2XINCAOExample1.5.Wedeneatranslationt(;)byt(x;y)=(x+;y+).Inotherw - PDF document

2XINCAOExample1.5.Wedeneatranslationt(;)byt(x;y)=(x+;y+).Inotherwords,thetranslationt(;)movestheorigin,O,to(;).Weverifythatatranslationisanisometrybyshowingthatitpreservesthesquareofthedistancebetweenanytwopointsontheeuclideanplane.Since,(x02�x01)2+(y02�y01)2=(x2+�(x1+))2+(y2+�(y1+))2=(x2�x1)2+(y2�y1)2;weconcludethatd(f(P1);f(P2))=d(P1;P2)forallP1;P22R2.Andbecause,t(;)
t(�;�)=t(�;�)
t(;)=1,theinverseoft(;)ist(�;�).Example1.6.Wedeneare ectioninthex-axisisby rx�axis(x;y)=(x;�y).Aquickcalculationshowsthatthere ectionsareisometries,what'smore,isthatanyre ectionisitsowninverse(i.e. r= r�1).Example1.7.Wedenearotationrbyr(x;y)=(xcos�ysin;xsin+ycos).Toshowthateuclideandistanceispreserved,wesquarethedistance,reducetheidentities,andsimplifytoget,d(f(P1);f(P2))=[(x2cos�y2sin)�(x1cos�y1sin)]2=[(x2�x1)cos�(y2�y1)sin]2+[(x2�x1)sin+(y2�y1)cos]2=(x2�x1)2cos2�2(x2�x1)(y2�y1)+(y2�y1)2sin2+(x2�x1)2sin2+2(x2�x1)(y2�y1)+(y2�y1)2cos2=(x2�x1)2+(y2�y1)2:Now,weshowthattherotationshaveinverses.Exercise1.8.Showthatrr=r+andhencethatr�1=r�Byexample1.7,afterrotationbyangleaboutO,wehavex0=xcos�ysin,y0=xsin+ycos.AndafterrotationbyangleaboutO,wehavex00=x0cos�y0sin=(xcos�ysin)cos�(xsin+ycos)sin=x(coscos�sinsin)�y(sincos+sincos)=xcos(+)�ysin(+);Likewise,y00=xsin(+)+ycos(+),whichshowsthatrr=r+.Furthersubstituting�for,weconcludethatr�=r�1.Havingfoundtheinversesforourexamplesthusfar,wecannowexpressrotationsandre ectionsatpointsandlinesotherthantheoriginorx-axis.Thisispossiblebytakingtheconjugateofaisometry.Conjugationisthemethodbywhichweperformthesameoperationbutinanewcoordinatesystem.Forinstance,there ectiont(;) rt�1(;)istheconjugateof rbyt(;).Theconjugatebyt(;)treatsthepoint(;)asifitweretheoriginoftheplane. 4XINCAO Figure1Second,takeL;Mtobethex-axisandtheliney==2.ThenifwetakeL0;M0tobethelinesy=!;y=!+=2,wendthat rM0 rL0=r= rM rL.Corollary1.11.Thesetoftranslationsandrotationsisclosedunderproduct.Proof.It'sclearthattheproductoftwotranslationswillalwaysbeatranslation.Tondtheproductoftworotationsaboutdierentxedpoints,wecandescribetheserotationsasre ectionsaboutlines.Usingtheabovecorollary,givenrotationsrP;andrQ;aboutpointsPandQrespectively,wecanchoosetorepresenttheproductoftheserotationsasproductsofre ectionsinlinesL;M;NshowninFigure1.ThenrP;
rQ;= rN rM
rM rL= rN rL,whichbyTheorem1.9,isarotationifNmeetsLandatranslationotherwise.Also,byasuitablechoiceoflines,theproductofatranslationandarotationcanalsobeexpressedastheproductoftwore ections.Weareabouttoshowthatallisometriesaretheproductsofatmostthreere ec-tions.Forsimplicity,weassumetheresultsofthefollowinglemmaandcorollary.Basically,thelemmastatesthateveryeuclideanisometryisdeterminedbyitseectonatriangle.Lemma1.12.AnyisometryfofR2isdeterminedbytheimagesf(A),f(B),f(C)ofthreepointsA,B,Cnotinaline.Corollary1.13.IfListhelineofpointsequidistantfrompointsPandQ,thenre ectioninLexchangesPandQ.Theorem1.14(ThreeRe ectionsTheorem).AnyisometryfofR2istheproductofone,two,orthreere ections.Proof.ConsiderthepointsA;B;Cnotinalineandtheirf-imagesf(A);f(B);f(C).Case(i)IftwopointsofA;B;Ccoincidewiththeirf-images,sayf(A)=Aandf(B)=B,thenthere ectioninthelineLthroughA;BsendsCtof(C)bycorollary1.13. EUCLIDEANISOMETRIESANDSURFACES5Case(ii)IfonlyonepointofA;B;Ccoincideswithitsf-image,sayf(A)=A,rstperformre ection ginthelineMofpointsequidistanttoBandf(B).Then, gsendsA;Btof(A);f(B)respectively.If g(C)=f(C),thenwearedone.Ifnot,thenbycase(i),weperformanotherre ection hinthelineLthroughf(A);f(B),whichsends gCtof(C).Case(iii)NoneofthepointsofA;B;Ccoincidewiththeirf-images.Weperformatmostthreere ections g; h; i,inlinesrespectivelyequidistantfromAandf(A), g(B)andf(B), h(C)andf(C).Bysimilarargumentsasintheprevioustwocases,weareensuredthattheproductofone,two,orthreeofthesere ectionsendsAtof(A),Btof(B),Ctof(C),thusprovingourtheorem.Itstillremainstobeseenhowwewillclassifytheeuclideanisometriesintotranslations,rotations,andglidere ections.Thiswillbecomepossiblebytherealizationthatwecanseparatetheisometriesintotwogroups,onegroupcontain-ingtheproductsofeven-numberedre ections,andasecondgroupcontainingtheproductsofodd-numberedre ections.Wenaturallyrecognizethegroupofeven-numberedre ectionsastheorientation-preservingisometries,andtheothergroupastheorientation-reversingisometries.Corollary1.15.TheisometriesofR2formagroupIso(R2),andtheproductsofevennumbersofre ectionsformasubgroupIso+(R2)ofindex2.Proof.Itisclearthatassociativityholdsforproductsofre ections,hence,alsoforproductsofisometries.Thereisalsoanidentityisometry.Thetheoremestablishesthateveryisometryalsohasaninverse.Sincere ectionsareself-inverses,theinverseofaisometry rL1::: rLnis rLn::: rL1.Thus,theisometriesformagroup,Iso(R2).Italsofollowsthattheproducts rL1::: rL2nofevennumbersofre ectionsformasubgroup,Iso+(R2)sinceproductsandinversesofsuchisometriesareofthethesameform.ToshowthatIso+(R2)isofindex2,weshowthatthecosetIso+(R2)
r=fproductsofoddnumbersofre ectionsgisnotIso+(R2).Thisisequivalenttoshowingthat r=2Iso+(R2).Bycorollary1.11,Iso+(R2)containsonlyrotationsandtranslations.Wededucethatre ectionscannotbelonginthissetsincethexed-pointsetofnon-trivialrotationsisonexedpoint,andthexed-pointsetoftranslationsiszero,whereasthexed-pointsetofre ectionsisalineofpoints.Wehavealreadyshownthatisometrieswhicharetheproductsofeven-numberedre ectionsaretranslationsandrotations.Itstillremainstoshowthattheprod-uctsofthreere ectionsandonere ectionareanothertypeofisometry:theglidere ection.Denition1.16.GlideRe ectionsAglidere ectionisanorientation-reversingisometry,describedastheproductofoneorthreere ections(theproductofatranslationandare ection).Fromthisdenition,re ectionscanalsobedescribedasglidere ections,inparticularthosewithtrivialtranslations.Nowweshowthattheremainingeuclideanisometriesareglidere ections.Theorem1.17.Aproduct rN rM rLofre ectionsinlinesL;M;Nisaglidere- ection. EUCLIDEANISOMETRIESANDSURFACES9Thus,anyneighborhoodDQincludesdistinctpointsR;gRinthesame�-orbit,contrarytothehypothesis.Theorem2.4.Adiscontinuous,xedpointfreegroup�ofisometriesofR2isgeneratedbyoneortwoelements.FollowingLemma2.3,weneedonlyconsidertranslationsorproperglidere ec-tionsaspossiblegeneratorsof�,sincebothre ectionsandrotationshavexedpoints.Weprovethistheoremfortranslationsonly,thoughtheideaissimilarifweincludeglidere ections.Proof.Supposethat�containstranslationsonly.ChooseapointP2R2.Since�isdiscontinuous,thereisaminimumdistance�0betweenPandanyotherpointinthe�-orbitofP.Wechooseourrstgeneratort12�asoneofthesenearestmembersof�PtoP.Weclaimthatthepowers:::t�11;1;t1;t21;:::oft1includealltranslationsin�withthesamedirectionast1.Ift2�isanothertranslationinthesamedirectionast1,andiftm1PistheclosestpossiblepointtotP,thent�1tm12�wouldbeatranslationinthesamedirectionast1,butshorter,contrarytothechoiceoft1.Thus,ifthethesetf:::t�11;1;t1;t21;:::gdoesnotexhaust�,thentheremainingtranslationshaveadierentdirection.Inthiscase,wechooseanothertranslationofminimallengthandcallitt2.Sincet1andt2areindierentdirections,theygeneratealatticeofpointsinR2{theverticesofatessellationofR2byequalparallelograms(seeFigure).Weshowthatthislatticeisthewhole�-orbitofP.Supposethatthepowersoft1t2=ftm1tn2jm;n2Zgdonotexhaust�,thenthereisatranslationt2�whichisnotalatticetranslation.Iftm1tn2PisthepointclosesttotP,thent�1tm1tn2isatranslationshorterthant1ort2,whichcontradictsourchoicesoft1andt2.ThisisequivalenttoshowingthatanypointQwithinaparallelogramissep-aratedfromatleastonevertexbyadistancelessthanthelengthofthelongestside.Thisisclearfromthegure(seegure).SinceQmustlieinone-halfoftheparallelogram,Qmustlieinsideacirclewiththelongsideoftheparallelogramastheradius.Corollary2.5.S=R2=�isacylinder,twistedcylinder,torus,orKleinbottle.Fromtheprevioustheorems,wearriveatthesesurfacesdependingonthegen-eratorsof�.1.(Cylinder)-Generatedbyasingletranslation.2.(TwistedCylinder)-Generatedbyasingleglidere ection.3.(Torus)-Generatedbytwotranslations.4.(KleinBottle)-Generatedbyaglidere ectionandtranslation(ortwoglidere ections).Wehavefoundallthepossiblecomplete,connected,euclideansurfaceswhichcanbegeneratedbydierentisometrygroups,butitremainstobeshownthatthesearetheonlysurfacespossible.Thus,werstshowthateuclideanplaneisacoverfortheanysuchsurface,andthenprovethatthereisadiscontinuous,xed-pointfreegroupofisometrieswhichgenerateanyofthesesurfaces.ToshowthattheeuclideanplaneR2coversacomplete,connectedeuclideansurfaceS,weuseadeviceknownasthepencilmap.Sincethepencilofapointis EUCLIDEANISOMETRIESANDSURFACES11 Figure4Exercise2.7.Supposethatg1;g2arecoveringisometriesforacoveringf.Werstshowthatg1;g2isacoveringisometry.Sinceg2P2R2,thenfg1(g2P)=fg2P,andhencefg2P=fP.Thisshowsthatfg1g2P=fP,whichmakesg1g2acoveringisometry.Toshowthatanycoveringisometryghasaninverse,webeginbywritinganyP2R2asg�1Q,usingthefactthatgisinvertible.Then,fgg�1Q=fg�1Q,i.e.,fQ=fg�1QforallQ2R2,whichsaysthatg�1isacoveringisometryaswell.Now,toprovethatpisa�-orbitmapunderthecoveringisometrygroup.Theorem2.8.IfpP=pQ,thenQ=gPforsomecoverpandcoveringisometryg(i.e.,P;Qareinthesame�-orbit).Proof.Bythelocalisometrypropertyofp,therearediscneighborhoodsD(P)andD(Q)mappedisometricallybypontoadiscp(D(P))=p(D(Q))ofS.Thus,D(P)p!pD(P)p�1!D(Q)isanisometryg:D(P)!D(Q).SinceD(P)containsthreepointsnotinaline,gisoneoftheeuclideanisometriesclassiedinsection1.Now,weshowthatgisacoveringisometryforp,thatpR=pgRforallR2R2.SupposeforacontradictionthatpR6=pgRforsomeR2R2.Then,considerthesetfRjpR=pgRg,whichweclaimisopenbythefactthatpandpgageeontheopendiscD(P).Then,thesetfRjpR6=pgRgisclosed.SincethefRjpR6=pgRgisclosed,theremustbealeastelement(theoneclosesttoP),whichweshallcallR0.GiventhatR0isthenearestpointtoPwithintheset,considerasequenceR01;R02;:::ofpointsbetweenPandR0whichconvergetoR0.Byhypothesis,pR0i=pgR0iandbycontinuityofpandg,p(limi!1Ri)=pg(limi!1Ri);whichshowsthatpR=pgR,andhenceisacontradiction.Thus,pR=pgRforallR2R2,andgisacoveringisometry. 12XINCAOCorollary2.9(Killing-Hopftheorem).Eachcomplete,connectedeuclideansurfaceisoftheformR=�,andhenceiseitheracylinder,twistedcylinder,torus,orKleinbottle(ifnotR2itself).Acknowledgments.Iwouldliketothankmymentor,DanielStudemund,forintroducingmetothetopicsofthispaper,andforhishelpthroughoutthepaper-writingprocess.IwouldalsoliketothankProfessorMayformakingtheREUanexcitingandinterestingexperienceforstudentsofallmathematicalbackgrounds.References[1]JohnStillwell.GeometryofSurfaces.1992Springer-VerlagNewYork,Inc.