Lie Groups and Geometric Aspects of Isometric Actions Marcos M. Alexandrino • Renato G. Bettiol Lie Groups and Geometric Aspects of Isometric Actions 123 MarcosM.Alexandrino RenatoG.Bettiol DepartamentodeMatemática DepartmentofMathematics InstitutodeMatemáticaeEstatística UniversityofPennsylvania UniversidadedeSãoPaulo Philadelphia,PA,USA SãoPaulo,Brazil ISBN978-3-319-16612-4 ISBN978-3-319-16613-1 (eBook) DOI10.1007/978-3-319-16613-1 LibraryofCongressControlNumber:2015936906 SpringerChamHeidelbergNewYorkDordrechtLondon ©SpringerInternationalPublishingSwitzerland2015 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade. Printedonacid-freepaper SpringerInternationalPublishingAGSwitzerlandispartofSpringerScience+BusinessMedia(www. springer.com) Dedicatedtoourfamilies. Preface This book is intended for advanced undergraduates, graduate students, and young researchersingeometry.Itwaswrittenwithtwomaingoalsinmind.First,wegivea gentleintroductiontotheclassicaltheoryofLiegroups,usingaconcisegeometric approach. Second, we provide an overview of topics related to isometric actions, exploringtheirrelationswiththeresearchareasoftheauthorsandgivingthemain ideas of proofs. We discuss recent applications to active research areas, such as isoparametricsubmanifolds,polaractionsandpolarfoliations,cohomogeneityone actions, and positive curvature via symmetries. In this way, the text is naturally dividedintwointerrelatedparts. Let us give a more precise description of such parts. The goal of the first part (Chaps.1 and 2) is to introduce the concepts of Lie groups, Lie algebras and adjoint representation, relating these objects. Moreover, we give basic results on closedsubgroups,bi-invariantmetrics,Killingforms,andsplittingofLiealgebras in simple ideals. This is done concisely due to the use of Riemannian geometry, whosefundamentaltechniquesarealsoquicklyreviewed. The second part (Chaps.3–6) is slightly more advanced. We begin with some results on proper and isometric actions in Chap.3, presenting a few research comments. In Chap.4, classical results on adjoint and conjugation actions are presented, especially regarding maximal tori, roots of compact Lie groups, and Dynkindiagrams.Inaddition,theconnectionwithisoparametricsubmanifoldsand polar actions is explored. In Chap.5, we survey on the theory of polar foliations, which generalizes some of the objects studied in the previous chapter. Finally, Chap.6 briefly discusses basic aspects of homogeneous spaces and builds on all thepreviousmaterialtoexplorethegeometryoflowcohomogeneityactionsandits interplaywithmanifoldswithpositive(andnonnegative)sectionalcurvature. Prerequisites expected from the reader are a good knowledge of advanced calculus and linear algebra, together with rudiments of calculus on manifolds. Nevertheless, a brief review of the main definitions and essential results is given intheAppendixA. This book can be used for a one-semester graduate course (of around 3h per week) or an individual study, as it was written to be as self-contained as possible. vii viii Preface PartofthematerialinChap.3,aswellasChaps.5and6,maybeskippedbystudents in a first reading. Most sections in the book are illustrated with several examples, designedtoconveyageometricintuitiononthematerial.Thesearecomplemented by exercises that are usually accompanied by a hint. Some exercises are labeled with a star ((cid:2)), indicating that they are slightly more involved than the others. We encourage the reader to think about them, in an effort to develop a good working knowledgeofthematerialandpracticeactivereading. The present book evolved from several lecture notes that we used to teach graduate courses and minicourses. In 2007, 2009, and 2010, graduate courses on Lie groups and proper actions were taught at the University of São Paulo, Brazil, exploring mostly the first four chapters of the text. Graduate students working in variousfieldsfollowedthesecourses,withverypositiveresults.Duringthisperiod, the same material was used in a graduate course at the University of Parma, Italy. Relevant contributions also originated from short courses given by the authors during the XV Brazilian School of Differential Geometry (Fortaleza, Brazil, July 2008), the Second São Paulo Geometry Meeting (São Carlos, Brazil, February 2009),andtheReyPastorSeminarattheUniversityofMurcia(Murcia,Spain,July 2009).In2009,apreliminarydraftofthistextwaspostedonthearXiv(0901.2374), which prompted instructors in various universities to list it as complementary study material. Since then, we have substantially improved and updated the text, particularlythelastchapters,featuringmanyrecentadvancesintheresearchareas discussed. Thereareseveralimportantresearchareasrelatedtothecontentofthisbookthat arenottreatedhere.Wewouldliketopointouttwoofthese,forwhichwehopeto givethenecessarybackground:first,representationtheoryandharmonicanalysis, for which we recommend Bröken and Tom Dieck [56], Deitmar [78], Fulton and Harris [90], Gangolli and Varadarajan [94], Helgason [125], Katznelson [136], Knapp[144],andVaradarajan[217],and,second,symmetriesindifferentialequa- tions and integrable systems, for which we recommend Bryant [57], Fehér and Pusztai[86,87],Guest[119],Noumi[175],andOlver[176]. Acknowledgements We acknowledge financial support from CNPq and FAPESP (Brazil) and NSF (USA). The authors are very grateful to Alexander Lytchak, Gudlaugur Thorbergsson, KarstenGrove,PaoloPiccione,andWolfgangZillerfortheirconstantsupportandseveralvaluable discussions.ItisalsoapleasuretothankAugustoRitterStoffel,DanielVictorTausk,DirkTöben, FábioSimas,FlausinoLucasSpindola,FranciscoCarlosJunior,IonMoutinho,IvanStruchiner, JohnHarvey,LászlóFehér,LeandroLichtenfelz,LeonardoBiliotti,MartinKerin,MartinWeilandt, Rafael Briquet, Renée Abib, and Ricardo Mendes for their helpful comments and suggestions duringtheelaborationofthisbook. SãoPaulo MarcosM.Alexandrino Philadelphia RenatoG.Bettiol June2015 Contents PartI LieGroups 1 BasicResultsonLieGroups................................................ 3 1.1 LieGroupsandLieAlgebras .......................................... 3 1.2 LieSubgroupsandLieHomomorphisms ............................. 7 1.3 ExponentialMapandAdjointRepresentation ........................ 13 1.4 ClosedSubgroupsandMoreExamples ............................... 18 2 LieGroupswithBi-invariantMetrics ..................................... 27 2.1 BasicFactsofRiemannianGeometry................................. 27 2.2 Bi-invariantMetrics.................................................... 38 2.3 KillingFormandSemisimpleLieAlgebras .......................... 41 2.4 SplittingLieGroupswithBi-invariantMetrics....................... 45 PartII IsometricActions 3 ProperandIsometricActions .............................................. 51 3.1 ProperActionsandFiberBundles..................................... 51 3.2 SlicesandTubularNeighborhoods.................................... 64 3.3 IsometricActions....................................................... 69 3.4 PrincipalOrbits......................................................... 73 3.5 OrbitTypes ............................................................. 76 4 AdjointandConjugationActions.......................................... 85 4.1 MaximalToriandPolarActions....................................... 85 4.2 NormalSlicesofConjugationActions................................ 92 4.3 RootsofaCompactLieGroup ........................................ 93 4.4 WeylGroup............................................................. 99 4.5 DynkinDiagrams....................................................... 102 5 PolarFoliations............................................................... 109 5.1 DefinitionsandFirstExamples........................................ 109 5.2 HolonomyandOrbifolds............................................... 111 ix