The theory of connections and G-structures. Applications to affine and isometric immersions Paolo Piccione Daniel V. Tausk DedicatedtoProf.FrancescoMercurionoccasionofhis60thbirthday Contents Preface..................................................................................................v Chapter1. Principalandassociatedfiberbundles.............................1 1.1. G-structuresonsets..............................................................1 1.2. Principalspacesandfiberproducts.......................................9 1.3. Principalfiberbundles...........................................................22 1.4. Associatedbundles................................................................36 1.5. Vectorbundlesandtheprincipalbundleofframes...............44 1.6. Functorialconstructionswithvectorbundles........................58 1.7. Thegroupoflefttranslationsofthefiber..............................72 1.8. G-structuresonvectorbundles.............................................73 Exercises..........................................................................................76 Chapter2. Thetheoryofconnections................................................96 2.1. Thegeneralconceptofconnection........................................96 2.2. Connectionsonprincipalfiberbundles.................................103 2.3. Thegeneralizedconnectionontheassociatedbundle...........115 2.4. Connectionsonvectorbundles..............................................119 2.5. Relatinglinearconnectionswithprincipalconnections........122 2.6. Pull-backofconnectionsonvectorbundles..........................127 2.7. Functorialconstructionswithconnectionsonvectorbundles130 2.8. Thecomponentsofalinearconnection.................................140 2.9. Differentialformsinaprincipalbundle................................144 2.10. Relatingconnectionswithprincipalsubbundles.................152 2.11. TheinnertorsionofaG-structure......................................157 Exercises..........................................................................................163 Chapter3. Immersiontheorems........................................................169 3.1. Affinemanifolds....................................................................169 3.2. Homogeneousaffinemanifolds.............................................170 iii iv CONTENTS 3.3. HomogeneousaffinemanifoldswithG-structure.................175 3.4. Affineimmersionsinhomogeneousspaces..........................177 3.5. Isometric immersions into homogeneous semi-Riemannian manifolds....................................................................191 Exercises..........................................................................................197 AppendixA. Vectorfieldsanddifferentialforms..............................199 A.1. Differentiablemanifolds.......................................................199 A.2. Vectorfieldsandflows..........................................................203 A.3. Differentialforms.................................................................207 A.4. TheFrobeniustheorem.........................................................209 A.5. Horizontalliftingsofcurves.................................................212 Exercises..........................................................................................215 AppendixB. Topologicaltools..........................................................217 B.1. Compact-OpenTopology......................................................217 B.2. Liftings..................................................................................219 B.3. CoveringMaps......................................................................222 B.4. SheavesandPre-Sheaves......................................................231 Exercises..........................................................................................238 Bibliography.........................................................................................239 ListofSymbols...............................................................................240 Index................................................................................................242 Preface Thisbookcontainsthenotesofashortcoursegivenbythetwoauthorsatthe 14thSchoolofDifferentialGeometry,heldattheUniversidadeFederaldaBahia, Salvador, Brazil, in July 2006. Our goal is to provide the reader/student with the necessary tools for the understanding of an immersion theorem that holds in the very general context of affine geometry. As most of our colleagues know, there is nobetterwayforlearningatopicthanteachingacourseaboutitand,evenbetter, writingabookaboutit. Thiswaspreciselyouroriginalmotivationforundertaking thistask,thatleaduswaybeyondourmostoptimisticprevisionsofwritingashort andconciseintroductiontothemachineryoffiberbundlesandconnections,anda self-containedcompactproofofageneralimmersiontheorem. Theoriginalideawastofindaunifyinglanguageforseveralisometricimmer- siontheoremsthatappearintheclassicalliterature[5](immersionsintoRiemann- ianmanifoldswithconstantsectionalcurvature,immersionsintoKa¨hlermanifolds ofconstantholomorphiccurvature), andalsosomerecentresults(seeforinstance [6,7])concerningtheexistenceofisometricimmersionsinmoregeneralRiemann- ian manifolds. The celebrated equations of Gauss, Codazzi and Ricci are well knownnecessaryconditionsfortheexistenceofisometricimmersions. Additional assumptions are needed in specific situations; the starting point of our theory was precisely the interpretation of such additional assumptions in terms of “structure preserving” maps, that eventually lead to the notion of G-structure. Giving a G- structureonann-dimensionalmanifoldM,whereGisaLiesubgroupofGL(Rn), means that it is chosen a set of “preferred frames” of the tangent bundle of M on whichGactsfreelyandtransitively. Forinstance,givinganO(Rn)structureisthe sameasgivingaRiemannianmetriconM byspecifyingwhicharetheorthonormal framesofthemetric. The central result of the book is an immersion theorem into (infinitesimally) homogeneous affine manifolds endowed with a G-structure. The covariant deriv- ative of the G-structure with respect to the given connection gives a tensor field on M, called the inner torsion of the G-structure, that plays a central role in our theory. Infinitesimallyhomogeneousmeansthatthecurvatureandthetorsionofthe connection,aswellastheinnertorsionoftheG-structure,areconstantinframesof the G-structure. Forinstance, considerthecase that M isa Riemannianmanifold endowed with the Levi-Civita connection of its metric tensor, G is the orthogo- nal group and the G-structure is given by the set of orthonormal frames. Since parallel transport takes orthonormal frames to orthonormal frames, the inner tor- sion of this G-structure is zero. The condition that the curvature tensor should be v vi PREFACE constantinorthonormalframesisequivalenttotheconditionthatM hasconstant sectional curvature, and we recover in this case the classical “fundamental theo- remofisometricimmersionsinspacesofconstantcurvature”. Similarly,ifM isa Riemannianmanifoldendowedwithanorthogonalalmostcomplexstructure,then one has a G-structure on M, where G is the unitary group, by considering the set of orthonormal complex frames of TM. In this case, the inner torsion of the G- structure relatively to the Levi-Civita connection of the Riemannian metric is the covariantderivativeofthealmostcomplexstructure,whichvanishesifandonlyif M isKa¨hler. Requiringthatthecurvaturetensorbeconstantinorthonormalcom- plexframesmeansthatM hasconstantholomorphiccurvature;inthiscontext,our immersion theorem reproduces the classical result of isometric immersions into Ka¨hlermanifoldsofconstantholomorphiccurvature. Anotherinterestingexample of G-structure that will be considered in detail in these notes is the case of Rie- mannian manifolds endowed with a distinguished unit vector field ξ; in this case, weobtainanimmersiontheoremintoRiemannianmanifoldswiththepropertythat both the curvature tensor and the covariant derivative of the vector field are con- stantinorthonormalframeswhosefirstvectorisξ. Thisisthecaseinanumberof important examples, like for instance all manifolds that are Riemannian products of a space form with a copy of the real line, as well as all homogeneous, simply- connected3-dimensionalmanifoldswhoseisometrygrouphasdimension4. These exampleswerefirstconsideredin[6]. Twomoreexampleswillbestudiedinsome detail. First,wewillconsiderisometricimmersionsintoLiegroupsendowedwith a left invariant semi-Riemannian metric tensor. These manifolds have an obvious 1-structure,givenbythechoiceofadistinguishedorthonormalleftinvariantframe; clearly,thecurvaturetensorisconstantinthisframe. Moreover,theinnertorsionof thestructureissimplytheChristoffeltensorassociatedtothisframe,whichisalso constant. Thesecondexamplethatwillbetreatedinsomedetailisthecaseofiso- metric immersions into products of manifolds with constant sectional curvature; in this situation, the G-structure considered is the one consisting of orthonormal framesadaptedtoasmoothdistribution. The book was written under severe time restrictions. Needless saying that, in its present form, these notes carry a substantial number of lacks, imprecisions, omissions, repetitions, etc. One evident weak point of the book is the total ab- senceofreferencetothealreadyexistingliteratureonthetopic. Mostthematerial discussed in this book, as well as much of the notations employed, was simply created on the blackboard of our offices, and not much attention has been given to the possibility that different conventions might have been established by previ- ous authors. Also, very little emphasis was given to the applications of the affine immersion theorem, that are presently confined to the very last section of Chap- ter 3, where a few isometric immersion theorems are discussed in the context of semi-Riemannian geometry. Applications to general affine geometry are not even mentioned in this book. Moreover, the reference list cited in the text is extremely reduced,anditdoesnotreflecttheintenseactivityofresearchproducedinthelast decades about affine geometry, submanifold theory, etc. In our apology, we must emphasize that the entire material exposed in these three long Chapters and two PREFACE vii Appendicesstartedfromzeroandwasproducedinaperiodofsevenmonthssince thebeginningofourproject. Ontheotherhand,weareparticularlyproudofhavingbeenabletowriteatext which is basically self-contained, and in which very little prerequisite is assumed on the reader’s side. Many preliminary topics discussed in these notes, that form the core of the book, have been treated in much detail, with the hope that the text might serve as a reference also for other purposes, beyond the problem of affine immersions. Particularcarehasbeengiventothetheoryofprincipalfiberbundles andprincipalconnections,whicharethebasictoolsforthestudyofmanytopicsin differential geometry. The theory of vector bundles is deduced from the theory of principalfiberbundlesviatheprincipalbundleofframes. Wefeelwehavedonea goodjobinrelatingthenotionsofprincipalconnectionsandoflinearconnections onvectorbundles,viathenotionsofassociatedbundleandcontractionmap. Acer- tainefforthasbeenmadetoclarifysomepointsthataresometimestreatedwithout many details in other texts, like for instance the question of inducing connections on vector bundles constructed from a given one by functorial constructions. The questionistreatedformallyinthistextwiththeintroductionofthenotionofsmooth naturaltransformationbetweenfunctors,andwiththeproofofseveralresultsthat allowonetogiveaformaljustificationformanytypesofcomputationsusingcon- nectionsthatareveryusefulinmanyapplications. Also,wehavetriedtomakethe expositionofthematerialinsuchawaythatgeneralizationstotheinfinitedimen- sional case should be easy to obtain. The global immersion results in this book have been proven using a general “globalization technique” that is explained in Appendix B in the language of pre-sheafs. An intensive effort has been made in ordertomaintainthe(sometimesheavy)notationsandterminologyself-consistent throughout the text. The book has been written having in mind an hypothetical reader that would read it sequentially from the beginning to the end. In spite of this,lotsofcrossreferenceshavebeenadded,andcomplete(andsometimesrepet- itive)statementshavebeenchosenforeachpropositionproved. Thanks are due to the Scientific Committee of the 14th School of Differential Geometryforgivingtheauthorstheopportunitytoteachthiscourse. Wealsowant tothankthestaffatIMPAfortakingcareofthepublishingofthebook,whichwas doneinaveryshorttime. Theauthorsgratefullyacknowledgethesponsorshipby CNPqandFapesp. ThetwoauthorswishtodedicatethisbooktotheircolleagueandfriendFrancesco Mercuri, in occasion of his 60th birthday. Franco has been to the two authors an example of careful dedication to research, teaching, and supervision of graduate students. CHAPTER 1 Principal and associated fiber bundles 1.1. G-structuresonsets A field of mathematics is sometimes characterized by the category it works with. Of central importance among categories are the ones whose objects are sets endowedwithsomesortofstructureandwhosemorphismsaremapsthatpreserve the given structure. A structure on a set X is often described by a certain number of operations, relations or some distinguished collection of subsets of the set X. FollowingtheideasoftheKleinprogramforgeometry,astructureonasetX can also be described along the following lines: one fixes a model space X , which 0 is supposed to be endowed with a canonical version of the structure that is being defined. Then, a collection P of bijective maps p : X → X is given in such 0 a way that if p : X → X, q : X → X belong to P then the transition map 0 0 p−1 ◦q : X → X belongstothegroupGofallautomorphismsofthestructure 0 0 ofthemodelspaceX . ThesetX thusinheritsthestructurefromthemodelspace 0 X via the given collection of bijective maps P. The maps p ∈ P can be thought 0 ofasparameterizationsofX. To illustrate the ideas described above in a more concrete way, we consider the following example. We wish to endow a set V with the structure of an n- dimensional real vector space, where n is some fixed natural number. This is usually done by defining on V a pair of operations and by verifying that such operations satisfy a list of properties. Following the ideas explained in the para- graphabove,wewouldinsteadproceedasfollows: letP beasetofbijectivemaps p : Rn → V suchthat: (a) forp,q ∈ P,themapp−1◦q : Rn → Rn isalinearisomorphism; (b) foreveryp ∈ P andeverylinearisomorphismg : Rn → Rn,thebijective mapp◦g : Rn → V isinP. ThesetP canbethoughtofasbeingann-dimensionalrealvectorspacestructure on the set V. Namely, using P and the canonical vector space operations of Rn, onecandefinevectorspaceoperationsonthesetV bysetting: (1.1.1) v+w = p(cid:0)p−1(v)+p−1(w)(cid:1), tv = p(cid:0)tp−1(v)(cid:1), for all v,w ∈ V and all t ∈ R, where p ∈ P is fixed. Clearly condition (a) above implies that the operations on V defined by (1.1.1) do not depend on the choice of the bijection p ∈ P. Moreover, the fact that the vector space operations ofRn satisfythestandardvectorspaceaxiomsimpliesthattheoperationsdefined on V also satisfy the standard vector space axioms. If V is endowed with the 1 2 1.PRINCIPALANDASSOCIATEDFIBERBUNDLES operationsdefinedby(1.1.1)thenthebijectivemapsp : Rn → V belongingtoP arelinearisomorphisms;condition(b)aboveimpliesthatP isactuallythesetofall linearisomorphismsfromRn toV. ThuseverysetofbijectivemapsP satisfying conditions (a) and (b) defines an n-dimensional real vector space structure on V. Conversely, every n-dimensional real vector space structure on V defines a set of bijectionsP satisfyingconditions(a)and(b);justtakeP tobethesetofalllinear isomorphisms from Rn to V. Using the standard terminology from the theory of groupactions,conditions(a)and(b)abovesaythatP isanorbitoftherightaction of the general linear group GL(Rn) on the set of all bijective maps p : Rn → V. ThesetP willbethuscalledaGL(Rn)-structureonthesetV. Letusnowpresentmoreexplicitlythenotionsthatwereinformallyexplained inthediscussionabove. Tothisaim,wequicklyrecallthebasicterminologyofthe theoryofgroupactions. LetGbeagroupwithoperation G×G 3 (g ,g ) 7−→ g g ∈ G 1 2 1 2 andunitelement1 ∈ G. Givenanelementg ∈ G,wedenotebyL : G → Gand g R : G → G respectively the left translation map and the right translation map g definedby: (1.1.2) L (x) = gx, R (x) = xg, g g forallx ∈ G;wealsodenotebyI : G → GtheinnerautomorphismofGdefined g by: (1.1.3) I = L ◦R−1 = R−1◦L . g g g g g GivenasetAthenaleftactionofGonAisamap: G×A 3 (g,a) 7−→ g·a ∈ A satisfyingtheconditions1·a = aand(g g )·a = g ·(g ·a),forallg ,g ∈ G, 1 2 1 2 1 2 andalla ∈ A;similarly,arightactionofGonAisamap: A×G 3 (a,g) 7−→ a·g ∈ A satisfyingtheconditionsa·1 = aanda·(g g ) = (a·g )·g ,forallg ,g ∈ G, 1 2 1 2 1 2 and all a ∈ A. Given a left action (resp., right action) of G on A then for every a ∈ A we denote by β : G → A the map given by action on the element a, i.e., a weset: (1.1.4) β (g) = g·a, a (resp.,β (g) = a·g),forallg ∈ G. Theset: a G = β−1(a) a a is a subgroup of G and is called the isotropy group of the element a ∈ A. The G-orbit (or, more simply, the orbit) of the element a ∈ A is the set Ga (resp., aG) given by the image of G under the map β ; a subset of A is called a G-orbit a (or, more simply, an orbit) ifit isequal to the G-orbit ofsome element of A. The setofallorbitsconstituteapartitionofthesetA. Themapβ inducesabijection a from the set G/G of left (resp., right) cosets of the isotropy subgroup G onto a a