to Conformal Geometry Lecture 1(保形几何讲义).pdf

《to Conformal Geometry Lecture 1(保形几何讲义).pdf》由会员上传分享，免费在线阅读，更多相关内容在教育资源-天天文库。

1、IntroductiontoConformalGeometryC.RobinGrahamLecture1FlatmodelforconformalgeometryConformalgroup:O(n+1;1)Lecture2CurvedconformalgeometryWeylandCottontensorsCharacterizationofconformal atnessLecture3AmbientmetricTractorbundleandconnection2Riemanniangeometry:M:smoothmanifoldg:metriconMCanmea

2、surelengthsoftangentvectors:jvj2=g(v;v);v2TpMIsometry:preserveslengthsConformalgeometry:Canonlymeasureangles:g(v;w)(v;w);cos=jvjjwjSameasknowingguptoscaleateachpointConformalmapping:preservesanglesSameaspreservingguptoscale3FlatModelRiemanniangeometry:PRn,Euclideanmetric:(dyi)2Isometries:E(n)=

3、groupofEuclideanmotionsy!Ay+b;A2O(n);b2RnCanviewE(n)GL(n+1;R)by:!Ab(A;b)$01!nn+1yEmbedR,!Rby:y$1!!!AbyAy+bThen=01114ConformalTransformationsofRnEuclideanmotionsDilationsy!sy;s2R+InversionsinspheresyForunitsphere,centeratorigin,gety!jyj2Denition:M•ob(Rn)=GroupgeneratedbyEuclideanmotions,dilatio

4、ns,inversionsM•ob(Rn)providesoneapproachtothestudyoftheconformalgroup.Butinversionsdon'tmapRntoRn:centerofspheregoesto1SuggestscompactifyingRnbyappending1UnnecessaryandinappropriateforEuclideanmotions5CompactifyRntoSn=Rn[1.MetriconRn+1inducesmetriconSn;henceaconformalstructureonSn.Snn1isconforma

5、llyequivalenttoRnviastereographicprojection.Fruitfulpointofview:describeconformalgeome-tryofSnintermsofMinkowskigeometryofRn+2:quadraticformQofsignature(n+1;1)x=(x0;x1;


;xn+1)2Rn+2XnQ(x)=(x)2(xn+1)2=0N=fx:Q(x)=0gRn+2nf0gNullconenoPn+1=`=[x]:x2Rn+2nf0glinesinRn+2Q=f`=[x]:x2NgPn+1QuadricQ=Sn

6、:Lety2Sn,soy2Rn+1;jyj=1."!#nyThemapS3y!2Qisabijection.1:N!Qprojection6Xn2n+12ge=(dx)(dx)Minkowskimetric=0!IJI0=geIJdxdx;wheregeIJ=01nX+1X=xI@n+2xIpositionvectoreldonRI=0Ifx2N,thenX(x)2TxNHavege(X;X)=geIJxIxJ=Q(x)=0onN.Claim:ge(X;V)=0forallV2TN.Proof:Q=0onN.SodQ(V)=0ifV2TN.NowQ=geIJxIxJ,sodQ=

7、2geIJxIdxJ.Gives0=dQ(V)=2geIJxIVJ=2ge(X;V).7Letx2N.Thengeisdegenerate:X?TxNTxN(x)ButgeinducesaninnerproductgonT`Q,TxNwhere`=(x)=[x].g(x)isdenedasfollows:Have:N!Q.Gives:TxN!T`QNow(X)=0,and:TxN=spanX!T`Qisanisomorphis