionof a sheaf of tori forwhich certain additional regularity conditions hold (" F9" standsfor "flat"). As in the case of a groupaction, an action of a sheafof groupsinduces a partitionof theunderlying space intoorbits. The main result of [CG4] assertsthe existence of an F-structureof positiverank (i.e., all orbits have positivedimension) on the sufficientlycollapsed part of a manifoldwith IKI < 1. Here, no assumptionis made concerningthe size of the manifold, whichmight even be infinite.In thisgenerality, the dimensionof the stalkof the F-structureis not alwayslocally constant. If not, the structureis called mixed;if so it is calledpure. The infinitesimalgenerator of thelocal actionof an F-structureis a sheafof abelianLie algebrasof vectorfields, which can be regardedas Killingfields for someRiemannian metric. For the F-structureconstructed in [CG4], thismetric can actuallybe chosenclose to the originalone. The Killingfields themselves point only in the "shortest"collapsed directions. As a consequence,this F- structuredescribes the local geometry of thecollapsed region only on itssmallest lengthscale, thatof the injectivityradius. This accountsfor the abelian (as opposedto nilpotent)character of thestructure. The existenceof an F-structureof positiverank does imposea globalcon- strainton thetopology of theunderlying space. For example,it impliesthat the Eulercharacteristic vanishes [CG3]. Example0.1. The needto considermixed F-structuresin cases wherediameter is not boundedis illustratedby themetric
dr + e d(R+rdO2+e- (R-r)d02 1 1~~~ on theset (-R, R) x S x S , (R >> 0) . By countingthe number of collapsed directions,it becomesclear thatin thisexample, the torithat act locallynear the ends are one-dimensional,while near the middle,a two-dimensionaltorus acts. Note however,that there is no completelycanonical way of choosingthe preciseset of pointsat whichthe transition takes place. Thereis also a converseto theexistence theorem for the F-structure.Namely, 330 JEFF CHEEGER, KENJI FUKAYA, AND MIKHAEL GROMOV associatedto everyF-structure of positiverank are sequencesof metricswith IKI < 1 , containingones thatare arbitrarilycollapsed, for which the action of thestructure is isometric(see [CG3]). This leads to theexistence of a "critical injectivityradius," which is analogousto the notionof "criticaldiameter" as mentionedabove. In theapproach due to thesecond author, the starting point is to considera manifold,Mn, with IKI < 1, which,as in the Bergerexample above, "to the nakedeye," closely resembles a lower-dimensionalmanifold Ym. Technically speaking,one requiresthat Mn is sufficientlyclose to yn in the Hausdorff distance(see [GLP]). The manifoldYn is assumedto have boundedgeometry but its diameterneed notbe finite.The conclusionis thatthere is a fibration zn-r - Mn ym whose fibre, znm, is an infranilmanifold. In case ym is a point,the assertionreduces to the theoremon almostflat manifolds (see [Fl] and ?2 of thepresent paper for details). Althoughthe contextof this fibrationtheorem might at firstseem rather special,it turnsout thatits equivariantgeneralization gives strong information on thestructure of arbitrarycollapsed regions of boundeddiameter. The reason is as follows.Suppose that for a givenmanifold, both the curvature tensor and its covariantderivative are bounded(the assumptionconcerning the covariant derivativeis actuallynot a seriousone, since by results of [BMR], [Shi],and [A], an arbitrarymetric can be approximatedby one forwhich this holds). Thenthe framebundle, FMn, equippedwith its naturalmetric, has boundedcurvature as well. If Un is a regionthat is sufficientlycollapsed relative to thesize of its diameter,one can showthat there exists ym as in thefibration theorem, such thatthe frame bundle, F Un, is sufficientlyHausdorff close to ym Moreover, in this case, the fibre,Z, of the fibration,Z -- FU n __Ym, is actuallya nilmanifold(and notjust infranil).(Ultimately, both assertions can be traced to thefact that an isometryof thebase space,which fixes a pointof theframe bundle,is the identitymap.) The fibration,z +n m FUn __ym,can be chosento be equivariantwith respect to theaction of O(n) on FUn . As a consequence,a partitioninto infranilmanifolds, in general not all of the same dimension,is inducedon Un . These "orbits"contain all collapseddirections, and so determineall possiblelength scales on whichcollapse takes place. The flat orbitsof the F-structurecan be thoughtof as lyinginside these nilpotent ones (in fact,they lie insidethe piecescorresponding to the centerof the nilpotent group);see [F3]. The fibrationtheorem is sharpenedin anotherdirection in [F2]. There,Ruh's theoremis used to obtaina smoothfamily of affineflat structures on thefibres. In fact,by Malcev's rigidity theorem, these are all affineequivalent to somefixed A\N; see Theorem3.7 and Proposition3.8. As a consequence,the structural groupof the fibrationreduces to thegroup of affineautomorphisms of A\N. The existenceof sucha reductionis a necessaryand sufficientcondition for the totalspace of a fibrationwith fibre diffeomorphic to A\N to collapseto thebase space keepingcurvature bounded. Thus thetheorem on the"critical diameter" generalizesto thefibration setting. We pointout thatthe resultof [F21 is obtainedwithout removing the de- pendenceon thebase pointin Ruh's construction.As a consequence,the con- NILPOTENT STRUCTURES AND INVARIANTMETRICS ON COLLAPSED MANIFOLDS 331 structionof [F2] is not G-equivariant.This point,which is importantfor the presentpaper, is dealtwith in ?3. We referto [CG3, CG4, FI-F3] forfurther background and examples.
1. STATEMENT OF MAIN RESULTS AND OUTLINE OF THEIR PROOF As alreadymentioned, the goal of thepresent paper is to synthesizethe two approachesto collapsethat were described in theprevious section. Thus, with- out assuminga bound on diameter,we will constructa nilpotentstructure, in generalof mixedtype, which is nontrivialon sufficientlycollapsed regions. The structureincorporates a descriptionof the local geometryon a fixedscale and notjust on the scale of the injectivityradius. It is called the nilpotentKilling structure.We will show thatits action is isometricfor a metricclose to the originalone. As mentionedearlier, there is also a second structure,called the nilpotent collapsingstructure. Although its orbitsare the same as thoseof the nilpotent Killingstructure, its constructionrequires a smallamount of additionalwork. This, togetherwith a descriptionof its role in collapsingwill be providedelse- where;see, however,Example 1.6 and compare[F2]. The existenceof a metricwhose symmetrystructure encodes the essential featuresof the geometrycan be made precisewithout reference to sheaves. However,the compatibility between this metric and thesheaf structure imposes a consistencycondition on the local symmetriesat neighboringpoints, which capturesthe purely topological aspect of thediscussion. Let (M, g) be a Riemannianmanifold. Let V c M be open and 7r: V V, a normalcovering with covering group, A. (1 1.1) Assumethat there exists a Lie group,H D A, withfinitely many com- ponents,and an isometricaction of H on V, extendingthat of A, suchthat (1.1.2) H is generatedby A and its identitycomponent, N, (1.1.3) N is nilpotent. A Riemannianmanifold (M, g) is called (p, k)-roundat p E M, if there exist V, V, H satisfying(1.1. )-(1.1.3) and the followingadditional condi- tions: (1.1.4) V containsthe metric ball, BP(p), of radius p centeredat p. ( 1.1.5) The injectivityradius at all pointsof V is > p . ( 1.1.6) (H/N) = t(A/A n N) < k . A metric,g, is called (p, k)-roundif it is (p, k)-roundat p, forall p. Modulo thechoice of (p, k), V has a normalcovering space withbounded geometryand a coveringgroup that is almostnilpotent. By (1.1.5), ifthe injec- tivityradius at p is < p/k, thenthe metric, g, has nontriviallocal symmetries near p; i.e., theorbit, H(p), of P E 7t l (p), underH, has positivedimension. If (M, g) is (p, k)-round,it followsthat the projectedorbit, 7r(H(J6)), containsthose sufficiently collapsed directions corresponding to shortgeodesic loops thatare homotopicallynontrivial in V. The (p, k)-roundmetrics con- structedin this paper actuallyhave a strongerproperty. Namely, the orbits 332 JEFFCHEEGER, KENJIFUKAYA, AND MIKHAEL GROMOV
(have smalldiameter and) actuallycontain all sufficientlycollapsed directions (again modulothe choice of k). One way of formulatingthis more precisely is to say thatthe orbit space has (p, k)-boundedgeometry in a suitablesense. (See Definition8.4 forthe conceptof (p, k)-boundedgeometry of the orbit space and Remark8.10; see also Appendix1.) Example1.2. Let G be a connectedLie groupand g be a leftinvariant metric. Then, for each discretesubgroup A of G, the quotientmetric, g, on the quotientspace M = A\G is (p, 1)-round,where p dependson g but is independentof A. Thisis a restatementof Zassenhaus's theorem, which asserts thatevery discrete subgroup of G generatedby smallelements is containedin a nilpotentsubgroup of G. (See [GLP, 8.44].) More generally,put M = GIK, forsome compactsubgroup K of G. Let g be a G-invariantmetric on M. Take a discretesubgroup A of G actingfreely on M. Then,by Zassenhaus's theorem,we concludethat the quotientmetric g on M = A\M is (p, k)- round,where p, k are independentof A. Let Vg denotethe Levi Civitaconnection of g. Our firstmain result is Theorem1.3 (Symmetrization).For all e > 0 and n E Z+, thereexists p > 0 and k e Z+ such thatif (Mn, g) is a completeRiemannian manifold with IKI < 1, thenthere is a (p, k)-roundmetric, gE, with
( 1.31) e-'g < g,< eEg ( 1.3.2) lVg- Vg-eI < 8 , (1.3.3) I(V&e)R)g I < c(n, i, e). One mightask whetherTheorem 1.3 can be strengthenedto the assertion thatin all instancesthere exists a (p, k)-roundmetric, g , such thateither p > p(n, e), k < k(n) or p > p(n), k < k(n, e). However,this turns out to be false;see Examples8.11, 8.12. Now let M be a smoothmanifold and let g be a sheafof connectedLie groupson M. Let g be theassociated sheaf of Lie algebras. Definition1.4. An actionof g is a (Lie algebra)homomorphism, h, of g into the sheafof smoothvector fields on M. A metric,g, is called invariantfor g if h(g) is a sheafof local Killingfields for g. Note thatif 7r:M -, M is a local homeomorphism,then there is an induced action, 7r*(h),of thepullback sheaf, r*(g). A curvec: (a, b) -* M is calledan integralcurve if c c V forsome open set, V, and c is everywheretangent to theimage, h(X), of somesection X E g(V) (i.e,. c'(s) = h(X)(c(s))). A set Z c M is called invariantif c c Z, forall such c with c n Z $ 0. The unique minimalinvariant set containingp is calledthe orbit, p, of p. Clearly,M is theunion of its orbits. Let h be an actionof a sheaf,n, of simplyconnected nilpotent Lie groups. Let g be a (p, k)-roundmetric and let No, V, etc.,be as in (l.l.1)-(1.1.6). NILPOTENT STRUCTURES AND INVARIANT METRICS ON COLLAPSED MANIFOLDS 333
Definition1.5. (n, h) definesa nilpotentKilling structure, for g, if forall p, we can choose H, V, V as follows.There is an invariantneighborhood U and normalcovering, U c V, suchthat: (1.5.1) lr*(h)is the infinitesimalgenerator of a (necessarilyunique) actionof thegroup, 7r*(n)(U), whose kernel, K, is discrete. No =7r*(n)(J)1K and theaction of NOIU is thequotient action. (1.5.2) For all W c U such that W n XG'(p) #&0, the structurehomomor- phism,lr*(n)(U) _ 7r*(n)(W)is an isomorphism. (1.5.3) The neighborhoodU and coveringU can be chosenindependent of p, forall p e 6 Clearly,the metric g in Definition1.5 is an invariantmetric for (n, h). A structureis calledpure if thedimension of thestalk is locallyconstant. Beforegoing to the nextexample, we will recallsome elementary(but con- fusing)facts. Let H be a Lie group.The diffeomorphismsof H obtainedby integrating rightinvariant vector fields are lefttranslations. Conversely, integrating left invariantvector fields yields right translations. In particular,given a leftinvariant metric on H, theright invariant fields are Killingfields but leftinvariant fields need notbe. Example1.6. Let N be a simplyconnected Lie groupand A c N a discrete subgroup. The quotient sheaf, n, of the constant sheaf, N x N -+ N, by the action, i: (no, n) -+ (An0 , An) has an action on A \ N inducedby leftmultiplication on N. The imagesheaf, h(n), is thesheaf of locallydefined rightinvariant vectorfields. Any left invariant metric on N inducesan invariant metricon A \ N forthe actionof thissheaf. It followsthat (n, h) definesa nilpotentKilling structure. Note,however, that the standardcollapsing construction for A\N involves inhomogeneousscaling of theleft invariant metric and henceof thelengths of the leftinvariant vector fields (see [BK]). The rightaction of N generatesthe leftinvariant fields and gives rise to the nilpotentcollapsing structure in this case. As indicatedabove, typically, the rightaction of N on A \ N does not giverise to a nilpotentKilling structure, because there is no metricthat it leaves invariant. Let M, g, g. be as in Theorem1.3. Our secondmain resultis Theorem1.7. The (p, k)-roundmetric, gE, can be chosensuch thatthere is a nilpotentKilling structure, 91, for g, whoseorbits are all compactwith diameter < EC. Remark1.8. The structuredescribed in Theorem1.7 can be viewedas gener- alizingthe system of fibrationswith nilpotent fibre and locallysymmetric base thatis knownto existnear infinity on a noncompactlocally symmetric space of finitevolume. Remark1.9. Theorem1.7 also providesan alternativemeans of obtainingan F-structureof positiverank on thecollapsed part of M. In fact,replacing each 334 JEFF CHEEGER, KENJI IFUKAYA, AND MIKHAEL GROMOV
Lie algebraof local sections,n(U), by its centerleads to the existenceof an F-structure. Open Problem1.10. Supposethat the original metric, g, in Theorems1.3 and 1.7 is Kahler,Einstein, etc. Can one take g, withinthe same category? In spirit,our constructionof thenilpotent Killing structure is similarto the constructionof the F-structurein [CG4]. But in carryingout the details,we use theframework of [F1-F3]. As in [CG4], we willfit together a collectionof locally defined pure structures. Initially,the collection is organizedin sucha waythat on nonemptyintersections of theirdomains, the structuresfit approximately, one inside another.Then usinga suitablestability property they are perturbedso as to fittogether exactly. In [CG4] thelocally defined pure structures are constructed on thescale of the injectivityradius, with the help of a resulton local approximationby complete flatmanifolds; see [CG4, ?3]. The stabilityproperty is a consequenceof the stabilityof compactgroup actions (in particularof torusactions); see [CG4, ?1]. Here,we workon lengthscales that, though small, may be arbitrarilylarge comparedto theinjectivity radius. We also workwith nilpotent groups, which typicallyhave no compactquotient groups. As a consequence,neither of the above-mentionedbasic toolsis available. Followingthe approachof [F1-F3], we will constructan O(n)-equivariant nilpotentKilling structure on the framebundle. This structureinduces the desirednilpotent structure on thebase. The requirementof maintaining0(n)- equivarianceat all stagesof the constructionintroduces some technicalprob- lems; see in particular?3. Theyare handledby averagingarguments, some of whichare verysimilar to thoseused to provethe stabilityof compactgroup actions. In addition,we use Malcev's rigiditytheorem for discrete cocompact subgroupsof nilpotentgroups, which serves as a partialreplacement for the stabilityof torusactions. The preliminarieson whichour construction is based are givenin ??2-4. The constructionis carriedout in ??5-8 and is organizedas follows. In ??5-6 we manufacturean 0(n)-equivariantcollection of local fibrations of the framebundle such thatif a pair of fibresfrom two of thesefibrations intersect,then one fibrecontains (or is equal to) theother. In ?7 flataffine structures are introducedon the fibres.Then, the fibrations are readjustedso thatfor a pairof fibresas above,the smaller is totallygeodesic in the larger,with respect to theseaffine structures. The affinestructures give riseto a nilpotentKilling structure on the framebundle. Those fibresthat are not containedin anyother are theorbits of thisstructure. Their images in the base are theorbits of thestructures we are seeking. In ?8 we checkthat the nilpotentstructure and metricon the framebundle do indeedinduce the desired objects on thebase. Beforegiving a more detailedsummary of the contentsof the paper,we explainthe following basic pointthat was alludedto in theprevious section. A manifoldis called A-regularif forsome nonnegativesequence A = {A }, we have NILPOTENT STRUCTURES AND INVARIANTMETRICS ON COLLAPSED MANIFOLDS 335
By the followingresult of Abresch[A] (see also [Ba, BMR, Shi]), we will ulti- mately(in ?8) be able to replacethe givenmetric in Theorems1.3 and 1.7 by one thatis A-regular,where
(1.11.2) Ai = Ai(n, e) and e is as in Theorems1.3 and 1.7. Thus, priorto ?8 we will alwayswork withmanifolds that are A-regularwith Ao = 1 . Theorem 1.12 (Abresch).On the set of completeRiemannian manifolds, (Mn, g), with IKI < 1, thereexists for all e > 0, a smoothingoperator, g -* S,(g) = k, such that (1. 12.1) e-eg < g < eeg (1.12.2) IV - VI < , (1.12.3) iV'Ri < Ai(n, e). Moreover,at p E Mn, thevalue of g dependsonly on gJBP(1). Finally,any isometryof g is also an isometryof g. Notethat since g(p) dependsonly on gIBp(I P , thecompleteness assumption can be removed,provided one staysaway from M \M (wherethe bar denotes metricspace completion). In ?2 we give a new proofof the fibrationtheorem of [F1], in the local equivariantform that we need. The projectionmap of thefibration is obtained by regularizinga Hausdorff approximation. Although there is some freedomin choosingthe scale on whichthe regularization is performed,for the application in ??5-8, it is importantthat the scale is chosento be thatof the injectivity radiusof thebase space. In ?3 we removethe dependenceon the base point in Ruh's theoremby averagingthe base pointdependent choices of the flatconnection that occur in the initialstep of the proof. Since this procedureand the remainderof Ruh's argumentsare bothcanonical, we immediatelyobtain an equivariantand parameterizedversion of his theorem. In ?4 we observethat the affine structures on thefibres introduced in ?3 allow us to definein a canonicalway, a pure nilpotentKilling structure on thetotal space of thefibration (we also constructa canonicalmetric that is invariantfor thestructure). Clearly, on a givenfibre we can speakof thelocal rightinvariant fields.But the issue is to definethese fields (locally) on the totalspace itself. For this,we notethat the affine equivalence that identifies neighboring fibres is unique up to elementsof the identitycomponent, Aff?(A\N), of Aff(A\N). One sees easilythat Aff?(A\N)C NR, the subgroupof Aff(N) consistingof righttranslations. Since NR acts triviallyon rightinvariant fields, it follows thatthere is a canonical1-1 correspondence between local rightinvariant fields (at nearbypoints) of neighboringfibres. In ?5 (usingthe resultsof ?2) we selecta systemof O(n)-equivariant,lo- cal fibrationsof the framebundle with almost flat fibres. On the intersections of theirdomains, these fit approximately, one insideanother. To achievethis requiresa suitablemechanism for picking out which directions are to be consid- eredcollapsed in cases whichmight otherwise appear to be ambiguous.Lacking 336 JEFFCHEEGER, KENJI FUKAYA, AND MIKHAEL GROMOV such a mechanism,we could windup withfibrations whose domains intersect, butwhose fibres do notsatisfy the above relationof approximatecontainment. Essentiallythe same pointhad to be dealt within [CG4, see ?5-b]. Here we employwhat amounts to a standarddevice from stratification theory. In ?6, by employingan inductiveargument that depends on the resultof Appendix2, the local fibrationsare modifiedO(n)-equivariantly, so thatthey fit,one insideanother, on theintersections of theirdomains. In ?7 we completethe constructionof the nilpotentKilling structure and invariantmetric on theframe bundle, using an inductiveargument like that of ?6. For theconstruction of theKilling structure, the main part of theinduction stepcan be describedas follows.Note that as a consequenceof ?3,each fibreof a local fibrationin ?6 is endowedwith a flataffine structure, affinely diffeomorphic to some A\N. Considera pair of fibrations,Y c 9 as in ?6 (i.e., the fibres of Y are containedin thoseof Y ). By ?4, the fibrescarry affine structures thatdetermine a local leftaction. However,the inclusion- c Y need not be compatiblewith the affinestructures. Using Malcev's rigiditytheorem, we finda unique subfibration,Y,' c Y, whosefibres are totallygeodesic for the affinestructure on the fibresof Y, and such thatfor each fibreof Y, there is a small motioncarrying it onto some fibreof 9'. Then,as in ?6, we find a small O(n)-equivariantdiffeomorphism that matches Y[ with Y' suchthat theaffine structure on i#7is carriedinto that of Y['. The remainingsections, ?8 and theAppendices, require no furtherdescription at thispoint. Withminor variations, we will employthe same notationas in [F3, see ?11 and [F4, see ?7]. In particular,we use: (1.13.1) d(., *): the distance function. (1.13.2) Bp(D) = {x E M I d(p, x) < D}. (1.13.3) TBp(D) = {v E TpM I lvl < D}. (1.13.4) 7rl(M, p; e) = {y: TBp(e) -f TBp(2e) I exppoy = expp}: thepseudo- fundamentalgroup. 7tI(M, p; e) has a pseudogroup structureand it acts on TBp(e) with TBp(e)/7r, (M, p; E) = Bp (E) . (1.13.5) dH(X, Y): the Hausdorffdistance between X and Y. When X and Y have isometricG-action, the G Hausdorffdistance is also denoted by dH (.I .) . (I1.13.6) z(e I a, b, ... ) denotes a positive numberdepending on the numbers in the parenthesesand satisfyinglime_O T(1 I a, b, c, ...) = 0, for each fixed a, b, c, .... (1. 13.7) If {Ai} is a positivesequence c(., A, *) willdenote a genericconstant depending on finitelymany of the A, (and possibly on some other parameters).
II. PRELIMINARIES
2. SMOOTHING HAUSDORFF APPROXIMATIONS In thissection we givea newproof of thefibration theorem of [Fl, F3] (see Theorem2.6). NILPOTENT STRUCTURES AND INVARIANTMETRICS ON COLLAPSED MANIFOLDS 337
A map h: X -. Y of metricspaces will be called an 3-Hausdorffapproxi- mationif forall x1, x2 (2.1.1) Id(x1,X2) - d(h(x1),h(X2))I < ; (2.1.2) the range of h is 3-dense. If G is a group acting by isometrieson X and Y, then h is called a G- 3- Hausdorffapproximation if in addition,for all g E G, x E X, (2.1.3) d(h(gx), gh(x)) < 3. Let V denote the completion of the metricspace V, and put (2.2) aV=V\V, (2.3) V,7= {v E V I d(v, aV) > t}I Now let Xn, Yj ( j < n ) be Riemannian manifolds such that for some sequence, {Ai}, with AO = 1, (2.4.1) Xn, Yj are {Ai}-regular. Assume in addition, that for all y E Y and some i < 1, (2.4.2) inj rady> min(F-1,d(y, aY)).
Let G act on AT, Y' by isometries. Let distances in X,,n,Yj be measured in XnT, Yj respectively.If the G-Hausdorffdistance, dH(X , Yj), satisfies
(2.4.3) dH(Xn, Yj)
(2.4.4) h: X( -* with h(Xan)D Y3, (see [F4, GLP, GrK]). In what follows it will be convenient simplyto assume the existenceof a continuous, G- 3-Hausdorffapproximation, h Xn __ y A fibration,f: X -- Y, of Riemannianmanifolds is called a 0-almostRie- manniansubmersion if for all y E Y, x E f (y), and V E TXx, normalto f-l(y),
(2.5.1) e 0lfI(V)l (2.6.2) f is a c(n, A)4-almostRiemannian submersion. (2.6.3) IIf i(y)I < c(n, A)' 1, for all y E f(Wn). (2.6.4) f is {C1(n, A)(1 +A2-n )Ill-i}-regular. (2.6.5) f is G-equivariant. (2.6.6) For c2(n, A)3F' -close.This does notsuffice forour purposessince it leads onlyto a boundon Vf and notto theassertion that f is an almostRiemannian submersion. It is in establishingthe required closenessof k and h thatthe geometry of our setupenters (in essentiallythe same wayas in [Fl, ?3]); see Lemmas2.16 and 2.19. In the lemmasthat follow, V denotesthe Levi Civita connectionof the pullbackmetric on TBx(1). Lemma2.11. For all x E Xn, thereis a function,k: TB (2A) Yi, suchthat (2.11.1) d(k, h) < c(n, A)A2. (2.11.2) k is a c(n)'112-almost Riemanniansubmersion, (2.11.3) IV2kl< c(n, A). Lemma2.12. (2.12.1) k, is a c(n)l-almostRiemannian submersion. (2.12.2) 1V2kj < c(n, A). Lemma2.13. (2.13.1) iVf - Vk2j < c(n, A)A. (2.13.2) IVt2f-V_k2I < c(n, A). Essentially,to get Lemma 2.13, we can estimatethe ith derivativeof the regularizationof (h - k) by A-' timesthe quantity in (2.11.1) (see (2.8)). Sim- ilarly,the properties(2.12.1), (2.12.2) are consequencesof the corresponding properties(2.11.2), (2.11.3). Indeed,Lemmas 2.12 and 2.13 wouldbe standardin thefamiliar case Xn Rn, Yj = RJ. In thepresent context, their proofs are straightforward,ifslightly 340 JEFFCHEEGER, KENJIFUKAYA, AND MIKHAEL GROMOV tedious,exercises in advancedcalculus. Hence, the proofs of theselemmas will be omitted. The proofof Lemma2.11 willbe givenat theend of thissection. By combiningLemmas 2.12 and 2.13 we see that (2.14) f is a c(n, A)A-almostRiemannian submersion if (2.6.2) and (2.6.3) holds. Clearly,we can choose Wn D Xn suchthat f (y) c W is compactfor all yef(Wn). We now prove(2.6.1). Note thatby (2.6.3), if some fibre,f I(yo), has diameter,gu5, then for all y E (!), BY 2 (2.15.1) dia(f1(y)) > c7(n, A)Lu5. In view of (2.14), it followsthat for a < c l(n), at least (2.15.2) c6 (n, A),u6 i balls of radius a are requiredto cover f l (By( But fromthe existence of the 6-Hausdorffapproximation h, it followsthat at most (2.15.3) c(n)F-j suchballs are required.Therefore we get (2.15.4) c(n)c(n, A) > u, whichgives (2.6.1). In orderto proveLemma 2.11 we willneed twoauxiliary lemmas (compare [F1, F3]). Lemma2.16. Let X, Y be as in Theorem2.6 and let a be a geodesicloop of length1 on p E Xl. Let y be a minimalgeodesic segment with 1i < Lly] < i and y(O)= p . Then (2.16.1) 4(Y'(O) '(O)) - 72< c max(li- , (5 - 1)1/2 Proof. By scaling,it sufficesto considerthe case i = 1. The inequality (2.17) 4(y'(0) a'(O)) > -2 - cl is a directconsequence of Toponogov'stheorem applied to thedegenerate isoce- les trianglewith sides y, y, a. On theother hand, put = (2.18.1) h(y(3)) exPh(p) (u) and choose p' suchthat (2.18.2) d(h(p') , exph(p)(-u))<5 By Toponogov'stheorem, if 4 is minimalfrom p to p', then (2.18.3) 14( '(O), y'(0)) - 7r1< c5/2. NILPOTENT STRUCTURES AND INVARIANTMETRICS ON COLLAPSED MANIFOLDS 341 By using(2.17) with C in place of y, we get(2.16.1) (compare[Fl, ?3]). Lemma 2.19. Let Xn, Yj be as in Theorem2.6. Let Yp, Y2 be minimal geodesicsin Xn joiningthe point z topoints q, 1 q2 E X1n respectively.Assume that !i < L[y, < -5 i and thatd(q1 , q2) = 1 << 'i. Then (2.19.1) 4(y'(0), y4(0)) < cmax(li-1, (i-51)1/2 Proof. By scalingit sufficesto considerthe case i = 1. Writeh(ql) = exph(Z) v and put y = exPh(z)(-v) . Usingthe facts that h is a 6-Hausdorffapproximation and that Yj has boundedgeometry, we findby a standardcomparison argument that (2.20.1) d(h(q2), h(z)) + d(h(z), y) - d(h(q2), y) < cl (l + 6)2. (This quantityis the excessof the trianglewith vertices h(q2), h(z), y.) Let w E X be suchthat d(h(w), y) (2.20.3) 4Q(y(0), -TI(0)) > JC- C3max(l, ( 1/2) Similarly, (2.20.4) d(q1, z) + d(z, w) - d(q1, w) < 36 and by Toponogov'stheorem, (2.20.5) 4(y'(0), T (0)) > 7n / Our claim followsfrom (2.20.3) and (2.20.5). Remark2.21. In the proofof Lemma 2.16 we used onlythe lowerbound on Kx; compare[Y]. But in Lemma2.19 we also use thetwo-sided bound on Ky. Proofof Lemma 2.11. Let e1, . .. , ej be an orthonormalframe at h(x) . Pick xl.. Xj E Xn suchthat (2.22) d(h(xi) 1eXPh( e) < a Let x' E TXtX-- , with expxxi = xi and expxtx1, t E [0, 1], a minimal segmentfrom x to xi. For p E Bx(2A)),p' E TBx(2)) c TXx, put (2.23.1) pxi(P) d(p, xi), (2.2 3.2) px (p ) := d(p, xi)i We claimthat Lemmas 2.16 and 2.19 imply (2.24) - PXI < CA. For themoment, let us grantthis. Then if we definek by (2.25) d(expI(X)( e,) k(p')) =px,() p1 , i we get (2.11.1). Moreover,(2.11.2), (2.11.3) are directconsequences of the definitionof k. 342 JEFFCHEEGER, KENJIFUKAYA, AND MIKHAEL GROMOV To verify(2.24), fix xi as above and let y1 be minimal from xi to x. Take p' E TBX(2).) c TXX and put p = expxp'. Let y2 be minimal from xi to p. Let Yl be the liftof y, runningfrom x' to 0 c TXX. Finally, let Y2 be the liftof Y2 with initial point x'. Since by Lemma 2.19, (2.26.1) 4 (y'(0), yI(0)) < CA, it follows that the end point, p", of Y2 lies in TB.,(cA). Let & be minimal from p" to p'. Then we have (2.26.2) L[d] < cA. The projection of & is a geodesic loop, a, on p, of length L[a] = L[d]. It follows fromLemma 2.16 that (2.27) 4(o'(0), -)4(l)) =4(&'(0), -) (l)) < + CA. Using (2.26.2), (2.27), and a standard comparison argument,we get (2.28.1) d (p', x') < d(p", x') + (cA)(cA), ii 2 2 =d(p ,x')+CA Since (2.28.2) d(p', X) =P(P (2.28.3) d(p", X') = P) this sufficesto complete the proof. Example 2.29. For a > 0, consider the annulus, a < r < 2, in R2. Let X2 (where a = 27/N) denote its quotient by the action of &Z/N; (r, 6) -* (r, 0 + 27/N). Let Y' be the open interval, (a, 2). Then the map h((r, 0)) := r is a Riemannian submersionand a 6-Hausdorffapproximation. But no matterhow small we take (, the second fundamentalform of the fibreshas norm 1/r, which blows up as a -O 0. Clearly, no smoothingprocedure will improve this situation. This confirmsthe necessityof restrictingf in Theorem 2.6 to points that are far from aXn (e.g., to Xn) independentof how small we take A. The reader may also wish to verifyLemma 2.11 directlyin the context of this example. For the application in ?5, we will need the followingsharpening of (2.6.2). Let f be as in Theorem 2.6 and let v be a tangent vector at x E f (y), with v orthogonalto f 1(y). Proposition2.30. Thereexists a geodesicy, withy(O) = x, yI [0, 31] minimal, and (2.30.1) 4(y'(0), v) < c(n, A)(bi-F)112 Let T be theminimal geodesic from f(y(0)) to f(yQ i)). Then (2.31.2) 4 (df(v), T (0)) < c(n, A)(o5i-1)112 NILPOTENT STRUCTURES AND INVARIANTMETRICS ON COLLAPSED MANIFOLDS 343 Proof. By Lemma 2.13, it sufficesto verifythe correspondingassertions for the map k, for which they are clear by inspection. 3. EQUIVARIANT AND PARAMETRIZED VERSION OF THE THEOREM ON ALMOST FLAT MANIFOLDS The main result of this section is concerned with fibrationssuch as those obtained in Theorem 2.6. To prove it, we show that one can canonically re- move the dependence on the base point in the initial step of the proof of Ruh's theorem[R] (see also [Ghl]). Thus, initiallywe will be concerned with a single almost flatmanifold. Let N be a nilpotentLie group (which need not be simplyconnected). The canonical connection, Vcan, on the tangent bundle of N, is, by definition, the unique connection that makes all the left invariantvector fields parallel. Let NL o Aut N be the skew product of NL and Aut N (NL denotes an isomorphic copy of N acting on N by left multiplication). It is easy to see that this group coincides with the group, Aff(N, Vcan), the group of all affine transformations of (N, Vcan). If A c Aff(N, Vcan) is a subgroup whose action can on N is properlydiscontinuous, we can definethe induced connection, V on the quotient space A\N. Remark 3.1. The subgroups, NL and NR, can be defined intrinsically,just using the affine structure of (N, Vcan). The group NL is the kernel of the holonomy homomorphism, i.e., the subgroup that acts trivially on all globally parallel fields. The group NR is obtained by integrating these fields. On the other hand, the subgroup, Aut N, can be described as the isotropy group of the identity element, e E N. Equivalently, it depends on a choice of base point in the affine homogeneous space (N, Vcan). Thus, the specific isomorphism, Aff(N, Vcan) NL oc Aut(N) depends on a choice of base points as well. Let Zm be an A-regular Riemannian manifold with Ao = 1 and diameter, 5 < ,(m) . In [R] Ruh observed that the results of [G] allow one to associate to each point, z E Zm, a flat orthogonal connection, Vz, such that: (3.2.1) For p, q E Zm, there is a gauge transformation, gP Iq, carrying Vq to VP . (3.2.2) gP,q can be chosen such that IgP, - Identl < c(m)5, IVigPpq, < c(A, i)5, where Ident denotes the identity element of the gauge group. (3.2.3) For all z, Vi(VL- V z)I < c(m, A, i)5, where VLC denotes the Levi Civita connection of the underlying metric. (3.2.4) The connection, Vz , depends smoothly on z, (with estimates like those above on derivatives with respect to z ). (3.2.5) The holonomy group of Vz has order < wm (see, however, Remark 3.9). Ruh went on to show that for some simply connected nilpotent Lie group, N, and discrete subgroup, A c Aff(N, Vcan), with #(An NL \A) equal to the order of the holonomy group, one can associate to each connection, Vz, a gauge transformation conjugating Vz into a connection isomorphic to the connection Vcan on A\N. The fact that A\N is actuallythe same for all z, follows from 344 JEFFCHEEGER, KENJIFUKAYA, AND MIKHAEL GROMOV Malcev's rigiditytheorem, (see [Rag, BK] and Theorem3.7). We now showthat by suitablyaveraging the family of connections,VZ, we can obtaina canonicalgauge equivalent flat connection associated to the Rie- mannianstructure (and notdepending on a choiceof base point).The connec- tionisomorphic to Vcan, associatedto thisone by Ruh's construction,depends smoothlyon the underlyingmetric and is automaticallyinvariant under all of its isometries.From this, the main resultof thissection follows immediately. We now explainthe averagingprocedure. Let 3" denotethe bundleasso- ciated to the framebundle, FZm, via the adjointrepresentation. Each fibre of 39 has a naturalgroup structure isomorphic to O(m). The gauge group is the space of sectionsof .9, equippedwith the groupstructure induced by pointwisemultiplication. It has a naturalaction on FZm, whichcommutes withthe action of O(m). Hence it also acts on thespace of connections. A connection,V, on FZm inducesa connectionon 39. The group,K(V), of gauge transformationsfixing V is easilyseen to be the groupof parallel sectionsof 39 withrespect to theinduced connection. Let %(V) c 9 be the bundlewhose fibreat z e Zm is gottenby evaluatingat z, the sectionsof K?(V), theidentity component of K(V). Then 5(V) is canonicallytrivial. Let gP.q be as in (3.2.1). Put (3.3.1) gP q(z) = hp q(z)kp q(Z) = eU(z) ev(z) where V(z) is in theLie algebraof X(V)z and forall z, (3.3.2) (U(z), V(z)) = 0. The innerproduct in (3.3.2) comesfrom the negative of theKilling form. Since gP,q is close to theidentity, it is uniquelydefined up to rightmultiplication by an elementof Ko(Vq). It followsthat hpq is independentof the particular choice of gP Iq. Also, there is a uniquechoice of gP,q that satisfies (3.3.3) fzm V(z) dz = 0, wheredz is the normalizedRiemannian volume element, for which (3.3.4) fzm dz = 1. This is an immediateconsequence of centerof mass constructionfor the com- pact Lie group K0(V); see [BK, ?8]. Suppose gp',gIq , gP W are normalizedas in (3.3.2), (3.3.3). Put w (3.4.1) gp,q= eulev, gq,w=e gP =eU3eV3 where VI E (Vq), V2, V3E %(Vw). Assume (3.4.2) IlUjll,9 IV-11< I << I1 Then sincethe product of elementsin a Lie groupthat are close to theidentity is commutativemodulo higher order terms, (3.4.3) gp qgq w=eutgq W(gq w)-'ev gq w = eU +U2eV2+Vl'+ 0(2). Here, NILPOTENT STRUCTURES AND INVARIANTMETRICS ON COLLAPSED MANIFOLDS 345 Thus, Vj' E X(Vw) . It followsthat (3.4.5) (U1 + U2, V2+ VI) = o(W), (3.4.6) fzm(V2 + VI') dz = 0, whicheasily implies that q,w (3.4.7) gP,q = gp,w + 0(12) For fixedw and variablep in (3.4.1), we write U3 = U3(P), V3 = V3(p). Set (3.5.1) gW = efZ U3(p) dp .efz v3(p) dp By using(3.4.7), we obtain (3.5.2) gqgq,W = gW + Q(12) Hence,if we put (3.5.3) Vw gw(vW) we get (3.5.4) Vq Vw + 0(?12). By iteratingthe above construction,we obtainconvergent sequences, VI Vq 5 ... suchthat for all q, w (3.5.6) V?? := limj VJ.liMJj- = CxOVJ is independentof thebase pointand, in particular,invariant under the isometry groupof Zm. We now turnto our mainresult, Proposition 3.6. Let Xn, Yj be A-regularRiemannian manifolds, with Ao = 1, and let f: Xn __YJ be a {C1i11'}-regular, 1-almostRiemannian submersion, where Ci = Ci(n, A). (This normalizationcorresponds to thatof Theorem2.6 butno assumptionon inj rady, y E YJ, is requiredhere.) Assumethat G acts on Xn, YJ by isometriesand that f is G-equivariant. Let Vy,LC denote the Levi Civita connectionfor the induced metricon I fI(y). Suppose that dia(fPl(y)) < 5 and III_,(y)I < Cl' , with cal < ,(n), wherec = c(n, A) and ,(n) is so smallthat f l(y) is almostflat. Let VY * denotethe affine flat connection on fl (y) obtainedby applyingthe con- structionof [R] (or [Gh1]) to the connection,Vy' 0?, associatedto Vy LC via (3.5.6). Thus, (f 1(y), Vy'*) is affinelydiffeomorphic to some (A \N, Vcan) with #(An NL \ A) < wn (see, however,Remark 3.9). Let y vary and regard, Vy LC _ Vy'?o, Vy'LC - * as tensor fields on putting V for V to X5, by LC - = 0, V, - = 0, orthogonal f'(y). Proposition3.6. (3.6.1) IVi(VY, LC _ VY'??) I< C.(n A)5-(2+i) (3.6.2) IV1(Vy'? - VY'*)I < Ci(n, A)65(2+i) . 346 JEFFCHEEGER, KENJIFUKAYA, AND MIKHAEL GROMOV (3.6.3) If h e G, then h: (f-' (y), VY.*) -. (h (f'-(y)), Vh(Y)*) is an affine diffeomorphism. Proof. By our previousdiscussion, Ruh's methodyields a familyof affineflat structureson thefibres and it is straightforwardto check that the conditions of the propositionhold. Each Zy is affineequivalent to some (Ay\ NY, Vcan). Moreover,since affine structures of thistype cannot occur in nontrivialfamilies, Ay\ NY = A\N is actuallyindependent of y. This is a weak generalization of the secondBieberbach theorem (the uniquenessof theaffine structure on a compactflat Riemannian manifold; see [Char]). For completeness,we givethe argument(see, however,Remark 3.9). A local trivializationof our fibrationover an open neighborhood,U, of y E Y, inducesisomorphisms, Ay/ , 7rX(Zy,)- 7rl(Zy) - Ay. The holonomy homomorphismsvary continuously and, by (3.6.1), havefinite image. It follows thatthe identifications,Ay/ - Ay, respectsthe kernels, Ay/ n NY,, Ayn NY of theholonomy homomorphisms. These are cocompactsubgroups of thegroups Nyl, NY. Bya theoremof Malcevthe isomorphisms Ayl n N1, - Ayn NY extend uniquelyto isomorphisms,NY, , NY Theorem3.7 (Malcev). Let N1, N2 be simplyconnected nilpotent Lie groups and A c N1 a cocompactsubgroup. Then a homomorphismfrom A to N2 extendsuniquely to N1. Now thefollowing consequence of theaffine center of mass constructionfor Lie groups[BK, ?8] impliesthe asserted rigidity of (AAy\ Ny, Vcan ). Proposition3.8. Let h,: GI -+ G2 be a continuousfamily of homomorphisms of Lie groupssuch thatfor some subgroupH c GI, offinite index, h,IH is independentof t. Then thereis a continuousmap, t -- k, E G2 such that h,=k,h0k-'. Proof. It sufficesto considert so smallthat the affinecenter of mass, kAc,of the finiteset {h71h0(g)jg E G1} is defined.As in Proposition8.1.7 of [BK], thischoice of k, has therequired property. Remark3.9. The resultsof thissection and the nextwill be applied in ?7 to thelocal fibrationsof theframe bundle constructed in ??5, 6. In thatcase, one actuallyhas A c NL; equivalently,the connection Vcan on A\N is globallyflat (see ?7 and Appendix1). Thus,Proposition 3.8 and theargument given in the proofof Proposition3.6 are not needed forthe constructionof the nilpotent Killingstructure. 4. NILPOTENT KILLING STRUCTURES ON FIBRATIONS Let Z -- XL Y be a fibrationacted on isometricallyby a compactgroup G, suchthat the assumptions of Proposition3.6 hold. By Proposition3.6, each fibrecarries a flataffine structure isomorphic to some A\N. Let NAff(N)Aand CAff(N)Adenote respectively the normalizer and centralizer of A in Aff(N). Then NILPOTENT STRUCTURES AND INVARIANTMETRICS ON COLLAPSED MANIFOLDS 347 (4.1.1) Aff(A\N) (NAff(N)A)/A and since A is discrete, (4.1.2) Aff?(A\N) (CAff(N)A)/A, where,Ae(A\N) c Aff(A\N)denotes the identity component. Also, An NL c NL is cocompact.Thus, by Malcev's theorem(3.7), (4-1-3) CAff(N)Ac CAff(N)NL. Moreover, (4.1.4) NR = CAff(N)NL= {(t5 Adt-,)} is just an isomorphiccopy of N actingby righttranslations. The identificationNR = CAff(N)NL dependsonly on the affinestructure of the affinehomogeneous space, (N, Vcan); compareRemark 3.1. However,an explicitisomorphism, NL - NR, or equivalentlythe representationNR = (t, Adt-,), does dependon a choiceof base point(which can thenbe viewed as theidentity element, e E N). More generally,corresponding to each normal subgroup,BL C NL, thereis a well-definedisomorphic subgroup, BR c NR. Again,a specificisomorphism, BL - BR, dependson a choiceof base point. Now let V be a locallydefined right invariant vector field on a neighborhood W c Z,y. As in ?3, we can finda local trivialization : U x Z -* X (over a smallneighborhood U of y ), withrespect to whichthe affine structure on the fibresis constant.Such a trivializationis uniqueup to a map U -* Aff0(A\N) . Since the group Aff (A \N) is containedin NR, it followsthat this group acts triviallyon local rightinvariant fields. Thus, V has a canonicalextension to q$(Ux W). In thisway, we obtaina sheaf n, of nilpotentLie algebrasof vectorfields on X and an actionof theassociated sheaf n, of simplyconnected nilpotentLie groups. The actionof G extendsin an obviousway to an actionon n and theactions of n and G on X commutein the obvious sense. In general,the actionof a groupon a sheafis called locallytrivial if foreach open set U, thereis a neighborhood,W, of theidentity in G suchthat for all g E W, = (4.2) Pg(u)nu, g(u) 9 Pg(u)nu, u Here, PB,A denotesthe restriction map fromA to B. Now thesame sortof argumentas was givenabove yields Proposition4.3. The actionof G on ntis locallytrivial. In case G acts freely,it followsdirectly from Proposition 4.3 thatthere is an inducedsheaf, nt, on X/G (see ?8 forthe detailed discussion). We now discussthe quantitativebehavior of the local rightinvariant fields constructedabove. This requiresa moreexplicit description of a local trivial- izationin whichthe affine structure on thefibres is constant. Let V be a tubularneighborhood of a fixedfibre, ZY,, such thatthe nor- mal exponentialmap of ZY providesa local trivialization b: U x ZY - V (where q(U x Zy) is the union of all fibrescontained in V). By the proof 348 JEFFCHEEGER, KENJIFUKAYA, AND MIKIHAELGROMOV of PropositionA2.2, the normalinjectivity radius of Zy is boundedbelow by c(n, A)min(i, d(y, aY)). Let pyl = 0((y', z)) e Zy,. The universalcovering space, (V, py), is fi- bred by universalcovering spaces (Z t, j3). y p The coveringgroups of all of thesespaces are canonicallyisomorphic to A. Let Aff(Zy1,V) denotea group of affineautomorphisms of Z y, withrespect to its canonicalflat affine con- nection, V. Let NL(ZYI) C Aff(ZY,,V) be the correspondingcanonically de- finedsubgroup. Then (up to naturalisomorphism), for all y', we can regard A c Aff(Zy, V). By : Malcev's theorem (3.7), there is a unique affineequivalence ZVY,- ZY , suchthat (4.4.1) Vy, p I and for all A e NL(ZY) (or equivalentlyfor all A e A n NL(ZYI)), (4.4.2) V/A = AV/ Giventhat such an affineequivalence exists, it is explicitlydetermined as follows. By integratingthe leftinvariant fields (i.e., the parallelfields for V* ) we obtain the group NR(ZYI), and hence,the rightinvariant vector fields. By integratingthese, we obtain the group NL(ZYI). Fix A e A n NL(ZY) - A n NL(ZYI). For each y ,there is a unique integral curve,c A, of a certain rightinvariant vector field on Z,, , suchthat cy",(O) = cy'A(tA) = cytA(tA) = )(fy). Here t, is independentof y'. Since V/, is an affineequivalence satisfying (4.4.1), (4.4.2), we get (4.5) dy/(c, A(O)) = cyt,A(O) Clearly,the collectionof vectors,{cI (0)}, spans the tangentspace at pp. Thus, (4.5) determinesthe linearmap, dy/1. Then VyY,itself is determined by the conditionthat it map a givenright invariant field on ZY to the right invariantfield on 2yt to whichit correspondsunder d . Now forall y we have injradp > c(n) > 0; see [BK, Proposition4.6.3]. On y theother hand, the points A(y) are c(n, A)5-densein Zm . Thus,we can find AI, ..., Am such that - (4.6) 4(cA (0), Cy A (0)) 2 < c(n, A)c . Fromthe preceding explicit description of Vy, togetherwith Proposition 3.6 and standardbounds on thelocal trivialization,X, we readilyobtain Proposition4.7. Let w bea rightinvariant field on VnB4p(26),with Iw(j)I = 1. Then (4.7.1) IVIwI < c(n, A, i)i'- NILPOTENT STRUCTURES AND INVARIANTMETRICS ON COLLAPSED MANIFOLDS 349 We now constructa canonical(and hence G-invariant)invariant metric for theaction of NL on U, or equivalentlyfor the action of n on X/G (in case G acts freely).Given such a metric,it is obviousthat the action of n determines a nilpotentKilling structure (see Definition1.5). We have #(An NL\A) < wC). Thus,it is clearthat we can reduceto thecase A c NL. Let v be a tangentvector at p3E V and let (, ) denotethe pullbackto V, of the originalmetric on V. Let h E NL and let hv denotethe imageof v underthe differentialof h. Then,the function,h -* (hv, hv), is constant on the leftcosets of A. Since the group NL is nilpotent,it is unimodular. Therefore,the space A\N inheritsa canonicalinvariant measure du, of total volume 1. The metric, (4.8) (v, v) = f (hv, hv) du, N/A is invariantunder NL and pushesdown to therequired metric on V. Clearly,our construction is independentof the choice of U and ofthe choice of base point used to define V. Thus it gives a canonical (and hence G- invariant)metric on X, whichis invariantfor the nilpotent Killing structure. Proposition4.9. The originalmetric, (, ), and invariantmetric, (, ), sat- isfy (4.9.1) 1V((, ) - (, ))I < c(n , A, i)5l Proof. The estimateson leftmultiplication that follow immediately from (4.7. 1) yield(4.9.1). Remark4.10. Note thatthe right-handside of (4.9.1) is small provideda is smallrelative to ii+ 1 Remark4.1 1. One can also constructan equivariantright action on X; it gives riseto thenilpotent collapsing structure. The construction,which will be carried out elsewhere,does makeuse of theinvariant metrics on fibres. III. THE NILPOTENT KILLING STRUCTURE AND INVARIANT ROUND METRIC 5. LOCAL FIBRATION OF THE FRAME BUNDLE In thissection we beginthe construction of thenilpotent Killing structure by constructinglocal fibrationsof theframe bundle. Let Mn be a completeA-regular Riemannian manifold. A standardcompu- tationshows that the frame bundle, FMn, withits natural metric, is B-regular (for B = B(A)). For fixedn and A, put 1 (5.5. 1) = {FB p(2) Mn is A-regular}. Note,on theright-hand side of (5.1.1), Mn is notfixed. (Also, we couldreplace 2 by anyfixed R > 0 in (5.1.1)). Let ta denotethe closure of a withrespect to the O(n)-Hausdorffdistance, dH. Then by [F3], ta consistsof B(n, A)- regularmanifolds, yi* It will be convenientto assumethat A is normalized 350 JEFFCHEEGER, KENJIFUKAYA, AND MIKHAEL GROMOV suchthat AO, BO(n,A) < 1. The inducedaction of 0(n) on Yi is D(n, A)- regularbut need not befree. Put (5.1.2) Qj : {Yj e Q:a I dim Yj = j}. Then f3 = Uj Ta5 determinesa stratificationof (Ea. This fact,although we do not use it explicitly,puts our constructionsin a naturalcontext. Clearly,(Ea is emptyfor j > n + n(n 1). One can also showthat (Ea is emptyfor j < n(n- 1) = dim 0(n); see Appendix1. Again,we do notuse this. It followsfrom [GLP, ?8] (togetherwith [CGT, Theorem4.3]) thatthere is a positivefunction, 0(b, n), with 0(b, n)/3 increasing,such that if (5.2.1) Yl/'E TFj , (5.2.2) inj rad < min(0(6, n), d(y, a YIJ'),for some y E Yl", thenthere exists Y2/2 e jwith (5.2.3) '2 < Jil (5.2.4) d(Y Y22)< Fromnow on we suppressthe dependence of X on n. [ Put 10 = I. Let 10 > II > .> n+n(n+1)]/2 and l3 > >3Z>...> 3n+[n(n+1)J/21be positivesequences, such that for > 1, (5.3.1) -'(ij) + j < j_I. Relation(5.3.1) can be satisfiedby taking (5.3.2) ij = 0( Ij_I) (5.3.3) cj < In thisand subsequentsections it willbe necessaryto assumethat 6 is small enoughrelative to I suchthat certain additional conditions are satisfied. Proposition5.4. Let FBp(l) E a and let j be thesmallest number such that thereexists Y' e fj with (5.4.1) dH(FBp()I Yj) < 3j. Thenfor any such Yj and y E Y' (5.4.2) injrady > min(i , d(y, aYj)). Proof. Note that since dH(FBP(1), FBp(1)) = 0, the set of Y' satisfying (5.4.1) is nonempty.If (5.4.2) failedto hold, thenby the definitionof the function4, we wouldhave (5.5. 1) dH(Y , Y,') < '6j- forsome Yl' withj < j - 1 . Then by (5.3.1) and (5.4.1), it followsthat (5.5.2) dH(FBP(1), Yj') < d1(FBp(1), Yj) + d( Yj, YrI) Fix A = A(n) < 1 to be determinedin ??6, 7. Proposition5.6. Let P, E s = 1, 2, with Proof. The existence of the fibrations,fs, satisfying(5.6.1), (5.6.3) follows from (5.4.1), (5.4.2). Using (2.4.3), (2.4.4), we constructan O(n)-equivariant Hausdorffapproxi- mation,h, withdomain f2(FBp,(1) n FBp2 ( 1)) and rangein Ylj . By regular- izing h, we obtain f12 satisfying(5.6.2), (5.6.4), and (5.6.5). Finally, (5.6.6) follows with the help of Proposition 2.30. Remark 5.7. The sets FBp, (1 )nFBp (1) are notnecessarily of the form fW1(U1) or V, 2 f2) 1(U2) and hence, are not unions of compact fibres.To obtain actual fibrationswe must restrictthe domain of a map, fs, to the set consistingof all compact fibreswhose intersectionwith FBp (2) is nonempty.This is a slightly smaller set. We will deal with this (minor) point when it arises in the proof of Proposition 6.1. But in the meantime, to simplifynotation, we will continue to referto "the fibrationfs ." More importantly,the maps, f1, f2 are not necessarilycompatible in the sense that the fibresof f2 need not be unions of the fibresof f, . Equivalently, f, 2f254 f1 in general. However, by (5.6.5) and (5.6.6), f1, f2 are almost compatible. This togetherwith the results of Appendix 2, will be used in ?6 to constructa collection of local fibrationsof FMn that are compatible in the above sense. Remark 5.8. The smaller the numbers, 5j, the more difficultthe condition (5.8.1) dH(FBp(2), Y') <(5. is to satisfy.In particular,the subsets of elements of j, for which thereexists a nontrivialfibration also gets smaller. Remark 5.9. If we fix e = 1 in Theorems 1.3 and 1.7 then we can work with a fixedsequence that is small enough for the argumentsof subsequent sections to go through. But if we let e -* 0, then necessarily (j -* 0 as well. As a consequence, for e verysmall, our structurewill be nontrivialonly on the part of Mn that is verycollapsed. 352 JEFFCHEEGER, KENJIFUKAYA, AND MIKHAEL GROMOV 6. MAKING THE LOCAL FIBRATIONS COMPATIBLE Let Mn be as in ?5. The fibrationsconstructed in thissection will be obtainedby slightlymodi- fyingthose constructed in ?5 and restrictingtheir domains. After this has been done, to simplifynotation, we will continueto denotethe modifiedfibrations by fS, fs, and theirbase spaces by Y1s. Let A = A(n) < 1 be a sufficientlysmall constant. The constraintson A(n) will be determinedin the courseof the proofof Proposition6.1. These and theconstraints entailed in theanalogous constructions of ?7 allow us to fixthe values of A(n). We willassume without further mention at theend of ?7 that thishas been done. Let b: [0, 1] -_ [0, 1] be an increasingfunction, with b(u) < u. Proposition6.1. Given b, thereexists a decreasingsequence, ii = IJ(b, n, A), suchthat the following holds. Thereis a covering,Mn = Us Bp( and 0(n)- equivariantfibrations, (6.1.1) f5: (2) - Ys5 FBp P~2 s suchthat for y E Yss, (6. 1.2) inj rady > min(iisI d (y, a Ysj-) ). $ < Moreover,if Bp5 (1) 2 n Bp,(')p2 0, Js j,, thenthere is an 0(n)-equivariant fibration, (6.1.3) fs,: f(FBp()~,2 n FBP(2, ))2f(fs(FBP2 (2)FnFBP,2" 1 suchthat (6-1.4) fs,tf,= fs Thefibrations, fs, satisfy: (6.1.5) dia(f I(y)) < b(ij). (6.1.6) fs is a c(n, A)A-almostRiemannian submersion. (6.1.7) fs is {C1(n, A)lii1}-regular. is (6.1.8) {IIIf-,_(Y) < c(n, A)i_1 (6.1.9) The maps fs, satisfy(6.1.5)-(6.1.8). (6.1.10) The (compact)fibres, fs-f(y), fs- 1(y) are diffeomorphicto nilmani- folds. Proof.The factthat the fibres,fs '(y), fs ,(y) are diffeomorphicto nilman- ifolds(and notjust infranilmanifolds)was mentionedin the introductionand is explainedfurther in ?7 and Appendix1. Pick a maximalcollection of points,ps, suchthat for all s, t, NILPOTENT STRUCTURES AND INVARIANT METRICS ON COLLAPSED MANIFOLDS 353 In particular, (6.2.2) Mn = UsBp( 1) Fix a decreasingsequence, (6-3-1) do > 61 > > 6n+[n(n+1)]12s to be determinedlater. As in (5.3.2), define (6.3.2) i = ( 1 ). Relativeto thesequence b I}, choosefor each ps, a fibration, (6.3.3) fS: FBp (1) - Ys , satisfying(5.4.1) withjs minimal.Let thecorresponding fibrations, fS be as in Proposition5.6. We can assumethat {1J} is suchthat (6.1.9) holds. In orderto make it clear thatwhen we repeatedlymodify our fibrations, approximatelycompatible fibrations do not eventuallybecome too far apart, we use a technicaldevice. As in Lemma 2.2 of [CGI], we partitionthe set {p5} intodisjoint subsets S, ..., SN(n)I suchthat if ps PU e Sk, then (6.4. 1 ) d (ps , pu) > 4. In particular,those balls, Bp (1), whoseintersection with a fixedball, Bp (1), is nonempty,all belongto differentsubsets, Sk . Thus,there are at most N(n) suchballs. Put (6.4.2) Sk j = {PS E SkI js =I}. There are T(n) = N(n)(n + 1 + n(n2+l)) of the Sk1j, some of which mightbe empty.Put (6.4.3) Sk j = Sk+N(n)j. Note that if Sa = Sk(a) j(a), then (6.4.4) a < a' impliesj(a) < j(a'). Also, Ps E S', fs:Bp (1) + Ysj,, implies (6.4.5) js = j(a). In orderto makeour fibrationscompatible, we nowmodify them in a totalof T(n)-stages, one for each Sa. Each stage, say ao, is divided into (2T(n)aOlo ) steps, one for each nonemptysubset, (al, ... , am), with a0 < al < ..< - am < T(n). Thus, there are N' (n) = 2T(n) - (T(n) - 1) steps in all. (The orderin whichthe steps are performedis specifiedbelow.) At a givenstep we mustalso decreasethe radii of the balls involvedby a definiteamount. The notationis simplestif at the end of each step a0, we actuallydecrease the radii of all balls (i.e., withcenters p1, P2, p3, ... ) by an amount -4-- Thus, at the beginningof a givenstep, every ball has radius, r = 1 - x 2N-4- (n) thenumber of stepsalready peformed. 354 JEFF CHEEGER, KENJI FUKAYA, AND MIKHAEL GROMOV Note thatsince at each stagewe decreasethe radii of our balls by exactly - we want = . 2N (,n) certainly A A(n) < 2N_r(n) At step (a1, ... , am) of stage ac, we modifyonly the fibrationsf , 1 < 1 < m, and f I < 11< 12 < m, oversets of theform FBp (r) n ...n FBp (r) (respectivelyf ,(FBP, (r) n ln FB (r))) where p, e Sc'. Note thatfor certain (a, ... , am) (e.g., unless S00, ..., S0' all belong to distinct Su) therewill be no suchnonempty intersections. However, if at anystep there are I no nonemptyintersections, we simplydecrease the radii of all balls by and proceedto thenext step. Now we can explainthe reason for introducing the sets Su. If Pi e S0' and (O ,) is distinctfrom (p, ... , p,i) then (by construction) (6.5) (FBp (r) n n FBp (r))n (FBp (r) n . FB (r))= 0. This guaranteesthat the various modifications performed at step (a. Cm) of stage a0 do not interactwith one another(and thata givenfibration is modifiedat most N'(n) times). As a consequence,fibrations that are initially almostcompatible do notgrow uncontrollably further apart as theconstruction progresses. It is importantthat in carryingout themodifications, the stages are arranged in descendingorder; i.e., we startwith stage T(n), thenpass to stage T(n) - 1, etc. It is also importantthat the steps of stagesao are arrangedas follows.First we do step (a0 + I, a0 + 2, ... , T(n)). Then, in some (arbitrary)order, we do the steps correspondingto subsets of (al , ..., am) of cardinality, T(n) - a0 - 1; then,in some (arbitrary)order, the stepscorresponding to subsetsof cardinality T(n) - a0 - 2, etc. Let r be thecommon radius of all ballsat thebeginning of step (a, ... , am) of stage a0 . At thebeginning of thisstep, we can assumeby inductionthat the followingholds. Let a' > ac and let (a'1, ... , am) be a step of stage a' that has already beencompleted (automatic unless ao = a0, m' < m ). Let pI e S' , 0 < 1' < ' in'. Then for O< /I < < m', our (previouslyredefined) fibrations satisfy (6.6.1) J;,,J; ; on FBp (r) n ..n FBp(r. 1? 12 -to In addition,we can assumeby inductionthat for all s, t, u with s< i < Ju (6.6.2) dia(fsj(y)), dia(fs[l(y)) < c(n, A). (6.6.3) fs, fs5, are c(n, A))A-almostRiemannian submersions. (6.6.4) f5,sfs, are {C,(n, A)Vi }-regular. (6.6.5) d(fs, f, fs) < c(n, A)A,. (6.6.6) IV(fs5,f,) - Vfsj < c(n, A)A. (6.6.7) d(fs,5f, us f5su) < c(n, A)Aij . (6.6.8) IV(fs ,f,,u) - Vfsul < c(n, A)A. NILPOTENT STRUCTURES AND INVARIANT METRICS ON COLLAPSED MANIFOLDS 355 We now definecertain 0(n)-equivariant self-diffeomorphisms (r)- (6.7.1) /i :FBp i,(r) 1< < m and - (6.7.2) (i fiI(FBp (r)) fi/(FBpl(r)), 1< < m. (We suppress the dependence of these maps on (a,, ... , am).) Eventually,we willredefine fi to be 1~ (6.7.3) T',fi , 1< < m, and redefinefi 12 to be ( 4) 1 < 12 < m il 12 12 Untilthis is done explicitly,fi fi i retaintheir previous meanings. We now use PropositionA2.2 to constructthe diffeomorphismVi . (Re- markssimilar to thosethat follow also apply to theconstruction of thediffeo- morphism(i below). SinceProposition A2.2 holdsfor fibrations with compact fibres,we firstrestrict the map fi] (whichis used in definingthe various Vi ) to thesubset of FBp (r) n... n FBp (r) consistingof theunion of all compact fibresof . The set contains (r)fn (the notation fi (FBp nfFBp, (r))C(.l A)61(,O) is as in (2.3)). In view of (6.6.3)-(6.6.6), by PropositionA.2.2 we can findan O(n)-equi- variantmap, v ( = Vil. ilm) suchthat: (6.8.1) V is a self-diffeomorphismof FBp (r)n n FBp (r). (6.8.2) V is theidentity near the boundary. = (6.8.3) fio/i (fi ) fi on FBp (r -) n...fn FBp (r-) (6.8.4) q/ is theidentity on thesubset of n (r p( (n) n, FBp ) on whichJi0i, f=f i (6.8.5) d(V, Ident)< c(n, A)Aii(o), (6.8.6) JVV/- Identl < c(n, A)A. (6.8.7) V is {C1(n, A)ld(-)}-regular. Define Vi as in (6.7.1) by yJ(X) x e (r)n ... n FBp (r), Jix FBp, I - x x i F Bp0FBp,i (r) n n (r). We now definethe diffeomorphisms, (i In view of (6.6.3), (6.6.4), (6.6.7), (6.6.8), by PropositionA2.2, we can find an O(n)-equivariantmap, (i ( I > 2 ) suchthat: (6.9.1) (i is a self-diffeomorphismof J, (FBp (r)fnn...FBp (r)). 356 JEFF CHEEGER, KENJI FUKAYA, AND MIKHAEL GROMOV (6.9.2) (i is the identitynear the boundary. (6-9.3) / . on (r- (r- ___ fi1o,1, f11,l11 =fi1fi i fi1I (FBpiP0 2N/(-n) -)n...nFBp l Nr-- (6.9.4) ( is the identityon the subset of fi (F BP, (r- I n ..n FBp (r-;) on which /i fi, =I >io.,- (6.9.5) d(4j, Ident) < c(n, A)AlJ(ao). (6.9.6) 1V4j -Identl < c(n, A)A. (6.9.7) (i is {Ci(n, A)l ,1a)}-regular. Extend (i to all of /i (FBp (r)) by definingit to be the identitymap off fi (FBp, (r) n -**n FBp (r)). Also define (i to be the identitymap. We now examine the effectof modifyingfi i fi 1 as in (6.7.3), (6.7.4) on our induction hypotheses. Firstof all, it followsfrom (6.6.1), (6.8.4), (6.9.4) that V,i ( is equal to the identityover the subset of FBp (r) nf. n FBpi,(r) that intersectsany FBp (r), where Pi e SP, ,B > ao, ,B #&al, ... , am. Moreover, the correspondingstate- ment holds for (i . As a consequence, in examining the effectof the proposed modificationson (6.6.1), we can assume that a, = al ,for some a, E SAlY u = 1, 2 (since otherwise,nothing changes). Next observe that for 1 < 11 < 12 < m, on FBp (r) n ...n FBp (r), by (6.6.1) we have ( 1, 1, 1i 112 112 112 112 ( 1 , j 12 12 (while outside FBp (r) n ... f FB (r), the maps V, are the identity). Now by construction, ... (6.1(~ 0.2) ~~~I fi *(C,4,tjYj)=foonFp( 'fVI)=f on FB (r ---4-'()fl..fFBn n FBp (r - ) (n) (recall (i is the identitymap). Finally, for 1 > 2, by (6.6.1), on FBp (r - 2(n - n FBp (r - ), we have (6.10.3) fio i(Ki fVi,)=fti.i, f*f * f,,i, fil,Vi i / NILPOTENT STRUCTURES AND INVARIANTMETRICS ON COLLAPSED MANIFOLDS 357 < ,1 used in definingig, is boundedby a constant - times il ratherthan by IJ, as is the case forthe map fi . Note that tims a1G) rahrtanb 4as sth1 171 >> , if j(ao) < j(a,). Also, care had to be takenin choosingthe methodof redefiningthe maps fi1 fi i , in orderto ensurethat control over Hessiansof relevantmaps was not lostin theinduction process. Remark6.12. By ??3,4, thefibres of ourmaps, f5,carry canonical affine struc- tures.However, the inclusions of fibresimplied by (6.1.4) (namely,ft-' (yt) c fs-l(ys) where Yt e fs' (ys)) need not be compatible with these affinestruc- tures. Arrangingthis is the subjectof ?7. However,if we pretendthat it is alreadythe case, thenProposition 6.1 summarizesmuch of what we aim to accomplishin thispaper. 7. MAKING THE LOCAL GROUP ACTIONS COMPATIBLE In thissection we completethe construction of thenilpotent Killing structure on theframe bundle. By ?6 we have a mutuallycompatible system of maps, (7.1) Zs -+FBP (-)Ls+Y'S suchthat Mn = UsBP (TA6) . As pointedout in theproof of Proposition6.1, to obtainactual fibrations with compact fibres, we mustreplace the sets FBP (2) in (7.1) by slightlysmaller sets, i.e., theunion of all compactfibres intersecting FBP (2). This is to be understood(sometimes without explicit mention) in whatfollows. We denoteby g thefibration corresponding to (7.1). By ?3, each fibre,Z, carriesa canonicalflat affine structure, affine isomor- phicto some (As\ Ns , Vcan), and a canonicalmetric, whose image under such an isomorphismlifts to a leftinvariant metric on N5 In our case, we actuallyhave (7.2) As C (Ns)L C Aff(Ns,cVa) 358 JEFF CHEEGER, KENJI FUKAYA, AND MIKHAEL GROMOV (and notjust #(Asn (NS)L \ AS) < (on). This followsfrom the factthat short closedloops on theframe bundle of an A-regularRiemannian manifold (in this case Zs) automaticallyhave small holonomy (compare [G, R] and Appendix1). If ( I ) n ( I ) is nonempty,then (say) each fibreZ, of - is containedin BpP~2 Bpp,2 some fibre,Zs of s. However,this inclusion need notbe compatiblewith the affinestructures. We now showthat Y lies closeto a unique O(n)-equivariant subfibration,< of Y, such thatthe tangentbundle to the fibres,TZ', is a totallygeodesic sub-bundle of ( TZs, Vcan). Given thisand an argumentthat replacesProposition A2.2, the constructionof the nilpotentKilling structure can be completedby argumentslike thosein ?6. Specifically,we will obtain modifiedfibrations such thaton nonemptyintersections of theirdomains, the inclusionsof fibresare compatiblewith affine structures. Then we construct nilpotentKilling structures and invartiantmetrics as in ?4. For each fibre,Zs of Y, thereis a fibration (7.3)- Z ZS W Usingthe fibration in (7.3) and theaffine structure on Zs, we willconstruct foreach Zs, a fibration, - (7.4) Z ZS W whichhas totallygeodesic fibres and thatlies close to theone in (7.3). Thenwe definethe fibration t' to be theone whosefibres are all Z,' in (7.4) (as Zs in (7.4) varies). Let Zs be theuniversal covering space of Zs. Althougha specificchoice of Zs is gottenby choosinga base point,the construction of t' thatfollows will not dependon theparticular choice of base point.Thus, our constructionwill automaticallybe O(n)-equivariant. WriteZs = As\ZSand let 7: Zs - Zs. The group irI(Zt) does dependon a choiceof base point. However,the existence of thefibration in (7.3) implies that (7.5) i(7rn(Zt)):=At C As the imageof 7rI(Zt) underthe map inducedby the inclusion,Z, -- Zs, is a well-definednormal subgroup. Lemma7.6. The map 'I (Zt) ir1(Zs) is an injection. Proof. Let Zs, Z, be the fibresof f5, f,, respectively.Then W is a fibreof fs,I. Hence,by (6.1.10), W is almostflat, and, in particular,aspherical. Then, by applyingthe homotopy sequence for fibrations to thefibration in (7.3), our claim follows. Let V denotethe pullback to Zs, of theflat affine connection on Zs. Then (7.7) (Z, V) (Ns, Vcan), wherewe view NS as an affinehomogeneous space; i.e., we do notdistinguish a base point. We regardthe invariantlydefined subgroup (NS)L as contained NILPOTENT STRUCTURES AND INVARIANTMETRICS ON COLLAPSED MANIFOLDS 359 , in Aff(Zs V); compareRemark 4.2. Then (7.8) AsC (Ns)L- By Malcev's theorem(3.7), thereis a uniquesimply connected subgroup, (7.9.1) (N;)L C (NS)L, whichcontains A, as a cocompactsubgroup. Since A, c As is normal,(N,')L C (NS)L is normal as well. Define A; D A, by (7.9.2) Al =As f(N )L. We willshow in Lemma7.13 that,in fact,A' = A,. Let the fibration,Z, -+ Zs W', in (7.4) be the one whosefibres are the orbits, (7.9.3) Z, = A;\(N:)L(Z), where z e Zs (compareRemark 3.9). Then the fibration,, is as specified after(7.4). We now showthat this is close to 7 over n ( ) . < Bp() p2 Bp 2 Note thatLemma 7.6 alreadyimplies (7.10) dimZ, = dim Z,. By (6.1.8), (7.11.1) IIIz? (7.14.3) d(q', q) < c(n, A )021 it . For 6(n, A) sufficientlysmall, this yields (2). (3) Finally, the angle between (Zt), and the parallel translate of (Zt),r (along, unique minimal geodesic from q' to q) is at most c(n, A)O. It is now clear that a normal projection to Z, definesa coveringmap from Z' to Zt. Since A c A', this must be a diffeomorphism. Remark 7.15. The principlebehind (2) above is the following.If two functions agree on an c-dense set and have derivativesup to order N bounded, then they Ni1 N-2 are close to order E -1, theirderivatives are close to order E-, etc. Now suppose that for 0 = O(n, A, io), io > 1 , in fact, (7.16. 1) b(u) (7.16.2) IV'((, - ( )) < io-i isI At this point, we can match the affinestructures on t, <', by adapting to our situation, the center of mass argumentof [GrK], used there to prove the stabilityof compact group actions. Let y e Yj1 . Put V = ft (lI )) . By shrinking V slightly,we obtain a l(Byy2] subdomain V c V such that if q e V., then the fibresof t and Ft' through q are both contained in V. Let (V, qc) denote the universal coveringspace of V. Then we have the followingpreliminary result. Lemma7.17. Thereexists c*(n, A) > 0 suchthat for 4, e V, (7.17.1) injradq > min(c*(n, A)tjS,d(qd,,aV) Proof. This follows easily from (7.11.1), Lemma 7.13 (1) and (7.14). The isomorphism, At At, extends to a canonical isomorphism (Nt)L, (Nt)L. Let ,u', 1u denote the actions of (Nt)L, (N:)L. Then for all A e At A II we have (7.18.1) ,t (A) = At(i). NILPOTENT STRUCTURES AND INVARIANTMETRICS ON COLLAPSED MANIFOLDS 361 Thus, the map M,(h)(,u)- (h) is well definedon (N,)LIA, (N:)L/A;. A slightvariant of theargument leading to Proposition4.7 showsthat for all h, (7.18.2) IV'(u,(h)(M,)-(h)) - Identl< c(n, A, i i Similarly,by integratingover (N,)L/A, _(N,)L/A (ratherthan all of N,) we can definethe center of massof the map 2u(2)41 ; compare[GrK] and (4.8). As in [GrK],this yields an 0(n)-equivariantdiffeomorphism, /, suchthat (7.19.1) >j,=g,g, (7.19.2) IV'@/- Identl < c(n, A, io) iJ. It followsfrom (7.17) and (7.19.1) that f/ commuteswith the action of A,. Thus, the collectionof maps, ,obtained by varyingy e Yj1, inducesa well-definedembedding, I - - (7.19.3) y/:FBp ( ) n FBP ( 2 ) FB ()fFBp,( X ) The map y sendsfibres of 7 to fibresof <', preservingaffine structures and correspondinglocal actions. Here, the number N'(n) is as in ?6. By modifyingy withthe aid of a cutofffunction, we obtainan 0(n)-equivariant map (also denoted yg) suchthat (7.20.1) v/ is a self-diffeomorphismof FBp p52(!) n FBpp2~ (1). (7.20.2) g is theidentity near the boundary. (7.20.3) d(V, Ident) < c(n, A, ij)10. (7.20.4) IV't - Identl = c(n, A, io)l 'o. (7.20.5) If g(Z,) n (FBP (' - n) FBp (' - 4)) $0, thenfor some Z', Y: Z, .Z4', preservingthe affine structure. Let fis,In, denotethe sheaves of local rightinvariant vector fields associated to the affinestructures on the fibresof '(Y), s, as in ?4. Let nis, nt,denote theassociated sheaves of simplyconnected nilpotent Lie groups.By arguingas in ?4, it is easy to checkthat on theircommon domain n-,is a subsheafof ins. I I Then thereis an obvioussheaf, fi U n,, over FBp I -)nFBp ( ), whose stalkat pointsof FBp (2 - rI-) coincideswith ns, and elsewhere, coincideswith that of n,. Now (assumingthat 0(n, A, io) is chosensufficiently small) as in theproof of Proposition6.1, we obtaina sheaf, nii,defined all of FMn, on whichthe naturalaction of 0(n) is trivial(in thesense of Proposition4.3). This sheafis associatedto a systemof (modified)maps, fs fst wheref: FB () Ys These maps satisfy(6.1.4)-(6.1.10) (withthe functionb as in (7.16.1) and A = A(n, A)). Moreoverthe affinestructures on the fibresof the associated fibrationsare mutuallycompatible. Let fs FBp(') - Y s be obtainedby restriction.Denote by ntthe sheaf of FM gottenby applyingthe above constructionto thesemaps. The sheaf nt will be shownin ?8 to inducethe desired structure on Mn. The factthat the domainsof itsdefining system of maps can be enlargedto thesets (1), is FBpPS4 362 JEFFCHEEGER, KENJIFUKAYA, AND MIKHAEL GROMOV requiredin orderto verifyTheorems 1.3 and 1.7. Finally,we observethat an invariantmetric close to theoriginal one, can be constructedfor our structure.Let S' be as in (6.4.3). Recall thatthere are T(n) of thesesets. Startwith those balls withcenters in ST(n). Over each such ball, B (i),4 1 constructan (0(n)-invariant)invariant metric on FB (1) ps PS as in ?4. Usinga suitablecutoff function, modify the original metric over each FBp ( ) \ FBp ( 8T (n) ), so as to obtain a new metricagreeing with the original one outsidethe union of the FBp (1) and withthe invariant one on theunion of the FBp (I- 8 n)) By applyingthis constructionsuccessively to ST(n) -, S T(n)-2,... we obtain an 0(n)-invariant metricthat is invariant,for the sheaf associatedto the coveringUs FBp ( ) and thus,for the sheaf n as well. More preciselywe getthe following: Let fi denotethe metric on FMn inducedby our original metric g on Mn. Proposition7.21. Givenc, io, thereexists 0(n, A, io, e) suchthat by choosing 0 = 0(n, A, io e) in (7.16.1), we obtaina metricgE thatis 0(n)-invariant and invariantfor n*, suchthat for i < io0, (7.21.1) |V1(g - ge)I 8. THE INDUCED STRUCTURE AND METRIC ON THE BASE In ?7 we constructedan 0(n)-invariantRiemannian metric, k, and an as- sociatednilpotent Killing structure, 91, on thetotal space, FMn, of theframe bundle. Here, we constructthe correspondingobjects, g, 91, on the base, Mn, and showthat the assertions of Theorem1.3 and 1.7 hold. The statements of Theorems1.3 and 1.7 are closelyrelated and it willbe convenientto prove themsimultaneously. Briefly, we must (1.3.A) constructthe metric, gE, satisfying(1.3.1)-(1.3.3), (1.3.B) show that g, is (p, k)-roundfor suitable p, k dependingon n, e (1. 1.1)-(1.1.6)), (1.7.A) constructthe nilpotent Killing structure 91, compatiblewith gE, and (1.7.B) showthat 91 has compactorbits of diameter< e. Proofof Theorems1.3 and 1.7. (1.3.A) The resultsof ??5-7 werestated for A-regularRiemannian manifolds, with the sequence, A, normalizedin ?5. However,the metric,g, in Theorem1.3 is assumedonly to satisfyIKI < 1. Therefore,given e as in Theorem1.3, we beginby replacingg by themetric, S,12(g) of Theorem 1.12. This metric is A(n, e/2)-regular. Althoughthe se- quence, A(n, e/2), is notnormalized as in ?5, by an obviousscaling argument the resultsof ??5-7 can still be applied to S,-2(g )' Fix io (large) and c'. Startingwith the metric S,12(g), constructa metric, ge, on FMn as in Proposition7.21. Since g, is 0(n)-invariant,there is a NILPOTENT STRUCTURES AND INVARIANTMETRICS ON COLLAPSED MANIFOLDS 363 __ unique metric, g,(,,1fl) on Mn, such that 7: FMn Mn is a Riemannian sub- mersion(see (1.13.6) forthe T( ) notation).Here thenotation is understood to be such that if T(E'In) = e, then g, satisfies(1.3.1)-(i.3.3). (1.7.B) The structure,9t, on FMn is 0(n)-equivariant.In particular,the actionof 0(n) mapsorbits to orbits,and hence,induces a partitionof Mn into compactsubmanifolds, {a} . Thesewill be seento be theorbits of the nilpotent Killing structureon Mn. Since 7r:FMn __Mn, is distance nonincreasing,it followsthat for all <, dia(&) < E, providedthat the same is truefor the orbits of 91. Clearly,if the constant0 = 0(n, c) in ?7 is chosensufficiently small, thiswill be thecase. (1.7.A) For theremainder of thissection we willuse a tildeto indicatethat a pointlies in the framebundle. Let p E Mn, p e r i(p) and let Z. denote theorbit of 91 throughpj. Choose I > 0 so smallthat (8.1.1) B1p(1) is simplyconnected, (8.1.2) B. (Ir)n 0(n) (B, (I)) is connected, (8.1.3) therestriction of ntto the t-tubularneighborhood, T,(Z.), of thefibre throughp is pure, (8.1.4) thenormal injectivity radius to theorbit &p is > . The space of local sections,i(B,(q)), is the nilpotentLie algebraof local rightinvariant fields. In viewof (8.1.1), (8.1.2), it followsfrom Proposition 4.3 thateach local vectorfield in B,(B(q)) is 7r-relatedto some vectorfield on 7r(B0(?1))= Bp(q). Thus we geta nilpotentLie algebraof local Killingfields on Bp(i), which,by Proposition4.3, is independentof thechoice of p e r I(p) . In a standardway, the collectionof Lie algebrason the various Bp(q) de- terminesa sheafof nilpotentLie algebrasof Killingfields on Mn. Let n be the associatedsheaf of simplyconnected Lie groupsand h its naturalaction. Obviouslythe orbits of thisaction are thoseconsidered in (1 .7.B). We now showthat n definesa nilpotentKilling structure. For p, ,1 as above,we put U = T(&p), where U is as in Definition1.5. Since it is clearthat we can choosethe same I forall pointson & , it follows that(1.5.3) holds. Recall thatthe local fibrationwith fibre, Z.p, is therestriction of a fibration definedon an open set containingT,18(Zp). (This fibrationcomes from91*; see the end of ?7). Let 91' denotethe correspondingpure nilpotentKilling structureon T,18(& ). By (8.1.3), 91' extends911U. An orbitof 91' will be denotedby &'. For the constructionof the neighborhood,V, appearingin (1.5.1), (1.5.2) and in (1.3.A), we need (91' and) Lemma 8.5 below,the statementof which requiressome terminology. Let W be a lengthspace (see [GLP]). Definition8.2. Bq(r) c W is starshaped if for all w e Bq(r), thereis a unique geodesicin Bq(r) joining w to q. 364 JEFFCHEEGER, KENJIFUKAYA, AND MIKHAEL GROMOV Let q e Wl = Ym/G,where ym is a Riemannianmanifold and G is a compactgroup of isometriesof W. Put Vol(C W) (8.3) Angq = Vol(Sq) where CqW is the set of unitvectors in thetangent cone at q E W. Definition8.4. W = Y/G has (p, k)-boundedgeometry at w e W if there exists q e W suchthat (8.4.1) Bw(p) c Bq(R) forsome starshaped Bq(R). (8.4.2) AngqW > k. Let (r, A) be the Hausdorffclosure of the collectionof FBp(r), such thatthe ball, Bp(r), is containedin an A-regularmanifold; compare ?5. Here we do not assumethat A is normalized.Let .Xm(r,A) be definedas in ?5. Lemma8.5. For all 3 > 0 thereexists p = p(r, A, 3) and k = k(r, A, 3) suchthat if Y E ?tm+i (r, A), satisfies (8.5. 1) dH (Y, ?@ m(r, A)) > 6, then Y/O(n) has (p, k)-boundedgeometry. Proof. This is essentiallya restatementof Theorem0.14 of [F3] (comparealso (4.6.2)). It followsfrom Theorem 10.1 of [F3]. Let FMn carrythe metric g(,'n)(- ' = '(e, n)) inducingthe metric g, on Mn. Fix p E Mn. For some Ys, as in ?7, we can identifyYj's/O(n) withthe orbitspace of theaction of n'. Let w e Y/O(n) be theprojection of p. Let q projectto q e Y,/O(n) as in (8.4.1). Remark8.6. By wayof explanation,we mentionthat the orbit, q, shouldbe thoughtof as one thathas minimaldimension among all orbitsin TR(@). It is an easy consequenceof Lemma 8.5 and (8.4.1) that A" has normal injectivityradius > R = R(n, e). For futurereference, we notethat since Mn is A(n, c)-regular,it followsthat the second fundamental form, satisfies II.,,q (8.7) jII, I < c(n, e) . q Put V = TR(6q), where V is theneighborhood occurring in (1.5.1), (1.5.2), and (I.1.1)-(I.1.6). Since V is a union of orbitsand 91' is pure,the actionof n' on V lifts to the actionof the simplyconnected nilpotent Lie group,fl*(n')(VV), on the universalcovering space, V- V. Let K be thekernel of theaction of fl*(n')(V) and put (8.8) No = 8*(n)(V)/K - NILPOTENT STRUCTURES AND INVARIANTMETRICS ON COLLAPSED MANIFOLDS 365 Apartfrom the assertionthat K is discrete(the proofof whichis givenin Appendix1) it is now clear that (1.5.1), (1.5.2) hold. Modulo the proofof (1.3.B), thiscompletes the proof of (1.7.A). (1.3.B) Let q e XG (q). The action of nt' on TR(Zq) liftsto the ac- tion of a simplyconnected Lie group, No, on the universalcovering space, TR(Z4) . Moreover,it followseasily from Proposition 4.3 thatthe natural map, T: = T,R(Zq) -* TR( ), intertwinesthe actions of No and No. Since 7r(Z) A c No, we have T(A) c No. Also, by Proposition4.3 and Lemma 8.5, Z-q fibres(with a torusas fibre)over a finitecovering of q, of indexat most k. Thenthe homotopy sequence for fibrations, implies that z(A) has indexat most k in 7r (,) = 7I (V) Let H of(I.1..)-(1.1.6) be theLie groupgenerated by 7r1(V) and No. Now (1.1. 1)-(1. 1.4) and ( 1.1.6) are obvious. To see (1.1.5), firstnote thatthe normalprojection, Vq: V - ' is well definedand by (8.7) increasesdistances by a factorof at most c = c(n, e). If y is a geodesic loop on q E q`, then y is homotopic to y c ;q' over curves of lengthat most c *L[y]. But since 6q' is isometricto a simplyconnected nilpotentLie groupwith left invariant metric, any closed curvein 6q'6q is con- tractibleto a pointover curves of shorterlength; see Proposition4.6.3 of [BK]. By Klingenberg'slemma on liftinghomotpies inside the conjugate locus, we get (8.9) L[y] > c(n, e)ir. This sufficesto completethe proof of (1.3.B). Remark8.10. The orbitspace Y/O(n) coincideslocally with the orbitspace of our structure'R. In thisconnection, Lemma 8.5 expressesthe fact that the orbitsof our structureabsorb all collapseddirections. The followingexamples show thatin general,the numersp, k of Theo- rem 1.3 cannotbe chosenindependent of e. Example8.11. We considercertain metrics on a nilmanifoldM , viewedas the totalspace of a fibration,S -* M 3- T2 (withnonzero Euler class). We assumethat all fibreshave lengthd', that Xr is a Riemanniansubmersion and thatthe metric on thebase is chosenas follows.Start with a metric,g, whichis theproduct of twoshort circles of length3. Deform g slightlyby introducing a small bump centeredat p such thatthe new metric,g', satisfies,IKg,I < 1, Kg,(p) = 1. Then the isometrygroup of R2 equippedwith the pullback metric,g', is discrete.Moreover, for some I > 0, anymetric that is I-quasi- isometricto g' also has thisproperty. Thus, for the correspondingmetric on M3, the isometrygroup of M3, the universalcovering of M3, containsthe skewproduct, R x 7 2 as a subgroupof finiteindex. Here, the center, R, acts I by translationin thedirection of theuniversal coverings of the S fibres. Now considera sequenceof such manifoldswhere d' -* 0, while 3 stays fixed,e = e(a) is as in Theorem1.3 and e << . Suppose we assumethat 366 JEFFCHEEGER, KENJIFUKAYA, AND MI1HAEL GROMOV for this sequence and p e M3, we can find p, k as in Theorem 1.3 with 2 12 3 p > dia(T , g') . Then neighborhood V of (1.1) must be Xr (T2) = M3, anand it is easy to check that forany group H satisfying(1 1. )-(1. 1.6) (the definiton of (p, k)-round)) we must have k - oo as 3' -a 0. Thus for (p, k) as in Theorem 1.3, we must have p < dia(T , g') < 33, as soon as 3' is sufficiently small. Finally, let 3 -a 0, 3'/3 -a 0 sufficientlyfast, and choose e = E(3) ?<< (a) as above. Then for such a sequence if p, k are as in Theorem 1.3, it follows that p - 0. On the other hand, if for each such manifold, we take p = inj radp, V = 7r (BP (p)), and V the universal coveringof V, we findthat we can choose k as in Theorem 1.3 equal to 1 (and H = N = R). Example 8.12. Start with flat R2 and introduce k mutually isometric tiny bumps as above centered at points with polar coordinates (r, +), where j = 0, ..., k - I and r > 0 is a small fixed number. Then Zk acts iso- metricallyon R2 by rotation about the origin. Moreover, for some I > 0, any metric on R2 that is a-quasi-isometricto the given one has at most k orientationpreserving isometries. Form R x R 2, with the product metric, where the metric on R2 is the one just described. Let (t, 92./k) denote the isometry of R3 that acts by translation by t units in the R factor and by rotation through an angle, 27r/k, about the origin 2 in R . Let A be the group of isometries generated by this transformation, and put M3 =A\R3. The image of the axis (x, 0, 0) is a circle, S,1 c M3, of length t . Fix e < I and take sufficientlysmall such that in particular, t *k < e . Then the group No -* will be the 1-parameter group s (st, Ident). (Note that apart from S,1, the orbits in M3 have length kt.) In this case, #(A/A n No) = k and k can be taken arbitrarily large. By taking products of the above manifolds with the ones in Example 8.1 1, we get examples for which necessarily, p -- 0, and k -+ oo as E -- 0. APPENDIX 1. LOCAL STRUCTURE OF MANIFOLDS OF BOUNDED CURVATURE The proof of Theorems 1.3 and 1.7 given in ?8 has as a consequence that every point p e Mn is contained in a neighborhood of the form TR('), for the metric g, . We now give a more explicit desription of the metric structure of TR(q). First of all, examination of the proof of Theorem 10.1 of [F3] shows that given R2 there exists R,(n, c, R2) such that we can choose R,(n, E, R2) < R < R2, provided we take p = p(n, e, R2) sufficientlysmall and k - k(n, E, R2) sufficientlylarge. For R2 = R2 (n, E) sufficientlysmall, we can replace g, ITR (6c') by the natu- ral metric on the tube of radius R in the normal bundle, v (6q), and obtain a metric that satisfies (1.3.1)-(1.3.3) (with E replaced by 2E). Here, we identify NILPOTENT STRUCTURES AND INVARIANTMETRICS ON COLLAPSED MANIFOLDS 367 TR(() and thistube via thenormal exponential map. Relations(1.3.1 )-( 1.3.3) continueto hold forthe new metric(and 2c), providedwe take (Al.1.1) p= p(n, c, R2(n, e)) (A1.1.2) k = k(n, c, R2(n, e)) (whichstill depend onlyon n, c). In thisway, we obtaina morecanonical local modelfor the geometry. The bundle v(6@') can be describedup to isometryas follows. It followseasily from (4.1) thatthere is a principalbundle, (A1.2) C Z4 ( RemarkAl.9. (A 1.9. 1) Vol(B (p)) > Ce d(nd) does not hold in general. The flat metric E2 dx2 + E4dy2 on T2 is a counterex- ample. Remark A 1.10. We say that n is filteredif there exists n(i) c n such that fln n(j) I c n(i+j) . Put d'(n) = Zdim n(j) . Then thereexists a locally homo- geneous metricon M = A \ N such that , (A 1.10. 1) Vol(B (p)) C Ed'(np) Proofof TheoremA1.7. We replace Bp(p) by a normal coveringspace FB (p) of FBp(p), of order < k. Then, in view of the local descriptionof the metric given at the beginningof this appendix, it sufficesto show the following Lemma Al.1l. Thereexist En and C', such thatif e < en thenthe following holds:Let (A \ N, g) be a compactn-dimensional nilmanifold equipped with a locallyhomogeneous metric. Suppose (A1.I 1. 1) dia(A \ N, gk)< E, (Al 11.2) |K(A\N,k)l < 1; thenwe have (A l .1l 1.3) Vol(A \ N) < C, Ed(n) wheren is theLie algebraof N. Proof. Let AI = [A, A] D D Ak+l = (A, Ak] . By (BK, 2.4.2], there exists C such that Ak is generatedby homotopyclasses of loops whose length is smaller than Ek. Let No D ... D Nk be the lower central series of N. Suppose Nk 54 1 , Nk+I = 1. Since Ak \ Nk is flat and since its fundamental group is generated by loops whose length is smaller than ek, it follows that (A1. 12.1) Vol(Ak \ Nk) < C E Similarly we have (A 1. 12.2) Vol(Akl \ Nk- INk) < CEC (A1.12.3) Vol(Akl \ Nkl) < CE dim N*+(k-I)dim Nk- Inductively we have (A1.12.4) Vol(Ak- \ Nk_) < C dim Nk_J+(k)dk- The lemma follows immediately. Remark A1. 13. The construction in [FY, ?7] is closely related to the lemma. NILPOTENT STRUCTURES AND INVARIANT METRICS ON COLLAPSED MANIFOLDS 369 Lastly,we returnto the pointin theproof of Theorem1.7 whoseproof was deferred.Namely, we show thatthe group K in (8.8) is discrete.For thisit sufficesto showthe following. Let Mn, c be as in Theorem1.7. Let p E Mn, p E 7r '(p) and let Z. be thecorresponding orbit of nt. PropositionA1.14. If c < e(n) is sufficientlysmall, thenfor all p E Mn and 3 > 0, thereexists p1 E Mn suchthat d(p, pi) < 3 and dim p = dimZ. Proof. The identitycomponent, C, of C' c 0(n), the isotropygroup of Z., is a toruswhose Lie algebralies in thecenter of nt.If dim C = m, then (Al1.15) dim Zp - m =dim p . The torusC actson a coveringspace TQ(6 ), oforder at mostk ,by rotation in thefibres normal to thelifted orbit, 6p . If thisaction is effective,its principal orbitshave dimensionm. It followseasily that if p, lies on a principalorbit sufficientlyclose to p, then (A 1.16) dim(p ) = dim Z. To see thatthe actionof C (or equivalently,that of n ) is effective,recall the followingstandard fact. Lemma A1.17. Thereexists cl (n, c), c2(n, e) such thatthe map of pseudo- - groups,7r *: 7rI (FM, p, cI ) 7rI (M, p,cI2), is an injection. As a consequenceof Lemma A 1.17, the uniqueinverse to the map 7r* is gottenby liftingelements of 7rI(M, p, c2) via theirdifferentials, to obtain elementsof 7r (FM, p7,cl) . Put 7r I(h) = h. Let d( , ) denotethe uniform norm. Clearly, there exists c(n) > 0, suchthat for all h E 7r,(FM, p, cl), (A 1.18) d(h, Ident)< c(n) d(h, Ident). (Recall thata local isometrythat fixes a pointand a frameat thatpoint is the identitymap.) Now identify np with the universalcovering, Zp, of Zp. Then we can regard7r1 (FM, p, cl) c A. Since A is b(i )-densein Z., it follows withthe help of (4.7) thatan estimatelike (A1.18) holdsfor all elementsof np (possiblywith a differentconstant c(n)). This sufficesto completethe proof. RemarkA1.19. In case Mn itselfis almost flat,the statementof Proposi- tion A1.14 yieldsa firstmain step in the proofof the theoremon almostflat manifolds;i.e., shortloops withnot too big holonomyhave holonomyat most comparableto theirlengths; compare [Gh2]. This argumentmight appear to be circularsince it dependson constructinga fibration of the framebundle with nilmanifoldfibres. However, in thiscase thefirst step of thealmost flat theorem 370 JEFFCHEEGER, KENJIFVKAYA, AND MIKHAEL GROMOV is trivialsince a shortloop in the framebundle automatically has holonomyat mostof size comparableto itslength. APPENDIX 2. FIBRATION ISOTOPY In thisappendix we givea versionof a well-knownresult to theeffect that if twofibrations are sufficientlyC'-close, thenone of themcan be deformedonto the other. For the applicationsin ?6 and ?7, it is importantthat the required degreeof closenessis independentof the injectivityradius of the totalspace and of theHessian of one of theprojection maps. Let X, Y be A-regularRiemannian manifolds with Ao = 1, on whicha compactgroup, G, acts by isometries.Assume that for all y e Y, (A2. 1) inj rady > min(i, d(y, d Y)) . (Here and belowwe use thenotation of (2.2), (2.3).) PropositionA2.2. Let f, g: X -- Y be G-equivariantmaps for whichthe followinghold: (A2.2.1) f is a 1-almostRiemannian submersion. (A2.2.2) f is {B'ill }-regular. (A2.2.3) g is C-regular. (A2.2.4) Forsome rI> 0, f, g are e-closein the C'-topology,with c(I )-' < flo(B1,B2), sufficientlysmall. Thenthere exists a G-equivariantself-diffeomorphism, VI,of X, suchthat (A2.2.5) f = gyl on X. (A2.2.6) V/is theidentity near AX. (A2.2.7) V is c(n, Bo, B1)e close to theidentity map in the C' -topology. (A2.2.8) VI is {D (A, B, C)(I1) 'i}-regular. Moreover,D, dependsonly on Bk, Ck, k < i (andfinitelymany Ad). Proof. By scaling the metricson X and Y we can assume i, Ij> 1. By a standardcomputation, for all y e Y, (A2.3.1 |IIfI - (y)I < cl (B2) l Moreover,for y e Y1, thenormal injectivity radius, 1, of f (y), satisfies (A2.3.2) 1 > c2(B2). Indeed,the normal exponential map is nonsingularon a tubeof radius c3(B2). Thus,by a standardargument, if I < c2(B2) thereexists a geodesicsegment, y, of length21, with y(0), y(21)e f i(y) and y'(0), y'(21) normalto f' (y). For I sufficientlysmall relative to c,, y'(t) is almostnormal to thefibre through y(t), forall 0 < t < 21. It followsthat f(y) is a shortloop on y of small geodesiccurvature. For I sufficientlysmall this contradicts inj rady> 1 . Note thatall fibresf '(y), g' (y) are ce-close,together with their tangent planes. For c-sufficientlysmall, normal projection from g-' (y) to f -' (y) is a coveringmap. Moreoveran argumentlike that of theprevious paragraph shows NILPOTENT STRUCTURES AND INVARIANTMETRICS ON COLLAPSED MANIFOLDS 371 thatthis map is actuallya diffeomorphism.The collectionof all such maps definesa G-equivariantmap f/ withdomain say, X112, satisfying(A2.2.5). Let X be a functionsuch that (A2.4. 1) XIX1--1, (A2-4-2) X IX \X120 constructedfrom a G-equivariantsmoothing of thedistance function as in ?2. For X, 10 as in (A2.2.4), sufficientlysmall, put (A2.4.3) V'= X@ + (1 - X)Ident. Then, V/is easilyseen to satisfy(A2.2.6)-(A2.2.8). ACKNOWLEDGMENT We are gratefulto Z. Shen and G. Wei forhelpful comments. REFERENCES [A] U. Abresch,Uber das glattenRiemannisher metriken, Habilitationsschrift, Reinischen Friedrich-Willhelms-UniversitatBonn, 1988. [Ba] S. Bando,Real analyticityof solutions of Hamilton's equation, Math. Z. 195 (1987), 93-97. [BMR] J.Bemelmans, Min-Oo, and A. Ruh,Smoothing Riemannian metrics, Math. Z. 188 (1984), 69-74. [BK] P. Buserand H. 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Yamaguchi, Collapsingand pinchingin lowercurvature bound, Ann. of Math. (2) 133 (1991), 317-357. DEPARTMENT OF MATHEMATICS, NEW YORK UNIVERSITY, COURANT INSTITUTE OF MATHEMAT- ICAL SCIENCES, NEW YORK, NEW YORK 10012 DEPARTMENT OF MATHEMATICS, FACULTY OF SCIENCE, UNIVERSITY OF TOKYO, HONGO, TOKYO, JAPAN 113 INSTITUTE DES HAUTES ETUDES SCIENTIFIQUES, (91440) BURES-SUR-YVETTE, FRANCE