Mappings from R^n to F^n which preserve unit Eucli

arXiv:math/0305232v4[math.MG]26Jun2003MappingsfromRntoFnwhihpreserveunitEulideandistane,whereFisaeldofharateristi0ApoloniuszTyszkaSummary.LetFbeaommutativeeldofharateristi0,ϕn:Fn×Fn→F,ϕn((x1,...,xn),(y1,...,yn))=(x1−y1)2+...+(xn−yn)2.Wesaythatg:Rn→Fnpreservesdistaned≥0ifforeahx,y∈Rn|x−y|=dimpliesϕn(g(x),g(y))=d2.Letf:Rn→Fnpreserveunitdistane.Weprove:(1)ifn≥2,x,y∈Rnandx6=y,thenϕn(f(x),f(y))6=0,(2)ifA,B,C,D∈R2,r∈Qand−−→CD=r−→AB,then−−−−−−→f(C)f(D)=r−−−−−−→f(A)f(B),(3)ifA,B,C,D∈R2and−→ABand−−→CDarelinearlydependent,then−−−−−−→f(A)f(B)and−−−−−−→f(C)f(D)arelinearlydependent,(4)ifA,B,C,D∈R2and−→ABisperpendiularto−−→CD,then−−−−−−→f(A)f(B)isperpendiularto−−−−−−→f(C)f(D),(5)ifA,B,C,D∈R2and|AB|=|CD|,thenϕ2(f(A),f(B))=ϕ2(f(C),f(D)).LetAn(F)denotethesetofallpositivenumbersdsuhthatanymapg:Rn→Fnthatpreservesunitdistanepreservesalsodistaned.LetDn(F)denotethesetofallpositivenumbersdwiththeproperty:ifx,y∈Rnand|x−y|=dthenthereexistsanitesetSxywith{x,y}⊆Sxy⊆Rnsuhthatanymapmapg:Sxy→Fnthatpreservesunitdistanepreservesalsothedistanebetweenxandy.Obviously,{1}⊆Dn(F)⊆An(F).Weprove:(6)An(C)⊆{d0:d2∈Q},(7){d0:d2∈Q}⊆D2(F).2000MathematisSubjetClassiation:51K05,51M05.Keywordsandphrases:Bekman-Quarlestheorem,Cayley-Mengerdeterminant,endomorphism(au-tomorphism)oftheeldofomplexnumbers,isometry,unit-distanepreservingmapping,unitEu-lideandistane.LetFbeaommutativeeldofharateristi0,ϕn:Fn×Fn→F,ϕn((x1,...,xn),(y1,...,yn))=(x1−y1)2+...+(xn−yn)2.Wesaythatf:Rn→Fnpreservesdistaned≥0ifforeahx,y∈Rn|x−y|=dimpliesϕn(f(x),f(y))=d2.Inthispaperwestudyunit-distanepreservingmappingsfromRntoFn.ByaomplexisometryofCnweunderstandanymaph:Cn→Cnoftheformh(z1,z2,...,zn)=(z′1,z′2,...,z′n)wherez′j=a0j+a1jz1+a2jz2+...+anjzn(j=1,2,...,n),theoeientsaijareomplexandthematrix||aij||(i,j=1,2,...,n)isorthogonali.e.satisestheonditionnXj=1aμjaνj=δμν(μ,ν=1,2,...,n)withKroneker’sdelta.Aordingto[6℄,ϕn(x,y)isinvariantunderomplexisometriesi.e.foreveryomplexisometryh:Cn→Cn(⋄)∀x,y∈Cnϕn(h(x),h(y))=ϕn(x,y).Conversely,ifh:Cn→Cnsatises(⋄)thenhisaomplexisometry;itfollowsfrom[4,proposition1,page21℄byreplaingRwithCandd(x,y)withϕn(x,y).Similarly,iff:Rn→Cnpreservesalldistanes,thenthereexistsaomplexisometryh:Cn→Cnsuhthatf=h|Rn.ByaeldendomorphismofCweunderstandanymapg:C→Csatisfying:∀x,y∈Cg(x+y)=g(x)+g(y),∀x,y∈Cg(x·y)=g(x)·g(y),g(0)=0,g(1)=1.Bijetiveendomorphismsarealledautomorphisms,formoreinformationonelden-domorphismsandautomorphismsofCthereaderisreferredto[10℄,[11℄and[16℄.Ifrisarationalnumber,theng(r)=rforanyeldendomorphismg:C→C.Proposition1showsthatonlyrationalnumbersrhavethisproperty:Proposition1([15℄).Ifr∈Candrisnotarationalnumber,thenthereexistsaeldautomorphismg:C→Csuhthatg(r)6=r.2Proof.NoterstthatifEisanysubeldofCandifgisanautomorphismofE,thenganbeextendedtoanautomorphismofC.Thisfollowsfrom[7,orollary1,A.V.111℄bytakingΩ=CandK=Q.Nowletr∈C\Q.IfrisalgebraioverQ,letEbethesplittingeldinCoftheminimalpolynomialμofroverQandletr′∈Ebeanyotherrootofμ.ThenthereexistsanautomorphismgofEthatsendsrtor′,seeforexample[9,orollary2,page66℄.IfristransendentaloverQ,letE=Q(r)andletr′∈EbeanyothergeneratorofE(e.g.r′=1/r).ThenthereexistsanautomorphismgofEthatsendsrtor′.Ineahase,ganbeextendedtoanautomorphismofC.Ifg:C→Cisaeldendomorphismthen(g|R,...,g|R):Rn→Cnpreservesalldistanes√rwithrationalr≥0.Indeed,if(x1−y1)2+...+(xn−yn)2=(√r)2thenϕn((g|R,...,g|R)(x1,...,xn),(g|R,...,g|R)(y1,...,yn))=ϕn((g(x1),...,g(xn)),(g(y1),...,g(yn)))=(g(x1)−g(y1))2+...+(g(xn)−g(yn))2=(g(x1−y1))2+...+(g(xn−yn))2=g((x1−y1)2)+...+g((xn−yn)2)=g((x1−y1)2+...+(xn−yn)2)=g((√r)2)=g(r)=r=(√r)2.Analogously,ifg:R→Fisaeldhomomorphismthen(g,...,g):Rn→Fnpreservesalldistanes√rwithrationalr≥0.Conjeture1.Eahunit-distanepreservingmappingfromRntoFn(n≥2)hasaformI◦(g,...,g),whereg:R→FisaeldhomomorphismandI:Fn→Fnisananemappingwithorthogonallinearpart.Theorem1([15℄).Ifx,y∈Rnand|x−y|2isnotarationalnumber,thenthereexistsf:Rn→Cnthatdoesnotpreservethedistanebetweenxandyalthoughfpreservesalldistanes√rwithrationalr≥0.Proof.ThereexistsanisometryI:Rn→RnsuhthatI(x)=(0,0,...,0)andI(y)=(|x−y|,0,...,0).ByProposition1thereexistsaeldautomorphismg:C→Csuhthatg(|x−y|2)6=|x−y|2.Thusg(|x−y|)6=|x−y|andg(|x−y|)6=−|x−y|.Therefore(g|R,...,g|R):Rn→Cndoesnotpreservethedistanebetween(0,0,...,0)∈Rnand(|x−y|,0,...,0)∈Rnalthough(g|R,...,g|R)preservesalldistanes√rwithrationalr≥0.Henef:=(g|R,...,g|R)◦I:Rn→Cndoesnotpreservethedistanebetweenxandyalthoughfpreservesalldistanes√rwithrationalr≥0.3LetAn(F)denotethesetofallpositivenumbersdsuhthatanymapf:Rn→Fnthatpreservesunitdistanepreservesalsodistaned.ThelassialBekman-Quarlestheoremstatesthateahunit-distanepreservingmappingfromRntoRn(n≥2)isanisometry,see[1℄-[4℄and[8℄.Itmeansthatforeahn≥2An(R)=(0,∞).ByTheorem1An(C)⊆{d0:d2∈Q}.LetDn(F)denotethesetofallpositivenumbersdwiththefollowingproperty:(∗)ifx,y∈Rnand|x−y|=dthenthereexistsanitesetSxywith{x,y}⊆Sxy⊆Rnsuhthatanymapf:Sxy→Fnthatpreservesunitdistanepreservesalsothedistanebetweenxandy.Obviously,{1}⊆Dn(F)⊆An(F).From[13℄and[14℄followsthatforeahn≥2Dn(R)isequaltothesetofpositivealg