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 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/action/showPublisher?publisherCode=ams. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. American Mathematical Society is collaborating with JSTOR to digitize, preserve and extend access to Journal of the American Mathematical Society. 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 the nilpotentKilling 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, whileleaving 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 injectivityradius 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 the presentpaper 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,