Semiparallel Submanifolds in Space Forms Ülo Lumiste Semiparallel Submanifolds in Space Forms 123 ÜloLumiste InstituteofPureMathematics UniversityofTartu Tartu50409 Estonia [email protected] ISBN: 978-0-387-49911-6 e-ISBN: 978-0-387-49913-0 DOI: 10.1007/978-0-387-49913-0 LibraryofCongressControlNumber:2007924353 MathematicsSubjectClassification(2000):53-02,53B25,53-C35,53C40 ©2009 SpringerScience+BusinessMedia, LLC Allrightsreserved. Thisworkmaynotbetranslatedorcopiedinwholeorinpartwithoutthewritten permissionofthepublisher(SpringerScience+BusinessMedia,LLC,233SpringStreet,NewYork,NY 10013,USA),exceptforbriefexcerptsinconnectionwithreviewsorscholarlyanalysis.Useinconnection withanyformofinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilar ordissimilarmethodologynowknownorhereafterdevelopedisforbidden. 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Printedonacid-freepaper springer.com Contents 0 Introduction................................................. 1 1 Preliminaries ................................................ 7 1.1 RealSpaceswithBilinearMetric ............................. 7 1.2 MovingFrames ............................................ 8 1.3 (Pseudo-)RiemannianManifolds.............................. 10 1.4 StandardModelsofSpaceandSpacetimeForms ................ 11 1.5 Symmetric(Pseudo-)RiemannianManifolds .................... 13 1.6 Semisymmetric(Pseudo-)RiemannianManifolds ................ 16 2 SubmanifoldsinSpaceForms.................................. 23 2.1 ASubmanifoldandItsAdaptedFrameBundle .................. 23 2.2 Higher-OrderFundamentalForms ............................ 26 2.3 FundamentalIdentities ...................................... 29 2.4 OsculatingandNormalSubspacesofHigherOrder .............. 29 3 ParallelSubmanifolds ........................................ 33 3.1 Parallelandk-ParallelSubmanifolds .......................... 33 3.2 Examples: SegreandPlückerSubmanifolds .................... 36 3.3 Example: VeroneseSubmanifold ............................. 40 3.4 ParallelSubmanifoldsandtheGaussMap ...................... 43 3.5 ParallelSubmanifoldsandLocalExtrinsicSymmetry ............ 44 3.6 CompleteParallelIrreducibleSubmanifoldsasStandardImbedded SymmetricR-Spaces........................................ 46 4 SemiparallelSubmanifolds .................................... 51 4.1 TheSemiparallelConditionandItsSpecialCases................ 51 4.2 TheSemiparallelConditionfromtheAlgebraicViewpoint ........ 54 4.3 DecompositionofSemiparallelFundamentalTriplets ............ 57 4.4 TripletsofLargePrincipalCodimension ....................... 59 4.5 SemiparallelSubmanifoldsasSecond-OrderEnvelopesofParallel Submanifolds.............................................. 63 vi Contents 4.6 Second-OrderEnvelopeofSegreSubmanifolds ................. 66 4.7 ANewApproachtoVeroneseSubmanifolds .................... 70 5 NormallyFlatSemiparallelSubmanifolds........................ 73 5.1 PrincipalCurvatureVectorsandtheSemiparallelCondition ....... 73 5.2 NormallyFlatParallelSubmanifolds .......................... 75 5.3 AdaptedFrameBundleforaSecond-OrderEnvelope ............ 78 5.4 Second-OrderEnvelopeasWarpedProduct..................... 80 5.5 SemiparallelSubmanifoldsofPrincipalCodimension1........... 84 5.6 Semiparallel Submanifolds of Principal Codimension 2 in EuclideanSpace ........................................... 89 5.7 Normally Flat Semiparallel Submanifolds of Principal Codimension2inNon-EuclideanSpaceForms.................. 93 6 SemiparallelSurfaces......................................... 97 6.1 SemiparallelSpacelikeSurfaces .............................. 97 6.2 TheCaseofRegularMetrics ................................. 99 6.3 VeroneseSurfaces .......................................... 101 6.4 Second-OrderEnvelopesofVeroneseSurfaces .................. 106 6.5 TheCaseofaSingularMetric ................................ 108 6.5.1 Thesubcaseswherespan{A,B}hassingularmetric ....... 109 6.5.2 Thesubcaseswherespan{A,B}hasregularmetric ........ 112 6.6 SemiparallelTimelikeSurfacesinLorentzSpacetimeForms ...... 114 6.6.1 Theprincipalcase.................................... 115 6.6.2 Theexceptionalcase ................................. 119 6.7 Spacelike2-ParallelSurfaces................................. 123 6.8 q-ParallelSurfacesasSemiparallelSurfaces .................... 130 7 SemiparallelThree-DimensionalSubmanifolds ................... 135 7.1 SemiparallelSubmanifoldsM3ofPrincipalCodimensionm ≤2.. 135 1 7.2 NonminimalSemiparallelM3ofPrincipalCodimensionm =3... 138 1 7.3 SemiparallelM3ofPrincipalCodimensionm =4 .............. 147 1 7.4 HigherPrincipalCodimensions: Conclusions ................... 154 8 DecompositionTheorems...................................... 157 8.1 DecompositionofSemiparallelSubmanifolds................... 157 8.2 DecompositionofParallelSubmanifolds ....................... 162 8.3 DecompositionofNormallyFlat2-ParallelSubmanifolds......... 164 8.4 StructureofSubmanifoldswithFlatvanderWaerden–Bortolotti Connection................................................ 168 9 Umbilic-LikenessofMainSymmetricOrbits ..................... 175 9.1 TwoKindsofSymmetricOrbits .............................. 175 9.2 Umbilic-LikenessofPlückerOrbits ........................... 178 9.3 UnitaryOrbitsofthePlückerAction........................... 181 9.4 Umbilic-LikenessofUnitaryOrbits ........................... 184 Contents vii 9.5 TheSegreActionandItsSymmetricOrbits..................... 195 9.6 TheVeroneseActionandItsSymmetricOrbits .................. 197 9.7 TheProblemofUmbilic-LikenessofVeroneseOrbits ............ 201 9.8 Umbilic-LikenessofVeronese–GrassmannOrbits ............... 205 9.9 DetailedAnalysisofaModelCase ............................ 214 10 GeometricDescriptionsinGeneral.............................. 219 10.1 ProductsofUmbilic-LikeOrbits .............................. 219 10.2 GeneralSemiparallelSubmanifoldsandTheirAdaptedFrame Bundles................................................... 223 10.3 WarpedProductsandImmersedFibreBundles .................. 227 10.4 SemiparallelSubmanifoldsofCylindricalorToroidalSegreType .. 229 10.4.1 Thecaseofumbilic-likeSegreorbits.................... 230 10.4.2 Thecaseofnonumbilic-likeSegreorbits................. 235 11 IsometricSemiparallelImmersionsofRiemannianManifoldsof ConullityTwo ............................................... 237 11.1 SemiparallelSubmanifoldswithPlaneGeneratorsof Codimension2............................................. 237 11.2 SomeParticularCases ...................................... 241 11.3 SemiparallelManifoldsofConullityTwoinGeneral ............. 242 12 SomeGeneralizations......................................... 249 12.1 k-SemiparallelSubmanifolds................................. 249 12.2 On2-SemiparallelSubmanifolds.............................. 252 12.3 2-SemiparallelSurfacesinSpaceForms........................ 253 12.4 RecurrentandPseudoparallelSubmanifolds .................... 261 12.5 SubmanifoldswithSemiparallelTensorFields .................. 263 12.6 Examples: TheSurfaces..................................... 266 12.6.1 H-semiparallelandH-parallelsurfaces.................. 267 ⊥ 12.6.2 R -parallelsurfaces.................................. 270 12.6.3 R-orRic-parallelsurfaces. ............................ 271 12.6.4 T-semiparallelsurfaces ............................... 271 12.7 Ric-SemiparallelHypersurfacesandRyan’sProblem............. 272 12.8 ExtendedRyan’sProblemforNormallyFlatSubmanifolds........ 279 12.9 R-SemiparallelbutNotSemiparallelNormallyFlatSubmanifolds ofCodimension2 .......................................... 282 References ...................................................... 287 Index........................................................... 303 0 Introduction AmongRiemannianmanifolds,themostinterestingandmostimportantforapplica- tions are the symmetric ones. From the local point of view, they were introduced independentlybyP.A.Shirokov[Shi25]andH.Levy[Le25]asRiemannianmani- foldswithcovariantlyconstant(alsocalledparallel)curvaturetensorfieldR,i.e.,with ∇R =0, (0.1) where∇ istheLevi-Civitaconnection[L-C17]. Anextensivetheoryofsymmetric Riemannian manifolds was worked out by É. Cartan in [Ca 26]. He showed that a Riemannian manifold M has parallel R if and only if every point x has a normal neirhbourhoodsuchthatallgeodesicsymmetrieswithrespecttox areisometries. Ifforeachpointx ∈Mthereexistsaninvolutiveisometrys ofMforwhichxis x anisolatedfixedpoint,thenMiscalleda(globally)symmetricspace. Theclosureof thegroupofisometriesgeneratedby{s :x ∈M}inthecompact-opentopologyisa x LiegroupGthatactstransitivelyonthesymmetricspace;hencethetypicalisotropy subgroupH atapointofM iscompact,andM =G/H. TheclassicalexamplesareconnectedcompleteRiemannianmanifoldswithcon- stantsectionalcurvaturec,calledspaceforms(see[Wo72],Section2.4). Later,asimilardevelopmenttookplaceinthegeometryofsubmanifoldsinspace forms, where a fundamental role is played by the first (or metric) form g (as the inducedRiemannianmetric)andthesecondfundamentalformh. BesidestheLevi- Civita connection ∇, with ∇g = 0, a normal connection ∇⊥ is also defined. The submanifoldswithparallelfundamentalform,i.e.,with ∇¯h=0, (0.2) where∇¯ isthepairof∇and∇⊥,deservespecialattention. DuetotheGaussidentity, eachofthemisintrinsicallyalocallysymmetricRiemannianmanifold. The first result here was given by V. F. Kagan [Ka 48], who showed that in Euclidean space E3, the surfaces with parallel h are open subsets of planes, round spheres, and circular cylinders S1 ×E1. All of these have nonnegative Gaussian Ü. Lumiste, Semiparallel Submanifolds in Space Forms, DOI 10.1007/978-0-387-49913-0_1, © Springer Science+Business Media, LLC 2009 2 0 Introduction curvature. The surfaces of negative constant Gaussian curvature in E3 are there- fore examples of submanifolds which are intrinsically locally symmetric, but have nonparallelh. The hypersurfaces with parallel h in En were determined by U. Simon and A.Weinstein[SW69]. SomenewexamplesofsurfaceswithparallelhinE4 were givenbyC.-S.Houh[Ho72]: theCliffordtoriS1×S1andtheVeronesesurfaces. The generaltheoryofsubmanifoldsMm withparallelhinEn wasinitiatedbyJ.Vilms [Vi 72], who showed, in particular, that each of them has totally geodesic Gauss image. NormallyflatsubmanifoldswithparallelhinEuclideanspacesandspheres wereclassifiedbyR.Walden[Wa73]. AproperlydevelopedtheorywasworkedoutbyD.Ferus[Fe74,80]. Heproved thatasubmanifoldMmwithparallelhinEnhasthepropertyoflocalextrinsicsym- metry,inthesensethateverypointhasaneighborhoodinvariantunderreflectionof En with respect to the normal subspace at this point; also conversely, an Mm with this property has parallel h. This was proved in general, for Mm in a Riemannian manifoldNn, byW.Strübing[St79]. Therefore, thesubmanifoldswithparallelh, especiallythecompleteones,werecalledsymmetricsubmanifoldsbyFerus(andthen by others); here extrinsically was meant, but often not explicitly stated. The other important result of Ferus was that a general symmetric submanifold in En reduces to a product of irreducible symmetric submanifolds, each of which (except possi- blyaEuclideansubspace)liesinasphere, isminimalinit, andcanbeobtainedas thestandardimmersionofaRiemanniansymmetricR-space. Conversely,eachsuch standardimmersiongivesasymmetricsubmanifold;andtheproductsoftheseimmer- sions(possiblyincludingaEuclideansubspace)exhaustallsymmetricsubmanifolds in En. These results gave a classification of such submanifolds in terms of special chaptersofthetheoryofLiegroupsandsymmetricspaces.Allofthesesubmanifolds canbeconsideredassymmetricorbits. This classification was then extended to submanifolds with parallel h in space forms by M. Takeuchi [Ta 81], who found it more suitable here to use the term parallel submanifolds. This term has become more popular, especially when the localpointofviewhasbeenconsidered. The theory of parallel submanifolds is concisely treated in recent monographic worksbyB.-Y.Chen[Ch2000](Chapter8),Ü.Lumiste[Lu2000](Sections5–7), andbyJ.Berndt,S.Console,andC.Olmos[BCO2003](Section3.7: “Symmetric submanifolds’’). AlreadyinthefirstinvestigationsofsymmetricRiemannianmanifolds[Shi25] and [Ca 26], it was noted that these manifolds must also satisfy the integrability condition R(X,Y)·R =0 (0.3) of the differential system ∇R = 0. (Here X and Y are tangent vector fields, and R(X,Y)isconsideredasafieldoflinearoperators,actingonR.) Riemannianman- ifoldswiththispoint-wiseconditionwereconsideredseparatelybyÉ.Cartanin[Ca 46]. HisinvestigationswerecontinuedbyA.Lichnerowicz[Li52,58]andR.Couty [Co57]. ThetermsemisymmetricforRiemannianmanifoldsM satisfyingthiscon- 0 Introduction 3 dition was introduced by N. S. Sinyukov [Si 56, 62], who showed the importance ofthisconditioninthetheoryofgeodesicmappingsofRiemannianmanifolds(see [Si79],Chapter2,Section3). AfruitfulimpulseforinvestigationsofmanifoldsofthisclasswasgivenbyK.No- mizuin[No68], whoconjecturedthatallcompleteirreduciblen-dimensionalRie- mannianmanifolds(n ≥ 3)satisfyingR(X,Y)·R = 0arelocallysymmetric,i.e., thattheymustalsosatisfy∇R =0. Thisconjecturewassupportedbytheresultthat foraRiemannianmanifold, ∇kR = 0withk > 1implies∇R = 0, provedforthe compactcasein[Li58],andforthecompletecasein[NO62];andthisisalsovalid ingeneral(cf.[KN63],Vol.1,Remark7). However,Nomizu’sconjecturewaseven- tuallyrefuted. Namely, in[Ta72]ahypersurfaceinE4 wasconstructedsatisfying R(X,Y)·R = 0 but not ∇R = 0. Acounterexample of arbitrary dimension was givenin[Sek72]. SemisymmetricRiemannianmanifoldswereclassifiedbyZ.I.Szabó,locally,in [Sza82]. HeshowedthatforeverysemisymmetricRiemannianmanifoldM,there existsaneverywheredenseopensubsetU ofM,suchthataroundeverypointofU, themanifoldislocallyisometrictoaspacethatisthedirectproductofanopensubset ofaEuclideanspaceandofinfinitesimallyirreduciblesimplesemisymmetricleaves, eachofwhichiseither(i)locallysymmetric, or(ii)locallyisometrictoanelliptic, a hyperbolic, a Euclidean, or a Kählerian cone, or (iii) locally isometric to a space foliatedbyEuclideanleavesofcodimension2(ortoatwo-dimensionalmanifold,in thecasedimM =2). TheseclassificationresultsofSzabówerepresentedbrieflyinthebook[BKV96], whosemainpurposewastosummarizerecentresultsonsemisymmetricRiemannian manifoldsofsubclass(iii);thesearenowcalledRiemannianmanifoldsofconullity two,andmaybeconsideredthemostinterestingamongsemisymmetricRiemannian manifolds. Parallelsubmanifoldswerelikewiselaterplacedinamoregeneralclassofsub- manifolds, generalizing the parallel ones in the same sense as locally symmetric Riemannianmanifolds(i.e.,with∇R =0)weregeneralizedbysemisymmetricRie- mannianmanifolds(i.e.,withR(X.Y)·R =0). Namely,theintegrabilitycondition ofthedifferentialsystem∇¯h=0is R¯(X,Y)·h=0, (0.4) whereR¯isthecurvatureoperatoroftheconnection∇¯ =∇⊕∇⊥,andX,Yaretangent vectorfields,asabove. Thisconditioninfactalreadycameupin[Fe74a]andthenin [BR83]. ThegeneralconceptofsubmanifoldsinEnsatisfying(0.4)wasintroduced by J. Deprez [De 85], who called them semiparallel. He proved that all of them are,intrinsically,semisymmetricRiemannianmanifoldsandgaveaclassificationof semiparallelsurfacesinEn. In[De86],healsoclassifiedsemiparallelhypersurfaces, andin[De89],summarizedthesefirstresults. The investigation of semiparallel submanifolds was continued by the author in [Lu 87a, 88a,b, 89a–c, 90a–e], etc., then by F. Dillen in [Di 90b, 91b], [DN 93], andA.C.Aspertiin[As93], [AM94]. Thefirstsummarieswerepublishedin[Lu 91f]andtheninthemonographicarticle[Lu2000a](whosereviewinMathematical 4 0 Introduction Reviews(see[MR2000j: 53071])isconcludedbyA.Buckiasfollows: “Theauthor’s contributiontothetheoryofsubmanifoldswithparallelfundamentalformwithhis more than forty papers on the subject is colossal’’). Currently the monograph [Lu 2000a]isnolongercompletelyuptodate;severalnewresultshavebeenaddedtothe theorysincethen. Thepresentbookwillgiveamorecompletesurveyofthetheoryofsemiparallel submanifoldsandofsomegeneralizationsinspaceforms. Semiparallelsubmanifolds aretreatedheremainlyassecond-orderenvelopesofsymmetricorbits. Thebookconsistsoftwelvechapters. Thefirstthreechaptersarepreparatoryin character. InChapter1,thenecessarybackgroundforsubsequentchaptersisgiven usingframebundles(i.e.,theCartanmovingframemethod)andexteriordifferential calculus, together with vector and tensor bundles. Basic facts from the theories of spaceformsandofsymmetricandsemisymmetricRiemannianmanifoldsarecovered. InChapter2,thegeneraltheoryofsmoothsubmanifoldsinspaceformsisdevel- oped. Thesecondfundamentalformhisintroduced, togetherwithitshigher-order generalizations, theirfundamentalidentities, andthecorrespondingnormalandos- culatingsubspacesarecovered. Thisisdonebyusingorthonormalframessuitably adaptedtothesubmanifold. InChapter3,thetheoryofparallelsubmanifoldsisdeveloped. Herethespecifics oftheirGaussmaps,theirlocalextrinsicsymmetry,Ferus’sdecompositiontheorem and its connection with symmetric R-spaces are presented. The most important examples of complete parallel submanifolds are also given: Segre, Plücker, and Veronesesubmanifolds. All of this is in preparation for the main subject, which is the investigation of semiparallel submanifolds. These are introduced in Chapter 4, where some char- acterizations for their class and several subclasses are given. It is emphasized that (0.4)isapointwiseconditionandthereforecanbetreatedpurelyalgebraically. The decompositiontheoremforsemiparallelsubmanifoldsisalsodealtwithinthesame manner. Theanalyticfact,thatthesesubmanifoldsarecharacterizedbytheintegrabil- ityconditionofthedifferentialsystem(0.2)forparallelsubmanifolds,isinterpreted geometricallyinthetheoremfrom[Lu90a],statingthateverysemiparallelsubman- ifoldisasecond-orderenvelopeofparallelsubmanifolds;suchenvelopesarefound forSegresubmanifolds,asexamples(extendingtheresultof[Lu91a]). Chapter 5 is devoted to normally flat semiparallel submanifolds. This class in- cludes all semiparallel submanifolds of principal codimension 1, in particular hy- persurfaces,andalsosemiparallelsubmanifoldsofprincipalcodimension2inspace formsofnonpositivecurvature.Ageneralgeometricdescriptionisgivenfornormally flatsemiparallelsubmanifoldsasimmersedwarpedproductsofspheres. SemiparallelsubmanifoldsoflowdimensionsareconsideredinChapters6(sur- faces) and 7 (three-dimensional submanifolds). They are all classified; the sub- manifolds of the most general class are described as second-order envelopes of Veronese submanifolds. It is shown that each two-dimensional holomorphic Rie- mannianmanifoldcanbeimmersedisometricallyinto(pseudo-)Euclideanspaceof dimension≥7,asasurfaceofthismostgeneralclassofsemiparallelsurfaces;butthis doesnotgeneralizetothreedimensions. Somegeneralclassesofsemiparallelthree-