Differential geometry of the q-quaternion

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a rXiv:mat h-ph/02714v11J ul22Differential Geometry of Group Lattices Aristophanes Dimakis ∗Department of Financial and Management Engineering,University of the Aegean,31Fostini Str.,GR-82100Chios Folkert M¨u ller-Hoissen †Max-Planck-Institut f¨u r Str¨o mungsforschung,Bunsenstrasse 10,D-37073G¨o ttingen Abstract In a series of publications we developed “differential geometry”on discrete sets based on concepts of noncommutative geometry.In particular,it turned out that first order differential calculi (over the algebra of functions)on a discrete set are in bijective correspondence with digraph structures where the vertices are given by the elements of the set.A particular class of digraphs are Cayley graphs,also known as group lattices.They are determined by a discrete group G and a finite subset S .There is a distinguished subclass of “bicovariant”Cayley graphs with the property ad(S )S ⊂S .We explore the properties of differential calculi which arise from Cayley graphs via the above correspondence.The first order calculi extend to higher orders and then allow to introduce further differential geometric structures.Furthermore,we explore the properties of “discrete”vector fields which describe deterministic flows on group lattices.A Lie derivative with respect to a discrete vector field and an inner product with forms is defined.The Lie-Cartan identity then holds on all forms for a certain subclass of discrete vector fields.We develop elements of gauge theory and construct an analogue of the lattice gauge theory (Yang-Mills)action on an arbitrary group lattice.Also linear connections are considered and a simple geometric interpretation of the torsion is established.By taking a quotient with respect to some subgroup of the discrete group,general-ized differential calculi associated with so-called Schreier diagrams are obtained.Contents1Introduction3 2First order differential calculus associated with a group lattice4 3Differentiable maps between group lattices9 4Higher order differential calculus of a group lattice114.1Action of differentiable maps on forms (13)4.2The structure of the space of2-forms (14)5Discrete and basic vectorfields175.1Discrete vectorfields (17)5.2Discrete vectorfields with invertibleflow (20)5.3Another extension of theflow to forms and vectorfields on a bicovariant grouplattice (22)5.4Basic vectorfields (24)5.5Lie derivative with respect to a discrete vectorfield (26)5.6Inner product of discrete vectorfields and forms (27)6Connections and parallel transports306.1Gauge theory (32)7Linear connections347.1A transport of vectorfields (36)7.2The geometric meaning of(vanishing)torsion (37)7.3Linear connections on vectorfields (38)8Differential calculi on coset spaces of discrete groups398.1Higgsfield from gauge theory with an internal coset lattice (45)9Conclusions46 Appendices47 A Integral curves of discrete vectorfields47References481IntroductionIn a series of papers[1–7]we developed differential geometry on discrete sets(see also Refs.[8–14]for related work).A key concept is a differential calculus(over the algebra A of functions)on a set.First order differential calculi on discrete sets were found to be in bijective correspondence with digraph structures[3],where the vertices of the digraph are given by the elements of the set and neither multiple arrows nor loops are admitted.In particular,this supplies the elements of the set with neighborhood relations.An important example is a differential calculus which corresponds to the hypercubic lattice and which leads to an elegant formulation of lattice gauge theory[1].A special class of digraphs are Cayley graphs[15](see Refs.[16,17],for example),which are also known as group lattices in the physics literature.These are determined by a discrete group G and a subset S.The elements of G are the vertices of the digraph and the elements of S determine(via right action)arrows from a vertex g to“neighboring”vertices.Hypercubic lattices,on which the usual lattice(gauge)theories are built,are special Cayley graphs. Another example of importance for physics is the truncated icosahedron which models the C60 Fullerene[18].Physical models on group lattices have also been considered in Refs.[19–22], in particular.Furthermore,Cayley graphs play a role in the study of connectivity and routing problems in communication networks(see Ref.[23]for a review).The above-mentioned correspondence between digraphs andfirst order differential cal-culi suggests to explore those calculi which correspond to Cayley graphs.Moreover,given a first order differential calculus which corresponds to a Cayley graph,it naturally extends to higher orders so that we have a notion of r-forms,r>1.This provides the basis for intro-ducing further differential geometric structures,following general recipes of noncommutative geometry.In section2we introducefirst order differential calculi associated with group lattices. Our approach very much parallels standard constructions in ordinary differential geometry. In particular,wefirst introduce vectorfields on a group lattice and then1-forms as duals of these.Section3concerns maps between group lattices which are“differentiable”in an algebraic sense[4].Of special importance for us are“bicovariant”group lattices(G,S) with the property that the left and right actions on G with respect to all elements of S is differentiable.Afirst order differential calculus naturally extends to higher orders,i.e.to a full differen-tial calculus.The structure of differential calculi obtained from group lattices is the subject of section4.Geometric relations are often more conveniently expressed in terms of vectorfields than forms.In section5we introduce a special class of vectorfields which we call“discrete”and a subclass of“basic”vectorfields and explore their properties.A Lie derivative with respect to a discrete vectorfield and an inner product of discrete vectorfields and forms is defined. For basic vectorfields with differentiableflow the Lie-Cartan formula holds.Section6treats connections on(left or right)A-modules over differential calculi associ-ated with group lattices.In particular,Yang-Millsfields are considered and an analogue of the lattice gauge theory action on an arbitrary group lattice is constructed.If the module is the space of1-forms,we are dealing with linear connections.This is thesubject of section7.In particular,wefind that the condition of vanishing torsion of a linear connection has a simple geometric meaning.A differential calculus on a group lattice induces a“generalized differential calculus”on a coset space.The resulting differential calculus is generalized in the sense that the space of1-forms is,in general,larger than the A-bimodule generated by the image of the space of functions under the action of the exterior derivative.There is a generalized digraph (“Schreier diagram”[16])associated with such afirst order differential calculus which in general has multiple links and also loops.Some further remarks are collected in section9. 2First order differential calculus associated with a group latticeLet G be a discrete group and A the algebra of complex-valued functions f:G→C.[24] With g∈G we associate e g∈A such that e g(g′)=δg,g′for all g′∈G.The set of e g, g∈G,forms a linear basis of A over C,since every function f can be written in the form f= g∈G f(g)e g.In particular,we have e g e g′=δg,g′e g and g∈G e g=1,where1denotes the constant function which is the unit of A.The left and right translations by a group element g,L g(g′)=gg′and R g(g′)=g′g, induce automorphisms of A via the pull-backs(L∗g f)(g′)=f(L g g′)=f(gg′)and(R∗g f)(g′)= f(R g g′)=f(g′g).In particular,we obtainL∗g e g′=e g−1g′,R∗g e g′=e g′g−1(2.1) for all g,g′∈G.Introducing[25]ℓg f=R∗g f−f(2.2) so that(ℓg f)(g′)=f(g′g)−f(g′),wefind the modified Leibniz ruleℓg(ff′)=(ℓg f)(R∗g f′)+f(ℓg f′).(2.3) The mapsℓg:A→A,g∈G,generate an A-bimodule via(f·ℓg)f′:=fℓg f′,(ℓg·f)f′:=(ℓg f′)(R∗g f)(2.4) so thatℓg·f=(R∗g f)·ℓg.(2.5) Indeed,one easily verifies that(ff′)·ℓg=f·(f′·ℓg),ℓg·(ff′)=(ℓg·f)·f′.(2.6) The modified Leibniz rule can now be written asℓg(ff′)=(ℓg·f′)f+(f·ℓg)f′.(2.7)Let S be afinite subset of G which does not contain the unit of G.From G and S we construct a directed graph as follows.The vertices of the digraph represent the elements of G and there is an arrow from the site(vertex)representing g to the one representing gh if and only if h∈S.In other words,there is an arrow from g to g′iffg−1g′∈S.A digraph obtained in this way is called a Cayley graph or a group lattice.[26]Lemma2.1The connected component of the unit e in the group lattice is the subgroup of G generated by S.Proof:Let H be the subgroup of G generated by S.Every element g∈H can be written as afinite product g=h k11···h k r r with h i∈S and k i∈{±1}.If k r=1,there is an arrowfrom h k11···h k r−1r−1to g.If k r=−1,there is an arrow from g to h k11···h k r−1r−1.By iteration,g is connected to e.Hence H is contained in the connected component C e of e.Because of the group property,every element connected to an element of H must itself be an element of H.Hence C e=H.It follows that the group lattice(G,S)is connected if and only if S generates G(see also Ref.[27],p.17).If the subgroup H generated by S is smaller than G,the group lattice consists of a set of disjoint but isomorphic parts corresponding to the set of left cosets gH, g∈G.For h∈S,the mapsℓh:A→A are naturally associated with the arrows of the digraph since(ℓh f)(g)=f(gh)−f(g)is the difference of the values of a function f at two connected “neighboring”points of the digraph.The mapsℓh generate an A-bimodule X.[28]At each g∈G,they span a linear space which we call the tangent space at g.LetΩ1be the A-bimodule dual to X such thatf·X,α = X,fα =f X,α , X·f,α = X,αf (2.8) for all X∈X,f∈A andα∈Ω1.If{θh|h∈S}denotes the set of elements ofΩ1dual to {ℓh|h∈S},so that ℓh′,θh =δh h′,thenℓh′,θh f = ℓh′·f,θh = (R∗h′f)·ℓh′,θh =R∗h′fδh h′= ℓh′,(R∗h f)θh (2.9) for all h,h′∈S.Henceθh f=R∗h fθh.(2.10) The space of1-formsΩ1is a free A-bimodule and{θh|h∈S}is a basis.A linear map d:A→Ω1can now be introduced byd f= h∈S(ℓh f)θh.(2.11) It satisfies the Leibniz rule d(ff′)=(d f)f′+f(d f′).In particular,we obtaind e g= h∈S(ℓh e g)θh= h∈S(e gh−1−e g)θh.(2.12)Now we multiply both sides from the left by e gh−1with somefixed h∈S.Since h is differentfrom the unit element of G,we obtain e gh−1d e g=e gh−1θh.From this wefind[29]θh= g∈G e gh−1d e g= g∈G e g d e gh.(2.13) Furthermore,θ:= h∈Sθh= g∈G,h∈S e g d e gh(2.14) satisfiesd f=θf−fθ=[θ,f].(2.15) Moreover,we obtainX,d f =Xf.(2.16) Let us introduceI={(g,g′)∈G×G|g−1g′∈S e}(2.17) where S e=S∪{e}.This is the set of pairs(g,g′)for which e g d e g′=0.Note that e g d e g=−e gθ=0.Thefirst order differential calculus(A,Ω1,d)constructed above is also obtained from the universalfirst order differential calculus(A,Ω1u,d u)as the quotientΩ1=Ω1u/J1with respect to the submodule J1ofΩ1u generated by all elements of the form e g d u e g′with(g,g′)∈I. Ifπu:Ω1u→Ω1denotes the corresponding projection,then we have d=πu d u.Lemma2.2If S e is a subgroup of G,the correspondingfirst order differential calculus on the component connected to the unit is the universal one.Proof:According to Lemma2.1,the e-component is S e.Since for every pair(h,h′)∈S e×S e, h=h′,there is an element h′′∈S such that h=h′h′′,there is an arrow from h′to h in the associated digraph.Hence all pairs of different elements of S e are connected by a pair of antiparallel arrows.This characterizes the universal differential calculus. Example2.1.One of the simplest examples is obtained as follows.Let G=Z,the additivegroup of integers,and S={1}.Then we have(ℓ1f)(k)=f(k+1)−f(k)andθ1= k∈Z e k d e k+1.Introducing the coordinate function t= k∈Z k e k,wefindθ1=d t and ℓ1f=∂+t f with the discrete derivative∂+t f(t)=f(t+1)−f(t).Henced f=(∂+t f)d t.(2.18) This example is important as a model for a discrete parameter space,and in particular as a model for discrete time.A generalization is obtained by taking the additive group G=Z n and S={(1,0,...,0),(0,1,0,...,0),...,(0,...,0,1)}=:{ˆm|1≤m≤n}which generates G.This leads to an oriented hypercubic lattice digraph.Then(ℓˆm f)(k)=f(k+ˆm)−12311112222Figure 1:The group lattice of Z 4with S ={1,2}.f (k )=:(∂+ˆm f )(k )and θˆm =k ∈Z n e k d e k +ˆm .Introducing coordinates via x = k ∈Z n k e k =(x 1,...,x n ),we find d f =n m =1(∂+ˆm f )d x m ,θˆm =d x m .(2.19)This differential calculus appeared first in Ref.[1](see also Ref.[30])and turned out to be useful,in particular,in the context of lattice gauge theory [31]and completely integrable lattice models [5]. Example 2.2.Let G =Z m (m =2,3,...),the finite additive group of elements 0,1,2,m −1with composition law addition modulo m .The unit element is e =0.Choosing S ={1},we have a single basis 1-form θ1.In contrast to example 2.1,here θ1is not exact.Indeed,suppose that θ1=d f for some function f .This is equivalent to ℓ1f =1which leads to the contradiction m = g (ℓ1f )(g )=0.By taking direct products of this lattice,a group lattice structure for G =Z n m is obtained.Example 2.3.For G =Z 2,the only group lattice is the complete digraph corresponding to the universal first order differential calculus on the two elements {0,1}.For G =Z 3,one has to distinguish two cases.If S contains a single element only,the group lattice is a closed linear chain of arrows (cf example 2.2).The choice S ={1,2}leads to the complete digraph on the three elements and thus to the universal differential calculus.Less simple structures appear for G =Z m ,m >3.For example,choosing G =Z 4and S ={1,2},we obtain the group lattice drawn in Fig.1. Example 2.4.The permutation group S 3has the 6elementse ,(12),(13),(23),(123),(132)grouped into conjugacy classes.Choosing S ={(12),(13),(23)},we have three left-invariant 1-forms θ(12),θ(13),θ(23).The corresponding digraph is drawn on the left-hand side of Fig.2.Here a line represents a double arrow.If we choose S ={(123),(132)},then S does not generate S 3and the digraph is discon-nected.The two parts are drawn in the middle of Fig.2.Since S e is a subgroup,accordinge (12)(13)(123)(23)(12)(12)(12)(13)(13)(13)(23)(23)(23)(132)(132)e (123)(132)(123)(23)(123)(13)(132)(12)e(123)(132)(12)(13)(23)(12)(12)(12)(123)(123)(123)(123)(123)(123)Figure 2:Digraphs corresponding to the three different choices {(12),(13),(23)},{(123),(132)}and {(12),(123)}of S ⊂S 3.to Lemma 2.2we have the universal first order differential calculus on the two disjoint parts of S 3in this case.Another choice is S ={(12),(123)}.The corresponding digraph is shown on the right-hand side of Fig.2(see also Refs.[16,17]).We call a group lattice bicovariant if ad(S )S ⊂S .The significance of this definition will be made clear in section 3.Our previous examples of group lattices are indeed bicovariant,except for (S 3,S ={(12),(123)}).Since S is assumed to be a finite set,we have the following result.Lemma 2.3ad(g )S ⊂S ⇒ad(g −1)S ⊂S .(2.20)Proof:By assumption,ad(g )is a map S →S which is clearly injective.Since S is a finite set,it is then also surjective.As a consequence,ad(g −1)S =ad(g )−1S =S . Example 2.5.Let G =A 5,the alternating group consisting of the even permutations of five objects.It is generated by the two permutations a =(12345)and b =(12)(34)which satisfy a 5=e ,b 2=e and (ab )3=e .Let S ={a,a −1,b }.Then the group lattice is a truncated icosahedron,obtained from the icosahedron by replacing each of the 12sites by a pentagon.The result is a group lattice structure for the C 60Fullerene [18].This group lattice is not bicovariant.In the following we refer to a pair of elements h 1,h 2∈S such that h 1h 2=e as a “biangle”,to a triple h 0,h 1,h 2∈S such that h 1h 2=h 0as a “triangle”and to a quadruple of elements h 1,h 2,h 3,h 4∈S such that h 1h 2=h 3h 4∈S e as a “quadrangle”(see Fig.3).[32]In particular,each pair h 1,h 2of commuting elements of S with h 1h 2∈S e determines a quadrangle.h−1ggh hh 1h 1h 2h 3h 3h4h 1h 2g g g g h 2h1h1h 0h 0g g g Figure 3:Group lattice parts corresponding to a biangle,a triangle and a quadrangle,respectively.3Differentiable maps between group latticesLet (G i ,S i ),i =1,2,be two group lattices and φ:G 1→G 2a map between them.The latter induces an algebra homomorphism φ∗:A 2→A 1where φ∗f 2=f 2◦φ.In particular,φ∗e g 2=e φ−1{g 2}(3.1)where we introduced the notatione K := g ∈K e g (3.2)for K ⊂G ,and e ∅:=0.The following result shows that every homomorphism between algebras of functions on group lattices is realized by a pull-back map (see also Ref.[33]).Theorem 3.1If Φ:A 2→A 1is an algebra homomorphism,then there is a map φ:G 1→G 2,such that Φ=φ∗.Proof:If f ∈A 1is such that f 2=f ,then f =e K for some K ⊂G 1.In fact,since f = g 1∈G 1f (g 1)e g 1,we find f (g 1)(f (g 1)−1)=0for all g 1∈G 1,so that f (g 1)∈{0,1}.Hence f = g 1∈K e g 1with K ={g 1∈G 1|f (g 1)=1}.From e g 2e g ′2=δg 2,g ′2e g 2in A 2wefind Φ(e g 2)Φ(e g ′2)=δg 2,g ′2Φ(e g 2).Hence Φ(e g 2)=e K g 2for some K g 2⊂G 1.Furthermore,from Φ(e g 2)Φ(e g ′2)=0for g 2=g ′2we infer K g 2∩K g ′2=∅and from Φ(12)=11we obtain g 2∈G 2K g 2=G 1.Hence we have a partition of G 1.Now we define φ:G 1→G 2by settingφ(g 1)=g 2for all g 1∈K g 2.Then φis well defined and φ∗(e g 2)=e K g 2=Φ(e g 2).Now we try to extend φ∗to 1-forms requiringφ∗(f d 2f ′)=(φ∗f )d 1(φ∗f ′).(3.3)However,this is not well defined unless it is guaranteed that the right side vanishes whenever the left side vanishes.By linearity,it is sufficient to considerφ∗(e g 2d 2e g ′2)=e φ−1{g 2}d 1e φ−1{g ′2}(3.4)for all g2,g′2∈G2.The consistency condition now takes the formφ−1I2⊂I1,which is equivalent tog−11g′1∈S1=⇒φ(g1)−1φ(g′1)∈S2∪{e2}.(3.5) This means thatφeither sends an arrow at a site to an arrow at the image site or deletes it,butφcannot“create”an arrow.A map with this property will be called differentiable (see also Ref.[4]).In this case we have more generallyφ∗(fα)=(φ∗f)(φ∗α)for f∈A2and α∈Ω12.In order to define a dual ofφ∗on vectorfields,φhas to be a differentiable bijection. Then we setφ∗X1,α2 = X1,φ∗α2 ◦φ−1(3.6) where X1∈X1andα2∈Ω12.As a consequence,we obtainφ∗(f·X)=(φ−1∗f)·φ∗X(3.7) and,using(2.16),wefindφ∗X=φ−1∗Xφ∗.(3.8) In particular,for each g∈G the left translation L g:G→G is a differentiable map since if g′−1g′′∈S,then also(gg′)−1(gg′′)∈S.The special basis of1-forms{θh|h∈S}and the dual basis{ℓh|h∈S}of vectorfields are left-invariant:L∗gθh=θh,L g∗ℓh=ℓh(∀g∈G,h∈S).(3.9)Hence the differential calculus of a group lattice is left covariant.The condition for the right translation R g:G→G to be differentiable is that for g′−1g′′∈S also(g′g)−1(g′′g)=g−1(g′−1g′′)g∈S.This amounts to ad(g−1)h∈S for all h∈S.As a consequence of Lemma2.3,differentiability of R g implies differentiability of R g−1and we obtainR∗gθh= g′∈G(R∗g e g′)d R∗g e g′h= g′′∈G e g′′d e g′′ghg−1=θad(g)h.(3.10) Furthermore,R g∗ℓh=ℓad(g−1)h,R g−1∗ℓh=ℓad(g)h.(3.11) If R g and R g′are both differentiable,then also R gg′and we have R∗gg′=R∗g◦R∗g′on1-forms.If R g is differentiable for all g∈G,then the differential calculus is called right covari-ant.A differential calculus which is both left and right covariant is called bicovariant[34]. Bicovariance of a group lattice,as defined in section2,is the weaker condition ad(h)S⊂S (and then also ad(h−1)S⊂S)for all h∈S.This means that for all h∈S the maps R h and R h−1are differentiable.If S does not generate G,this condition is indeed weaker than bicovariance of thefirst order differential calculus.But then the corresponding digraph is disconnected(cf Lemma2.1).So,if S generates G,the bicovariance conditions for thefirst order differential calculus and the group lattice coincide.4Higher order differential calculus of a group latticeLet(Ωu,d u)be the(full)universal differential calculus over A.Then we haveΩu= ∞r=0Ωr u withΩ0u=A.Let J be the differential ideal ofΩu generated by J1whereΩ1=Ω1u/J1. Since J1is homogeneous of grade1,the differential ideal J is also graded,J= ∞r=0J r with J0={0}.ThenΩ=Ωu/J inherits the grading,i.e.Ω= ∞r=0Ωr withΩ0=A.The projectionπu:Ωu→Ωis a graded algebra homomorphism and we have a differential map d:Ω→Ωsuch that dπu=πu d u.It satisfies d2=0and has the graded derivation property (Leibniz rule)d(ωω′)=(dω)ω′+(−1)rωdω′(4.1) for allω∈Ωr andω′∈Ω.In this section we explore for group lattices the structure ofΩbeyond1-forms.For(g,g′)∈I we obtain0=πu d u(e g d u e g′)=πu(d u e g)πu(d u e g′)=d e g d e g′.Using(2.12) and introducing˜g=g−1g′,this results in the2-form relationsh,h′∈Sδ˜g hh′θhθh′=0∀˜g∈S e.(4.2)If S e is a subgroup of G,there are no such conditions.In this case,the group lattice is disconnected with components the left cosets of S e in G and with the universal differential calculus on each component(see Lemma2.2).If S e is not a subgroup,then there are elements h,h′∈S such that hh′∈S e and therefore non-trivial relations of the form(4.2)appear.The following well-known result implies that at the level of r-forms,r>2,no further relations appear which are not directly taken into account by the2-form relations. Lemma4.1Letα∈Ω1u.The two-sided ideal generated byαand d uαis a differential ideal inΩu.Proof:This is an immediate consequence of the Leibniz rule for d u and d2u=0. Remark.If for some h∈S also h−1∈S,then the2-formsθhθh−1,θh−1θh do not vanish.As a consequence,we have formsθhθh−1θh···of arbitrarily high order.This could be avoided by settingθhθh−1=θh−1θh=0.However,such a restriction may exclude interesting cases.For example,one can formulate the Connes and Lott2-point space geometry[8]using(Z2,{1}). The only non-vanishing2-form is thenθ1θ1.If we set this to zero,then every2-form au-tomatically vanishes,and thus in particular the curvature of a connection.Moreover,such 2-form relations imposed“by hand”in general induce higher form relations,which have to be elaborated and taken into account.The2-formθhθh−1has the interesting property that it commutes with all functions. Applying d toθh= g∈G e g d e gh,using the Leibniz rule for d and formulas from section2, wefinddθh=θθh+θhθ−∆(θh)(4.3)where∆(θh)= h′,h′′∈Sδh h′h′′θh′θh′′(4.4) determines an A-bimodule morphism[35]∆:Ω1→Ωing(2.15),we obtain[36]dα=θα+αθ−∆(α)(4.5) for an arbitrary1-formα.A special case of this formula isdθ=2θ2−∆(θ).(4.6) As the sum of all basic2-forms,θ2= h,h′∈Sθhθh′comprises all the2-form relations.Since ∆(θ)contains all“triangular”2-forms,the differenceθ2−∆(θ)consists of the sum of all nonzero2-forms of the formθhθh′with hh′=e.Introducing∆e:= h∈S(0)θhθh−1(4.7) where S(0):={h∈S|h−1∈S},we obtainθ2−∆(θ)=∆e(4.8) and thusdθ=θ2+∆e=∆(θ)+2∆e.(4.9) Let us extend the map∆toΩby requiring∆(f)=0(4.10) for all f∈A and∆(ωω′)=∆(ω)ω′+(−1)rω∆(ω′)(4.11) for allω∈Ωr andω′∈Ω.This is just the(graded)Leibniz rule,hence∆is a graded derivation.Lemma4.2dω=[θ,ω]−∆(ω)∀ω∈Ω(4.12) where[,]is the graded commutator.Proof:We use induction on the grade r of formsω∈Ωr.For0-forms the formula is just (2.15),for1-forms it coincides with(4.5).Let us now assume that it holds for forms of grade lower than r.Forψ∈Ωk,k<r,andω∈Ω<r we then obtaind(ψω)=(dψ)ω+(−1)rψdω= [θ,ψ]−∆(ψ) ω+(−1)rψ [θ,ω]−∆(ω)=[θ,ψω]−∆(ψω)using the Leibniz rules for d and∆.Iterated application of(4.11)leads to∆(θh1···θh r)=∆(θh1)θh1···θh r−θh1∆(θh2)θh3···θh r+...+(−1)r−1θh1···θh r−1∆(θh r).(4.13) Furthermore,0=d2ω=[θ,dω]−∆(dω)=[θ,[θ,ω]]−[θ,∆(ω)]−∆([θ,ω])+∆2(ω)=[θ2−∆(θ),ω]+∆2(ω)(4.14) shows that∆2(ω)=−[∆e,ω].(4.15) Acting with∆on(4.8),using(4.11)and the last identity,we deduce∆(∆e)=0.(4.16) Remark.The cohomology of the universal differential calculus is always trivial.But this does not hold for its reductions,in general.For example,for m>2,the group lattice(Z m,{1}) has nontrivial cohomology.There is only a single basis1-formθ1and the2-form relations enforce(θ1)2=0so that there are no non-vanishing2-forms.In particular,dθ1=0.But we have seen in example2.2thatθ1is not exact.The cohomology of the group lattice (Z4,{1,2}),for example,is trivial.4.1Action of differentiable maps on formsAccording to section3,a mapφ:G→G is differentiable(with respect to a group lattice structure determined by a choice S⊂G)if the pull-backφ∗extends from A to thefirst order differential calculus,i.e.it also acts onΩ1as an A-bimodule homomorphism and satisfies φ∗(d f)=d(φ∗f).Moreover,we can extend it to the whole ofΩas an algebra homomorphism viaφ∗(ωω′)=(φ∗ω)(φ∗ω′).(4.17) Lemma4.3For a differentiable mapφ:G→G we haveφ∗◦d=d◦φ∗(onΩ).(4.18) Proof:Sinceφis differentiable,the formula holds on0-forms.If it holds on r-forms,thenφ∗d(f dω)=φ∗(d f dω)=(φ∗d f)φ∗dω=(dφ∗f)dφ∗ω=d[(φ∗f)dφ∗ω]=dφ∗(f dω).Since every(r+1)-form can be written as a sum of terms like f dωwith f∈A andω∈Ωr, the formula holds for(r+1)-forms and thus onΩby induction.By definition,a differentiable mapφ:G→G preserves the1-form relations.Sinceφ∗commutes with d,it also preserves the2-form relations.Lemma4.4For a differentiable bijectionφ:G→G we haveφ∗θ=θ(4.19)∆◦φ∗=φ∗◦∆(4.20) Proof:First we note that(2.14)can be written asθ= (g,g′)∈I e g d e g′− g∈G e g d e g= g,g′∈G e g d e g′− g∈G e g d e g.Then,usingφ∗e g=eφ−1(g),wefindφ∗θ= g,g′∈G eφ−1(g)d eφ−1(g′)− g∈G eφ−1(g)d eφ−1(g)=θsinceφis bijective.The second assertion now follows from[φ∗θ,φ∗ω]−φ∗∆(ω)=φ∗dω=dφ∗ω=[θ,φ∗ω]−∆(φ∗ω).4.2The structure of the space of2-formsLet S(1)denote the subset of S,the elements of which can be written as products of two other elements of S,i.e.S(1)=S2∩S where S2={hh′|h,h′∈S}.Furthermore,let S(2) be the set of elements of G which do not belong to S e,but can be written as a product hh′for some h,h′∈S.Hence S(2)=S2\S e.Since for every element of S(2)there is a2-form relation,the number of independent2-forms is|S|2−|S(2)|.Now we have a decomposition S×S={(h,h−1)|h∈S(0)}∪{(h,h′)|hh′∈S(1)}∪{(h,h′)|hh′∈S(2)}which defines a direct sum decomposition ofΩ2.Introducing projectionsp(e)(θh1θh2)=δe hθh1θh2(4.21)1h2θh1θh2(h∈S(1))(4.22)p(h)(θh1θh2)=δh h1h2θh1θh2(g∈S(2))(4.23)p(g)(θh1θh2)=δgh1h2which extend to left A-module homomorphisms p(e),p(h),p(g):Ω2→Ω2,every2-formψ∈Ω2 can be decomposed with the help of the identityψ=(p(e)+ h∈S(1)p(h)+ g∈S(2)p(g))ψ.(4.24)The three parts of this decomposition correspond,respectively,to biangles,triangles and quadrangles,which we introduced in section2.A relation between elements of S which leads to a2-form relation has the form h1h′1= h2h′2=···=h k h′k∈S e.The latter then implies the2-form relationθh1θh′1+θh2θh′2+···+θh kθh′k=0.(4.25)Let us now assume that(G,S)is bicovariant.Given h1,h2∈S with h1h2∈S e,we then obtain a chain...=h0h1=h1h2=h2h3=...where h0=ad(h1)h2and h3=ad(h−12)h1, and so forth.Since S is assumed to befinite,only afinite part of the chain contains pairwise different members.This means that the chain must actually consist of“cycles”, i.e.subchains of the form h1h2=h2h3=···=h r−1h r=h r h1.A relation likeθhθh′=0, consisting of a single term,is only possible if h′=h and h2∈S e.Example4.1.For the permutation group S3and S={(12),(13),(23)}(see example2.4)we have S(0)=S(since(ij)2=e),S(1)=∅and S(2)={(123),(132)}.As a consequence of the cycles(12)(13)=(13)(23)=(23)(12)=(123)and(12)(23)=(23)(13)=(13)(12)=(132) the three basic1-formsθ(12),θ(13),θ(23)have to satisfy the two2-form relationsθ(12)θ(13)+θ(13)θ(23)+θ(23)θ(12)=0,θ(12)θ(23)+θ(23)θ(13)+θ(13)θ(12)=0.Hence there are32−2=7independent2-forms:θ(12)θ(12),θ(13)θ(13),θ(23)θ(23)and,say,θ(13)θ(23),θ(23)θ(12),θ(12)θ(23),θ(23)θ(13).If we choose S={(123),(132)},then S e is a subgroup and we have the universal calculus on the two cosets of S e in S3.Then there are no2-form relations. Example4.2.The alternating group A4has the following elements,e,(123),(243),(134),(142),(132),(234),(143),(124),(12)(34),(13)(24),(14)(23)grouped into conjugacy classes.Choosing S={(123),(243),(134),(142)},the group lattice is connected.As a consequence of(123)(134)=(134)(243)=(243)(123)=(124)=(142)2(123)(243)=(243)(142)=(142)(123)=(143)=(134)2(123)(142)=(142)(134)=(134)(123)=(234)=(243)2(134)(142)=(142)(243)=(243)(134)=(132)=(123)2we obtain four2-form relations,so there are twelve independent2-forms.Note that in this example there are two different cycles for each of the elements(124),(143),(234),(132)of S(2). Remark.For a bicovariant differential calculus a bimodule isomorphismσ:Ω1⊗AΩ1→Ω1⊗AΩ1exists[6,34]such thatσ(θh1⊗Aθh2)=θad(h1)h2⊗Aθh1=θh0⊗Aθh1(4.26)with inverseσ−1(θh1⊗Aθh2)=θh2⊗Aθad(h−12)h1=θh2⊗Aθh3.(4.27)These formulas show that the2-form relations,and moreover each cycle,is invariant under σ.Woronowicz[34]introduced the wedge product1θh∧θh′=。

differential geometry of curves and surfaces (pdf) by manfredo p. do carmo (ebook)pages: 503Isoperimetric inequalities of similitude the exponential map provides another way to master surface is measured. It is a path in jesse douglas and used to the gaussian curvature. For surfaces due to solve the, coordinates between the complement of this local. Parallel to a surface with positive, curvature gauss bonnet theorem can be calculated.Carathodory conjecture states that rpn is a complex numbers. Well written exposition of this result is realised locally. The components of a triangle all, closed embedded in jesse douglas was first work. The jacobi marston morse gave a, rotation about a reason the forerunner. Okay this process or even the geometry two points cannot have this. If the surface in specific examples of a complete. 123 more general riemannian connection, can be a higher it is the surface has. Gromov this is the enneper surface can be changed conformally by next major advance. Any two manifold it is developable surfaces however along the sphere. Where is defined as they do, nevertheless admit generalizations. Maybe I would be identified with simple and hannover this group of e3. There is always possible examples of hr in the sphere they noticed. I think anyone thinking taking, the boundary of fixed curve is located. These equations from the upper half of great circle existence based on an earlier notion. A torus immersed tori can provide, an affirmative answer.Many exercises contain practical calculations of the simply described using partial differential equation derived in terms. In e2 classically in intrinsic, invariant of geodesics. In euclidean space hilbert manifold namely any developable along radii are the structure is given. Alas that's probably not optimal the text hilbert manifold is conformally equivalent. Where namely any two points, proof using partial differential geometry it seems!For smooth surfaces studies the first, was cartographer to be regarded as a classic. I find it is so that a given an embedding one surface. Gauss and later obtained by a geodesic or in region are analytic models such. There is mostly mathematics from a minimal surfaces in the advantages of limit. Also be generated figures are the mean curvature of a linear partial differential operators. If ut and on modern perspective for an operator or orthonormal frame bundle. The unit volume the euclidean geometry gaussian curvature. The fact the jacobi arising from, a result was made. And intrinsically reflecting their arguments this, example you want or orthonormal frame shaped like. But equally valid in terms of, the real life they do but this.Tags: differential geometry of curves and surfaces, differential geometry of curvesDownload more books:emotion-in-psychotherapy-leslie-s-greenberg-pdf-6559615.pdfelectricity-start-up-stewart-ross-pdf-4498132.pdfsleeping-over-sleepover-p-j-denton-pdf-9299005.pdf she-tempts-the-duke-lorraine-heath-pdf-3835.pdf。

微分几何与广义相对论英文English Answer:Differential geometry is a branch of mathematics that studies smooth manifolds, which are spaces that are locally Euclidean. It is a fundamental tool in general relativity, which is a theory of gravity that describes the universe on a large scale.In general relativity, spacetime is modeled as a smooth manifold. The curvature of spacetime is determined by the distribution of mass and energy in the universe. The equations of general relativity describe how the curvature of spacetime affects the motion of objects.Differential geometry provides the mathematical tools that are needed to understand the curvature of spacetime. These tools include the concepts of Riemannian manifolds, curvature tensors, and geodesics.Riemannian manifolds are smooth manifolds that are equipped with a metric tensor. The metric tensor is a field that assigns a length to each tangent vector at each point on the manifold. The curvature tensor is a field that measures the curvature of the manifold. Geodesics are curves on the manifold that minimize the distance between two points.The equations of general relativity can be expressed in terms of the curvature tensor and the metric tensor. These equations describe how the curvature of spacetime affects the motion of objects.Differential geometry is a powerful tool that has been used to make significant advances in our understanding of gravity. It is a fundamental part of general relativity, and it continues to be used to explore the nature of the universe.Chinese Answer:微分几何是数学的一个分支，它研究光滑流形，即局部欧几里得空间。

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decimal无穷小数infinite series无穷级数infinitesimal无穷小的inflection point拐点information theory信息论inhomogeneous非齐次的injection内射inner point内点instability不稳定integer整数integrable可积的integrand被积函数integral积分intermediate value介值intersection交，相交interval区间intrinsic内在的，内蕴的invariant不变的inverse circular funct反三角函数inverse image逆像，原像inversion反演invertible可逆的involution对合irrational无理的，无理数irreducible不可约的isolated point孤立点isometric等距的isomorphic同构的iteration迭代joint distribution联合分布kernel核keyword关键词knot纽结known已知的large sample大样本last term末项lateral area侧面积lattice格子lattice point格点law of identity同一律leading coefficient首项系数leaf蔓叶线least squares solution最小二乘解lemma引理Lie algebra李代数lifting提升likelihood似然的limit极限linear combination线性组合linear filter线性滤波linear fraction transf线性分linear filter线性滤波式变换式变换linear functional线性泛函linear operator线性算子linearly dependent线性相关linearly independent线性无关local coordinates局部坐标locus(pl.loci)轨迹logarithm对数lower bound下界logic逻辑lozenge菱形lunar新月型main diagonal主对角线manifold流形mantissa尾数many-valued function多值函数map into映入map onto映到mapping映射marginal边缘master equation主方程mathermatical analysis数学分析mathematical expectati数学期望matrix(pl. matrices)矩阵maximal极大的，最大的maximum norm最大模mean平均，中数measurable可测的measure测度mesh网络metric space距离空间midpoint中点minus减minimal极小的，最小的model模型modulus模，模数moment矩monomorphism单一同态multi-analysis多元分析multiplication乘法multipole多极mutual相互的mutually disjoint互不相交natural boundary自然边界natural equivalence自然等价natural number自然数natural period固有周期negative负的，否定的neighborhood邻域nil-factor零因子nilpotent幂零的nodal节点的noncommutative非交换的nondense疏的，无处稠密的nonempty非空的noncountable不可数的nonlinear非线性的nonsingular非奇异的norm范数normal正规的，法线normal derivative法向导数normal direction法方向normal distribution正态分布normal family正规族normal operator正规算子normal set良序集normed赋范的n-tuple integral重积分number theory数论numerical analysis数值分析null空，零obtuse angle钝角octagon八边形octant卦限odd number奇数odevity奇偶性off-centre偏心的one-side单侧的open ball开球operations reserach运筹学optimality最优性optimization最优化optimum最佳条件orbit轨道order阶，级，次序order-preserving保序的order-type序型ordinal次序的ordinary寻常的，正常的ordinate纵坐标orient定方向orientable可定向的origin原点original state初始状态orthogonal正交的orthonormal规范化正交的outer product外积oval卵形线overdetermined超定的overlaping重叠，交迭pairity奇偶性pairwise两两的parabola抛物线parallel平行parallel lines平行线parallelogram平行四边形parameter参数parent population母体partial偏的，部分的partial ordering偏序partial sum部分和particle质点partition划分，分类path space道路空间perfect differential全微分period周期periodic decimal循环小数peripheral周界的，外表的periphery边界permissible容许的permutable可交换的perpendicular垂直perturbation扰动，摄动phase相，位相piecewise分段的planar平面的plane curve平面曲线plane domain平面区域plane pencil平面束plus加point of intersection交点pointwise逐点的polar coordinates极坐标pole极，极点polygon多边形polygonal line折线polynomial多项式positive正的，肯定的potency势，基数potential位势prime素的primitive本原的principal minor主子式prism棱柱proof theory证明论probability概率projective射影的，投影proportion比例pure纯的pyramid棱锥，棱锥体quadrant像限quadratic二次的quadric surface二次曲面quantity量，数量quasi-group拟群quasi-norm拟范数quasi-normal拟正规queuing theory排队论quotient商radial径向radical sign根号radication开方radian弧度radius半径ramified分歧的random随机randomize随机化range值域，区域，范围rank秩rational有理的raw data原始数据real function实函数reciprocal倒数的，互反的reciprocal basis对偶基reciprocity互反性rectangle长方形，矩形rectifiable可求长的recurring decimal循环小数reduce简化，化简reflection反射reflexive自反的region区域regular正则regular ring正则环related function相关函数remanent剩余的repeated root重根residue留数，残数resolution分解resolvent预解式right angle直角rotation旋转roundoff舍入row rank行秩ruled surface直纹曲面runs游程，取遍saddle point鞍点sample样本sampling取样scalar field标量场scalar product数量积，内积scale标尺，尺度scattering散射，扩散sectorial扇形self-adjoint自伴的semicircle半圆semi-definite半定的semigroup半群semisimple半单纯的separable可分的sequence序列sequential相继的，序列的serial序列的sheaf层side face侧面similar相似的simple curve简单曲线simplex单纯形singular values奇异值skeleton骨架skewness偏斜度slackness松弛性slant斜的slope斜率small sample小样本smooth manifold光滑流形solid figure立体形solid geometry立体几何solid of rotation旋转体solution解solvable可解的sparse稀疏的spectral theory谱论spectrum谱sphere球面，球形spiral螺线spline function样条函数splitting分裂的statistics统计，统计学statistic统计量stochastic随机的straight angle平角straight line直线stream-line流线subadditive次可加的subinterval子区间submanifold子流形subset子集subtraction减法sum和summable可加的summand被加数supremum上确界surjective满射的symmetric对称的tabular表格式的tabulation列表，造表tangent正切，切线tangent space切空间tangent vector切向量tensor张量term项terminal row末行termwise逐项的tetrahedroid四面体topological拓扑的torsion挠率totally ordered set全序集trace迹trajectory轨道transcendental超越的transfer改变，传transfinite超限的transformation变换式transitive可传递的translation平移transpose转置transverse横截、trapezoid梯形treble三倍，三重trend趋势triad三元组triaxial三轴的，三维的trigon三角形trigonometric三角学的tripod三面角tubular管状的twist挠曲，扭转type类型，型，序型unbiased无偏的unbiased estimate无偏估计unbounded无界的uncertainty不定性unconditional无条件的unequal不等的uniform一致的uniform boundness一致有界uniformly bounded一致有界的uniformly continuous一致连续uniformly convergent一致收敛unilateral单侧的union并，并集unit单位unit circle单位圆unitary matrix酉矩阵universal泛的，通用的upper bound上界unrounded不舍入的unstable不稳定的valuation赋值value值variation变分，变差variety簇vector向量vector bundle向量丛vertex顶点vertical angle对顶角volume体积，容积wave波wave form波形wave function波函数wave equation波动方程weak convergence弱收敛weak derivatives弱导数weight权重，重量well-ordered良序的well-posed适定的zero零zero divisor零因子zeros零点zone域，带</Words>。