Topics in modern differential geometry PDF
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Atlantis Transactions in Geometry Stefan Haesen Leopold Verstraelen Editors Topics in Modern Differential Geometry Atlantis Transactions in Geometry Volume 1 Series editor Johan Gielis, Antwerpen, Belgium The series aims at publishing contemporary results in geometry including large partsofanalysisandtopology.Theserieswillpublishbooksofboththeoreticaland appliednature. Theoreticalvolumeswill focus among other topicson submanifold theory, Riemannian and pseudo-Riemannian geometry, minimal surfaces and submanifolds in Euclidean geometry. Applications are found in biology, physics, engineering and other areas. More information about this series at http://www.atlantis-press.com/series/15429 Stefan Haesen Leopold Verstraelen (cid:129) Editors Topics in Modern Differential Geometry Editors StefanHaesen Leopold Verstraelen Department ofTeacher Education Department ofMathematics ThomasMore University College University of Leuven Vorselaar Leuven Belgium Belgium Atlantis Transactionsin Geometry ISBN978-94-6239-239-7 ISBN978-94-6239-240-3 (eBook) DOI 10.2991/978-94-6239-240-3 LibraryofCongressControlNumber:2016955687 ©AtlantisPressandtheauthor(s)2017 This book, or any parts thereof, may not be reproduced for commercial purposes in any form or by anymeans, electronic or mechanical, including photocopying, recording or any information storage andretrievalsystemknownortobeinvented,withoutpriorpermissionfromthePublisher. Printedonacid-freepaper Preface In 2008 and 2009, the Simon Stevin Institute for Geometry participated in the organization of Ph.D. courses at the universities of Leuven (Belgium), Kragujevac (Serbia), Murcia (Spain) and Brasov (Romania). Besides the main course lectures on “Natural geometrical intrinsic and extrinsic symmetries”, there were invited shortlecturesonavariedselectionoftopicsindifferentialgeometry.Severalofthe lecturers were able to find the time to prepare their talks for the publication in this book. Oursincerethanksgotoallthestudentswhoparticipatedatthevariouscourses, toProf.Dr. R.Deszcz andDr.A.Albujerwho taughtseveralofthemainlectures, toProf. Dr. L. Alías, Prof. Dr. F. Dillen,Dr. J. Gielis, Prof. Dr. I.Mihai,Prof. Dr. M. Petrović-Torgašev and Prof. Dr. E. Stoica for the local organization of the courses and to all the invited lecturers. Vorselaar, Belgium Stefan Haesen Leuven, Belgium Leopold Verstraelen February 2010 v Contents The Riemannian and Lorentzian Splitting Theorems... .... ..... .... 1 José Luis Flores Periodic Trajectories of Dynamical Systems Having a One-Parameter Group of Symmetries . .... .... .... .... ..... .... 21 Roberto Giambò and Paolo Piccione Geometry and Materials. .... ..... .... .... .... .... .... ..... .... 37 Bennett Palmer On Deciding Whether a Submanifold Is Parabolic or Hyperbolic Using Its Mean Curvature ... ..... .... .... .... .... .... ..... .... 49 Vicente Palmer Contact Forms in Geometry and Topology... .... .... .... ..... .... 79 Gheorghe Pitiş Farkas and János Bolyai .... ..... .... .... .... .... .... ..... .... 95 Mileva Prvanović Spectrum Estimates and Applications to Geometry .... .... ..... .... 111 G. Pacelli Bessa, L. Jorge, L. Mari and J. Fábio Montenegro Some Variational Problems on Curves and Applications.... ..... .... 199 Angel Ferrández Special Submanifolds in Hermitian Manifolds .... .... .... ..... .... 223 Ion Mihai An Introduction to Certain Topics on Lorentzian Geometry. ..... .... 259 Alfonso Romero vii The Riemannian and Lorentzian Splitting Theorems JoséLuisFlores Abstract In these notes we are going to briefly review some of the main ideas involved in the formulation and proof of the Riemannian and Lorentzian Splitting Theorems.Wewilltrytoemphasizethesimilaritiesanddifferencesappearedwhen passing from the Riemannian to the Lorentzian case, and the way in which these difficultiesareovercomebytheauthors. 1 Introduction ThesplittingprobleminRiemannianandLorentzianGeometryiscloselyrelatedto the idea of “rigidity” in Geometry. So, in order to introduce this problem, first we aregoingtodedicatesomelinestorecallthisimportantnotion. Assume that we are interested in studying some Riemannian manifold (M,g). Usually,itisveryusefultocompareitwithsomemodelspaceM ,i.e.acomplete1- K connectedRiemannianmanifoldofconstantsectionalcurvatureK.Infact,therearea seriesofresultswhichensurethat(M,g)willretainglobalgeometricalpropertiesof M undercertainstrictcurvatureboundsfor(M,g)intermsofK.Evenmore,under K theseconditions,itisusuallypossibletoconcludethatMwillalsoretaintopological propertiesofM .Anaturalquestionwhicharisesfromthissituationis,whathappen K when one relaxes the condition of “strict” curvature inequality to some “weak” curvatureinequality?Itisnotdifficulttorealizethat,underthesenewhypotheses, the conclusion may not hold any more. This is clearly illustrated by the following simpleobservation:thereisacrucialdifferencebetweenthetopologyofthesphere TheauthorisgratefultotheorganizersoftheInternationalResearchSchoolonDifferential GeometryandSymmetry,celebratedatUniversityofMurciafromMarch9to18,2009,for givinghimtheopportunitytodeliveralectureonwhichthesenotesarebased.Partiallysupported bySpanishMEC-FEDERGrantMTM2007-60731andRegionalJ.AndalucíaGrantP06-FQM- 01951. B J.L.Flores( ) DepartamentodeÁlgebra,GeometríayTopologíaFacultaddeCiencias, UniversidaddeMálagaCampusTeatinoss/n,29071Málaga,Spain e-mail:fl[email protected] ©AtlantisPressandtheauthor(s)2017 1 S.HaesenandL.Verstraelen(eds.),TopicsinModernDifferentialGeometry, AtlantisTransactionsinGeometry1,DOI10.2991/978-94-6239-240-3_1 2 J.L.Flores (K >0) and that of the Euclidean space (K ≡0), even for spheres of radius very big,andso,withcurvatureveryclosetothenullcurvatureoftheEuclideanspace. However,arelevantpropertystillholds:aconclusionwhichbecomesfalsewhenone relaxesthe“strict”curvatureconditiontoa“weak”curvatureconditionusuallycan beshowntofailonlyunderveryspecialcircumstances!Thisimportantidea,usually referredas“rigidity”inGeometry,isroughlysummarizedinthefollowingprototype result: PrototypeRigidityTheorem:IfM satisfiesa“weak”curvaturecondition,andthe geometric restriction derived from the corresponding “strict” curvature condition doesnotholdanymore,then M mustbe“veryspecial”. Inorder torelatethesplittingproblemtothisprototyperigiditytheorem,letus recallthefollowingresultbyGromollandMeyer[17]: Theorem1.1 (Gromoll,Meyer)AcompleteRiemannianmanifold(M,g)ofdimen- sionn ≥2suchthat Ric(v,v)>0forallv ∈TM isconnectedatinfinity. This is a typical result where a strict curvature inequality (Ric(v,v)>0) implies atopologicalrestrictiononthemanifold(connectednessatinfinity).Now,suppose thatwereplacethestrictcurvaturecondition Ric(v,v)>0bytheweakcurvature condition Ric(v,v)≥0and,consequently,weassumethat(M,g)failstobecon- nectedatinfinity.Since M iscomplete,nowonecanensuretheexistenceofaline joining any two different ends of M. Under these new hypotheses, Cheeger and Gromoll proved that (M,g) must be isometric to a product manifold. So, this is a typicalrigiditytheoreminthesensedescribedabove.ThisresultanditsLorentzian versionconstitutethecentralsubjectofthesenotes. In the next section we will establish with precision the Riemannian Splitting Theorem. We will also provide some brief comments about the initial motivation andtheprecedentsofthetheorem.Finally,wewillintroducesomebasicnotionsand resultswhichwillbeusedlaterintheproof.InSect.3wewilloutlinethemainideas involvedintheoriginalproofbyCheegerandGromoll.Thisproofstronglyusesthe theory of elliptic operators, and, indeed, it is stronger than actually needed. So, in Sect.4wewilldescribeanalternativeproofofthesameresult,givenbyEschenburg andHeintze,whichminimizestheuseoftheelliptictheory.Thissecondapproach willberelevantforusbecauseitintroducesanewviewpointusefulfortheproofof theLorentzianversionofthetheorem.InSect.5,wewillrecallsomebasicnotions andresultsfromLorentzianGeometry.Afterthat,wewillestablishtheLorentzian Splitting Theorem in Sect.6, providing some brief comments about the main hits inthehistoryofitssolution.TheproofofthisresultwillbestudiedinSect.7.We willessentiallyfollowtheargumentsgivenbyGallowayin[14]:aftersomeprevious technical lemmas in Subsects.7.1–7.3, the proof will be delivered in six steps in Subsect.7.4.Finally,inSect.8wewillrecallarelatedopenproblemwithaphysical significance,theBartnik’sConjecture. TheRiemannianandLorentzianSplittingTheorems 3 2 RiemannianSplittingTheorem TheRiemannianSplittingTheoremcanbestatedinthefollowingway[7]: Riemannian Splitting Theorem (Cheeger, Gromoll) Suppose that the Riemannianmanifold(M,g),ofdimensionn ≥2,satisfiesthefollowingconditions: (1) (M,g)isgeodesicallycomplete, (2) Ric(v,v)≥0forallv ∈TM, (3) M has a line (i.e. a complete unitary geodesic γ :R→(M,g) realizing the distancebetweenanytwoofitspoints). Then M is isometric to the product (M,g)∼=(Rk ×M ,g ⊕g ), k >0, where 1 0 1 (M ,g )containsnolinesandg isthestandardmetriconRk. 1 1 0 This is a very important result which has been extensively used in Riemannian Geometry in the last decades. An important precedent of this result is due to Topogonov [23], who obtained the same thesis under the more restrictive curva- tureassumption of nonnegative sectional curvature. The proof of theTopogonov’s result lies on the Triangle Comparison Theorem by the same author. The original motivationfortheCheegerandGromoll’sresultwasthenecessitytoextendtheexist- ingresultsconcerningthefundamentalgroupofmanifoldsofnonnegativesectional curvature[8]tothecaseofnonnegativeRiccicurvature.Inparticular,theyneeded asplittingtheoremundertheweakerhypothesisofnonnegativeRiccicurvature,for whichtheTopogonov’sTriangleComparisonTheoremdoesnotwork.Afirstsplit- tingresultofthistypewereobtainedbyCohn-Vossenin[9].However,thegeneral resultrequiredtotallynewarguments,whichwerenotdevelopedtillthepublication oftheremarkablepaper[7]. InordertodescribetheproofoftheCheeger–GromollSplittingTheorem,firstly weneedtointroducesomepreviousnotions,whichareofinterestbyitself: Byarayγwewillunderstandanunitarygeodesicdefinedon[0,∞)whichrealizes thedistancebetweenanyofitspoints.Then,theBusemannfunction(associatedto γ)isdefinedasthefunctionb : M →R3obtainedfromthelimit γ b (·):= lim(r −d(·,γ(r))), (1) γ r→∞ whered isthedistanceassociatedtotheRiemannianmetricg.Itisnotdifficultto provethatpreviouslimitalwaysexists(isfinite)andtheresultingfunctioniscontin- uous.Infact,thelimit(1)existsandisfinitebecause,fromthetriangleinequality, themap r (cid:9)→b (p)=r −d(p,γ(r)) r isnondecreasing r −r =d(γ(r ),γ(r ))≥d(p,γ(r ))−d(p,γ(r )) if r ≤r 2 1 1 2 2 1 1 2 andboundedabove
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