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A CONNECTION WHOSE CURVATURE IS THE LIE BRACKET KENTE.MORRISON 9 0 0 ABSTRACT. LetGbeaLiegroupwithLiealgebrag.Onthetrivialprinci- 2 palG-bundleovergthereisanaturalconnectionwhosecurvatureisthe n Liebracketofg. TheexponentialmapofGisgivenbyparalleltransport a of this connection. If Gisthe diffeomorphismgroupof amanifold M, J thecurvatureofthenaturalconnectionistheLiebracketofvectorfields 7 on M. In the case that G = SO(3) the motionof asphererollingona 2 planeisgivenbyparalleltransportofapullbackofthenaturalconnec- tionbyamapfromtheplanetoso(3). Themotionofasphererollingon G] anorientedsurfaceinR3canbedescribedbyasimilarconnection. D . h t a 1. A NATURAL CONNECTION AND ITS CURVATURE m Samelson[4]hasshownthatthecovariantderivativeofaconnectioncan [ be expressedas a Lie bracket. It is the purposeof this article to show that 2 v theLiebracketofaLiealgebracanbeexpressedasthecurvatureformofa 1 naturalconnection. 2 3 Thesettingforthisresultisthefollowing. Letπ :P → X bearightprin- 3 cipal G-bundle with the LiegroupG as thestructuregroup. A connection . 3 0 onP isasmoothG-equivariantdistributionofhorizontalspacesinthetan- 8 gentbundleTP complementarytotheverticaltangentspacesofthefibers. 0 : Thecurvature ofaconnectionisag-valued 2-formonthetotalspaceP. A v i goodreferenceisBleecker’sbook[2]. X LetGbeaLiegroupandgitsLiealgebra. LetP = g×Gbethetotalspace r a of the trivial right principal G-bundle with projection P → g : (x,g) 7→ x. TherightactionofGonP isgivenby(x,h)·g = (x,hg)andletR denote g theactionofg onP;thatis,R : P → P : (x,h) 7→ (x,hg). Let1 ∈ Gbethe g identityelementandletι:g → g×G :x 7→ (x,1)betheidentitysectionof thebundle. LetL : G → G : h 7→ gh andR : G → G : h 7→ hg betheleftandright g g multiplication byg. Thecontextshouldmake itclear whetherR is acting g Date:August24,2008. 2000MathematicsSubjectClassification. Primary53C05;Secondary53C29. Keywordsandphrases. Principalbundles,connections,curvature,Liebracket,holonomy, rollingsphere. 1 2 KENTE.MORRISON onP oron G. Theadjoint action ofG on g is the derivative at the identity oftheconjugationactionofGonitself. Thatis, Adg : g → g : v 7→ T1(Lg ◦Rg−1)(v). Finally,werecallthedefinitionofafundamentalvectorfieldonaprinci- palG-bundleP. Forξ ∈ g,letξ˜bethevectorfieldonP whosevalueatpis givenby d ξ˜(p) = p·exp(tξ) . dt t=0 (cid:0) (cid:1)(cid:12) The vectorfield ξ˜is called a fundamental vect(cid:12)orfield. In the case that P is thetrivialbundleg×G,itisroutinetocheckthat ξ˜(x,g) = (0,T L (ξ)) ∈g×T G. 1 g g Theorem1. ThereisanaturalconnectiononthetrivialbundleP = g×Gwhose local curvature 2-form (with respect to the identity section ι) is the constant g- valued 2-form on g whose value on a pair of tangent vectors ξ,η ∈ g is the Lie bracket[ξ,η]. Proof. We define the connection on P by choosing the horizontal spaces. For(x,g) ∈ P letH bethesubspaceofT P definedby (x,g) (x,g) H := {(v,T R (v))|v ∈ g}. (x,g) 1 g It is easy to check that the distribution is right equivariant and that H (x,g) iscomplementarytotheverticaltangentspaceat(x,g). Letαbetheg-valued1-formonP definingthisconnection. Itischarac- terizedby having thehorizontal spaceH as thekernelofα and by (x,g) (x,g) satisfyingtheconditions (i) α (ξ˜(x,g)) = ξ, (x,g) ∗ (ii) Rg α= Adg−1 ◦α. Fromthesepropertiesonecancheckthatαisdefinedby α(x,g)(v,ξ) = TgLg−1(ξ)−Adg−1(v). The curvature of a connection is the g-valued 2-form on the total space P definedby 1 dα+ [α,α]. 2 ∗ Pullingαbacktogbyιgivesthelocalconnection1-formω = ι α. ACONNECTIONWHOSECURVATUREISTHELIEBRACKET 3 ThelocalcurvatureformΩisthen 1 Ω = dω+ [ω,ω]. 2 Computingω weseethat ω (v) = α (v,0) x (x,1) = −v. Hence ω is a constant form, and dω = 0. We evaluate [ω,ω] on a pair of tangentvectorsξ,η ∈ T g ∼= gasfollows: x [ω,ω](ξ,η) = [ω(ξ),ω(η)]−[ω(η),ω(ξ)] = [−ξ,−η]−[−η,−ξ] = 2[ξ,η]. Therefore, the local curvature form Ω is the constant g-valued 2-form that maps apair oftangentvectorsξ,η, which are justelementsofg, to theLie bracketofξ andη. 1 Ω(ξ,η) = (dω+ [ω,ω])(ξ,η) = [ξ,η] 2 (cid:3) For a Lie group homomorphism φ : G → G , let Lφ : g → g be the 1 2 1 2 associatedLie algebra homomorphism. (Notethat Lφis T φ whenthe Lie 1 algebras are viewed as the tangent spaces at the identities of the groups.) Thenthemap φ˜: g ×G → g ×G :(x,g) 7→ (Lφ(x),φ(g)) 1 1 2 2 is a morphism of principal bundles, which means that it commutes with therightactionsofG andG : 1 2 φ˜(x,hg) = (Lφ(x),φ(hg)) = (Lφ(x),φ(h)φ(g)). Furthermore, it is a morphism that preserves the horizontal spaces of the natural connections defined in Theorem 1. More precisely, Tφ˜ maps the horizontalsubspacesinT(g ×G )tothehorizontalsubspacesinT(g ×G ) 1 1 2 2 4 KENTE.MORRISON asfollows. Let(v,T R (v))beinH ⊂ T (g×T G ). 1 g (x,g) (x,g) g 1 Tφ˜(v,T R (v)) = (T Lφ(v),T φ(T R (v))) 1 g x g 1 g = (Lφ(v),T (φ◦R )(v)) 1 g = (Lφ(v),T (R ◦φ)(v)) 1 g = (Lφ(v),T R (T φ(v))), 1 φ(g) 1 whichisanelementofthehorizontalspaceat(Lφ(x),φ(g)) inT(g ×G ). 2 2 WealsonotethatforacompositionofLiegrouphomomorphismsφ◦ψ, the principal bundle map φ]◦ψ = φ˜◦ ψ˜, and that I˜is the identity on the principal bundle for the identity I : G → G. Hence, we have proved the followingtheorem. Theorem2. ThereisacovariantfunctorF fromthecategoryofLiegroupstothe category of principal bundles with connection, which is defined on objects so that F(G)isthetrivialprincipalbundleg×Gwithitsnaturalconnectionanddefined onmorphismsbyF(φ) = φ˜. Nextweconsidertherelationshipbetweentheconnection1-formsofthe twobundles. Proposition3. Thefollowingdiagram commutes Tφ˜ T(g ×G ) −−−−→ T(g ×G ) 1 1 2 2 α1 α2    Lφ  gy −−−−→ gy 1 2 Equivalently, φ˜∗α =Lφ◦α . 2 1 ACONNECTIONWHOSECURVATUREISTHELIEBRACKET 5 Proof. Startingwiththecompositionα ◦Tφ˜wehave 2 (α2)φ˜(x,g)Tφ˜(v,ξ) = (α2)(Lφ(x),φ(g))(Lφ(v),Tgφ(ξ)) = Tφ(g)Lφ(g)−1(Tgφ(ξ))−Adφ(g)−1(Lφ(v)) = Tg(Lφ(g)−1 ◦φ)(ξ)−Adφ(g)−1(Lφ(v)) = Tg(φ◦Lg−1)(ξ)−Adφ(g)−1(Lφ(v)) = T1φ(TgLg−1(ξ)−Lφ(Adg−1(v)) = Lφ(TgLg−1(ξ)−Lφ(Adg−1(v)) = Lφ(TgLg−1(ξ)−Adg−1(v) = Lφ((α ) (v,ξ). 1 (x,g) (cid:3) Proposition4. Thefollowingdiagram commutes T(Lφ) Tg −−−−→ Tg 1 2 ω1 ω2    Lφ  gy −−−−→ gy 1 2 Equivalently, ∗ (Lφ) ω = Lφ◦ω . 2 1 Proof. BecauseLφislinear,T (Lφ) = Lφforallx ∈g . Thus,forv ∈ T g = x 1 x 1 g , 1 ω (T (Lφ)(v)) = ω (Lφ(v)) = −Lφ(v)= Lφ(−v) = Lφ(ω (v)). 2 x 2 1 (cid:3) Also,thelocalcurvatureformscommutewiththeLiealgebrahomomor- phismLφ. Proposition 5. Let Ω , i = 1,2, be the local curvature 2-form for the natural i connectionong ×G . ThenLφ◦Ω = Ω ◦Lφ. i i 1 2 Proof. This is simply restating the fact that Lφ is a homomorphism of Lie algebras: Lφ([ξ,η]) = [Lφ(ξ),Lφ(η)]. (cid:3) Remark 6. Let G be the diffeomorphism group of a manifold M with g beingthespaceofvectorfieldsonM. AlthoughGisnot,strictlyspeaking, a Lie group, the natural connection on g×G still makes sense, and so the curvatureofthisconnectionisgivenbytheLiebracketofvectorfields. 6 KENTE.MORRISON Givenasmoothcurvec: [0,1] → g,paralleltransportalongcishorizon- tal lift (c(t),g(t)) with initial condition g(0) = 1. Therefore, g is a solution ofthedifferentialequation ′ ′ g (t) = T R (c(t)), 1 g(t) ′ ′ whichsimplysaysthat(c(t),g (t))isinthehorizontalsubspaceatthepoint (c(t),g(t)). Theorem 7. Let ξ be an element of the Lie algebra g and define c(t) = tξ. Then paralleltransportalongcisgivenbyg(t) = exp(tξ). Proof. It suffices to show that g(t) = exp(tξ) satisfies the differential equa- ′ ′ tiong (t) = T R (c(t))with initial conditiong(0) = 1. Thederivativeof 1 g(t) exp(tξ)isgivenby d d exp(tξ)= exp((t+s)ξ)| s=0 dt ds d = (exp(sξ)exp(tξ))| s=0 ds d = R exp(sξ) | exp(tξ) s=0 ds (cid:0) (cid:1) = T R (ξ) 1 exp(tξ) ′ = T R (c(t)) 1 g(t) (cid:3) Remark8. With thisresultthereis anotherway toseethattheLiebracket isthecurvature,since 1 d2 [ξ,η] = exp(tξ)exp(tη)exp(−tξ)exp(−tη) 2dt2 (cid:12)t=0 (cid:12) istheinfinitesimalparalleltransportaroundtheparallelogr(cid:12)amspannedby ξ andη,which,inturn,isthecurvaturetensorappliedtoξ andη. Corollary 9. Let η and ξ be elements of g and define c(t) = η +tξ. Let g(t) be thehorizontal liftofcwithg(0) = g ∈ G. Theng(t) = exp(tξ)g . 0 0 ′ Proof. Since c(t) = ξ, the value of η does not matter. Therefore, let η = 0. Bythetheoremγ(t) = exp(tξ)is the horizontallift ofcwith γ(0) = 1. The right-invarianceofthehorizontalspacesimpliesthatg(t) = γ(t)g . (cid:3) 0 Remark 10. Parallel transport along c is also known as the time-ordered ′ (or path-ordered) exponential of c. One of the equivalent ways to define ACONNECTIONWHOSECURVATUREISTHELIEBRACKET 7 the time-ordered exponential of a curve a(t) ∈ g is to define it as the so- ′ lution of the differential equation g (t) = T R (a(t)) with g(0) = 1. For 1 g(t) ′ a(t)= c(t)thisisthedifferentialequationdefiningparalleltransportalong c. Tounderstandtheuseofthephrase“time-orderedexponential,”weuse apiecewiselinear approximation tocstartingat c(0) and consistingofthe linesegmentsconnectingc(ti−1)andc(ti)whereti = (i/n)t,i = 0,1,...,n. Let ∆t = t/n. Then by repeated use of the corollary g(t) is approximated by g(t) ≈ exp(c(tn)−c(tn−1))exp(c(tn−1)−c(tn−2))···exp(c(t1)−c(t0)) ′ ′ ′ ≈ exp(∆tc(tn−1))exp(∆tc(tn−2))···exp(∆tc(t0)). Therefore,g(t) is the limit as n goesto infinity ofthis productofexponen- tials,whichareorderedaccordingtheparametervalue. Remark 11. The holonomysubgroup (at a point x in the base space)of a 0 connection on a principal G-bundle is the subgroupof G consisting of the resultsofparalleltransportingaroundclosedcurvesstartingandendingat x . The null holonomy group is the subgroupresulting from transporting 0 around null-homotopic curves. By the Ambrose-Singer Theorem [1] the Lie algebra of the null holonomy group is generated by the values of the curvature tensor. Now g is simply connected and so the null holonomy group is the holonomy group and its Lie algebra is the derived algebra [g,g]. AssumingGisconnected,theholonomygroupisthederivedgroup ofG. 2. EXAMPLES 2.1. The Special Orthogonal Group. Let G be the rotation group SO(3) withg = so(3), theLiealgebraofinfinitesimalrotations. Weidentifyso(3) with R3 in the standard way so that the vector v in R3 correspondsto the infinitesimal rotation with axis v that goes counter-clockwise with respect to an observer with v pointing towards him. The angular velocity is the magnitude of v. If v = (v ,v ,v ), then the corresponding matrix ρ in 1 2 3 v so(3)is 0 −v v 3 2 ρ =  v 0 −v . v 3 1  −v2 v1 0    Then ρv×w = [ρv,ρw], so that the map ρ is an isomorphismof Lie algebras (R3,×) → (so(3),[, ]). 8 KENTE.MORRISON Given a curve c : [0,1] → R3, parallel transport along c is given by the curve g : [0,1] → SO(3), which is the unique solution to the differential equation ′ g (t) = ρc′(t)g(t), g(0) = I. Inotherwords,theinfinitesimalrotationatthasforitsaxisofrotationthe ′ vector c(t). One can visualize this as a sphere of radius 1 rotating at the head of a screw (with right hand threads) that is tunneling through space following thetrajectorygivenby thecurve c. Notethat thesphericalhead isnotrigidlyattachedtothescrewbecausetheaxisofrotationmustbefree tovary. 2.2. A Rolling Sphere on a Plane. A variation of this natural connection can be used to describe the geometry of a sphere rolling without slipping onahorizontalplane. TheplaneisR2 embeddedinR3 asthesetofpoints {(x ,x ,0)}andasphereofradius1sitsontopoftheplane. Letc :[0,1] → 1 2 R2 be a smooth curve with initial point c(0) = (x (0),x (0)). Roll the 1 2 spherealongthecurvecuntilitreachestheendpointc(1) = (x (1),x (1)). 1 2 Asthesphererolls along, thepointofcontact isc(t) and theconfiguration of the sphere is given by a curve g(t) in SO(3). At each point c(t) the infinitesimal rotation is about the axis J(c′(t)), where J : R2 → R2 is π/2 rotation counter-clockwise defined by J(x ,x ) = (x ,−x ). We can for- 1 2 2 1 mulatethedifferentialequationsatisfiedbyg(t)as ′ g (t)= ρJ(c′(t))g(t), g(0) = I. (Briefly,g(t+dt) = (I+ρJ(c′(t)))g(t)dt,fromwhichthedifferentialequation follows.) In order to see this differential equation as the parallel transport equationforthecurvecwithinitialconditiong(0) = I,wedefineaconnec- tiononR2×SO(3)withhorizontalspaceatthepoint(x,g)inR2×SO(3) givenby H = {(v,ρ g)|v ∈ R2} ⊂R2 ×T SO(3). (x,g) J(v) g Theconnection1-formαisdefinedby α (v,ξ) = g−1ξ−g−1ρ g. (x,g) J(v) ∗ Letω = ι αbethelocalconnection1-form. Since ω (v) = α (v,0) = −ρ , x (x,I) J(v) ACONNECTIONWHOSECURVATUREISTHELIEBRACKET 9 weseethatω isconstant,i.e.,dω = 0. Thenthelocal curvature2-formacts onapairoftangentvectorsu,v ∈ T R2 = R2 ⊂ R3 by x 1 1 (dω+ [ω,ω])(u,v) = 0+ [ω,ω](u,v) 2 2 1 = ([ω(u),ω(v)]−[ω(v),ω(u)]) 2 = [ω(u),ω(v)] = [−ρ ,−ρ ] J(u) J(v) = [ρ ,ρ ] J(u) J(v) = [ρ ,ρ ]. u v For the last step note that [ρJ(u),ρJ(v)] = ρJ(u)×J(v) because ρ is an iso- morphism of Lie algebras. Also, J(u)×J(v) = u×v from the geometric propertiesofthecrossproduct. Finally,ρu×v = [ρu,ρv]. The derived algebra of so(3) is so(3) because each of the basis elements e ,e ,e inR3 isacrossproduct. ThegroupSO(3)isconnected,andso,by 1 2 3 the Ambrose-Singer Theorem, the holonomy subgroup is all of SO(3). In otherwords,anyrotationofaspherecanbeachievedbyrollingthesphere around some closed path in the plane. An elementary proof (without the apparatus of modern differential geometry) of this old result has recently appeared[3]. The connection just defined is actually just a pull-back of the natural connectiononso(3)×SO(3). Theorem12. Define 0 0 x 1 f : R2 → so(3) : x7→ ρ =  0 0 x . J(x) 2  −x1 x2 0    ThentheconnectiononR2×SO(3)associatedtotherollingsphereisthepullback byρ ofthenaturalconnectiononso(3)×SO(3). J Proof. Thepullbackbundleofthetrivialbundleisalsotrivial. Letfˆdenote thebundle map R2 ×SO(3) → so(3)×SO(3) : (x,g) 7→ (f(x),g). Thenit sufficestocomputefˆ∗α,thepullbackoftheconnection1-formαonso(3)× SO(3), to see that it is the connection 1-form of the rolling sphere. At a 10 KENTE.MORRISON point(x,g) ∈R2×SO(3)wehave (fˆ∗α) (v,ξ) = α (Df(x)(v),ξ) (x,g) (f(x),g) = α (f(v),ξ) (sincef islinear) (f(x),g) = g−1ξ−g−1f(v)g = g−1ξ−g−1ρ g. J(v) (cid:3) 2.3. A Rolling Sphere on a Surface. Now more generally, we consider a sphere rolling on a surface in R3. We will construct a connection on the trivial SO(3)-bundle over the surface whose parallel transport describes therotationofthesphere. However,inthisgeneralitytheconnectionneed not be a pull-back of the natural connection on so(3) × SO(3). Let X be a smoothorientable surface in R3 and let n be the unit normal vectorfield pointingtothesideonwhich thesphererolls. LetJ be theautomorphism ofthetangentbundleTX thatrotateseachtangentspacecounterclockwise π/2withaxisofrotationgivenbytheunitnormaln. Withthenaturaliden- tification ofT X withR2, J (v) = n(x)×v. Ascomparedwith theplanar x x surfaceitisnowmorecomplicatedtodescribetheinfinitesimalrotationat apointx∈ X inthedirectionv ∈ T X becauseofthetwistingandturning x ofthetangentspacesofthesurface. The unit normal vectorfield is a map n : X → R3. The derivative of n at x is a linear map Dn(x) : T X → R3. Differentiating the constant x function hn(x),n(x)i = 1 shows that hn(x),Dn(x)(v)i = 0. Therefore, Dn(x)(v) ∈ T X andhencev+Dn(x)(v)alsoliesinT X. Thenthevector x x J(v +Dn(x)(v)) is the axis of the infinitesimal rotation of the sphere at x inthedirectionv. WedefineaconnectiononthetrivialbundleX ×SO(3) whosehorizontalspaceat(x,g)is H(x,g) ={(v,ρJ(v+Dn(x)(v))g)|v ∈TxX} ⊂ TxX ×TgSO(3). Theconnection1-formαisgivenby α(x,g)(v,ξ) = g−1ξ−g−1ρJ(v+Dn(x)(v))g. ∗ Letω = ι αbethelocalconnection1-form. Thus, ωx(v) = α(x,I)(v,0) = −ρJ(v+Dn(x)(v)).