Controllability of rolling without twisting or slipping in higher dimensions

CONTR OLLABILITY OF ROLLING WIT HOUT TWISTING OR SLIPPING IN HIGHER DIMEN SIO NS ERLEND GR ONG Abstract. W e describe how t he dynamical syste m of rolling t w o n -dimensional connect ed, oriente d Riemannian manifolds M and c M witho ut t wisting or slip- ping, can be lif ted to a nonholonomic system of elemen ts in the product of the orient ed orthonormal frame bundles belonging to the manifolds. By consider- ing the lifted problem and using prop erties of the elements in the respective principal Ehresmann connections, we obtain sufficient conditions for the lo cal con trollability of the system i n terms of the curv ature tensors and the sec- tional curv atures of the manifol ds inv olv ed. W e also give some results for the particular cases when M an d c M are l o cally s ymmetric or complete . 1. Introduction The r o lling of tw o s urfaces, without twisting or s lipping, is a go o d illustr ation of a nonholonomic mec hanical system, whose pr o p erties are intimately c o nnected with geometry . It has therefore received muc h interest, and we can mention [1, 4, 5, 17] and [2, Chapter 24] as examples of research pro duced in this area. In particular, the treatment o f rolling in [2, 4] was done by formulating it as an intrinsic problem, independent o f the imbedding o f the surfaces in to Euclidean space. The gener alization of this concept to that of an n -dimensional manifold rolling without twisting or slipping on the n -dimensio nal E uclidean space, is w ell known (see e.g. [13, p. 26 8], [10, Chapter 2.1]). It is usually formulated intrinsically , in terms of frame bundles, and is an imp or tant to ol in sto chastic calculus on mani- folds. A definition for tw o arbitra r y n -dimensional manifolds r olling on each other without twisting or slipping, first appe a red in [20, App. B], how ever, this only dealt with manifolds imbedded into Euclidean space. An in tr insic definition for r o lling of higher dimensional manifolds, that connected the definition in [20] with the in trin- sic approach in [2, 4], w as pr esented in [8]. Apart from app ea ring as mechanical systems, rolling of higher dimensional manifolds ca n be a lso used as a too l in in- terp olation theo r y . F or demons tr ation of the “rolling and wr apping”-technique, we refer to [11]. See a lso [2 1] for an example where this is a pplied in r ob ot motion planning. F or the ro lling of tw o 2-dimensiona l manifolds, there is a b ea utiful corresp o n- dence be tw een the degree of co nt rol and the geometr y of the manifolds [2, 4]. Essentially , we hav e complete control over o ur dynamical system if the resp ective Gaussian curv atures M and c M do not coincide. Con tr ollability in hig her dimen- sions has b een a ddr essed in some sp ecia l ca ses [16, 2 3]. The firs t genera l result 2000 Mathematics Subje ct Classific ation. 37J60, 53A55, 53A17. Key wor ds and phr ases. Roll ing maps, cont rollability , br ack et-generating, frame bundles, non- holonomic constrain ts. 1 2 E. GRONG on controllabilit y in higher dimension is presented in [6 ], where it is shown tha t the curv atur e tensors determine the brackets of the distr ibution obtained from the constraints giv en b y neither t wisting, no r slipping. It is a lso proved that being able to find a rolling to an arbitrary clo se configuration, connecting the same t wo p oints, is a sufficient condition for complete controllabilit y . The ob jective of this pap er will b e to describ e the connectio n b etw een geome- try and co nt rollability for rolling o f higher dimensional manifolds. W e do this b y connecting the earlier men tioned vie w p o in t from stochastic calculus with the one presented in [8]. This pa p e r is o rganized a s fo llows. In Section 2, we sta te the intrinsic defini- tion of ro lling. W e present some of the theory of frame bundles, and develop our notation there. W e end this se ction b y showing how we can lift our pr oblem to the oriented orthono rmal frame bundles of the in volv ed manifolds. W e contin ue in section Sectio n 3, by do ing computatio ns on the lifted problem. By using prop er ties of the s ections in the Ehresma nn connections, we obtain for mulas for co mputation of the brack ets of the rolling distribution. In Section 4 we pro ject the res ults back to our configur ation s pace of r elative po sitions of the manifolds . Section 4.2 con- sist of conditions for controllability in terms o f the Riemann curv ature tenso r and the s ectional curv ature of the manifolds inv olved. W e end this section with s o me examples. Sectio n 5 fo cuses on results concerning the rolling of lo c a lly symmetric and complete manifolds. Section 6 contains a brief comment on how to general- ize the c o ncept of rolling without twisting or slipping to manifolds with an affine connection, and why the res ults pres ented here a lso holds for a rolling of manifolds with a torsion fre e affine connection. The author would like to expr e s s his gra titude to Maur ic io Go doy Molina for many fr uitful discuss io ns conc e rning this sub ject. 2. Intrinsic definition of rolling an d its rela tions to frame bundl es 2.1. In trinsi c definition of rolling without t wis ting or s lipping . Thro ughout this pap er, M and c M will denote connected, orient ed, n -dimensio nal Riemannian manifolds. Since in the special case n = 1, the co nditions of rolling without twisting or slipping become holonomic (see [8]), we will always as s ume that n ≥ 2. W e adopt the conv ention to e quip ob jects (po int s, pro jection, etc.) r e lated to c M with a hat (ˆ). Ob jects rela ted to b oth o f them a re usually denoted b y a ba r (¯), while ob jects connected to M a re not given any sp ecial dis tinctio n. The exception to this r ule is the Riema nnia n metric a nd the affine Levi-Civita connection whic h a re r e s pe ctively denoted by h· , ·i and ∇ on b o th M a nd c M . The co nt ext will make it cle a r which manifold these ob jects ar e related to. F or a vector field X on M , we will write X | m rather than X ( m ), a nd we will use simila r notation for other sections of bundles. F or a ny pa ir of oriented inner pro duct spaces V and b V , we let SO( V , b V ) deno te the s pace of all or ient ation preser ving linear isometries from V to b V . This allows us to define the SO( n )-fib er bundle Q ov er M × c M by Q = n q ∈ SO  T m M , T b m c M  : m ∈ M , b m ∈ c M o . W e c an be sur e that this fib e r bundle is pr inc ipa l in the case when n = 2 , but not in gener al. The spa ce Q r epresents a ll configura tions or r elative p os itio ns of M and c M , so that the tw o manifolds lie tangent to each other at so me pair of p o int s. CONTR OLLABILITY OF ROLLING IN HIGHER DIM. 3 The iso metry q : T m M → T b m c M , represents a configuration where M a t m lies tangent to c M at b m . The re la tive positio ning of their tangent spaces is given b y how q maps T m M in to T b m c M . A rolling then b eco mes a curve in the space o f these configuratio ns. Definition 1. L et π and b π denote the r esp e ctive natu ra l pr oje ctions fr om Q to M and c M . A r ol ling without twisting or slipping is an absolutely c ont inuous curve q : [0 , τ ] → Q, with m ( t ) := π ( q ( t )) , b m ( t ) := b π ( q ( t )) , satisfying the fol lowing c onditions: No slip c ondition: ˙ b m ( t ) = q ( t ) ˙ m ( t ) for almost every t , No twist c ondition: an arbitr ary ve ctor field X ( t ) is p ar al lel alo ng m ( t ) if and only if q ( t ) X ( t ) is p ar al lel along b m ( t ) . F rom now on w e will mostly refer to a ro lling q ( t ) without t wis ting or slipping as simply a ro lling. These tw o conditions ca n be describ ed in terms of a distribution D of ra nk n on Q . Consider the pro blem of finding the r olling q ( t ) b etw een tw o different configuratio ns q 0 and q 1 such that m ( t ) (and thereby also b m ( t )) has minimal length. Here, by a distribution on Q , we mea n a sub-bundle of T Q . This type of curves can be viewed as o ptimal curves in an input-linear drift free optimal control problem or length minimizers in a sub-Riemannian manifold. W e will describ e this structure in mo re deta il in Section 2.3. First, how ever, we will review some facts ab o ut connections and frame bundles. 2.2. F rom pri ncipal bundles to oriented orthonormal frame bundles. Con- sider a general principal G -bundle τ : P → M , where the Lie gr oup G a cts on the right. W e call the s ub- bundle V := ker τ ∗ of T P t he vertic al sp ac e of P . If g is the Lie algebra of G , then for a ny element A ∈ g , we hav e a vector field σ ( A ) defined by (1) σ ( A ) | p φ = d dt     t =0 φ ( p · exp G ( tA )) , for a ny p ∈ P, φ ∈ C ∞ ( P ) . Here, exp G : g → G is the gro up exp onential. W e remark that σ ( A ) is a s ection of V a nd for any p ∈ P , the ma p g → V p , A 7→ σ ( A ) | p is a linear isomorphism. A sub- bundle E o f T P is calle d a princip al Ehr esmann c onne ction if T P = E ⊕ V and satisfy r g E p = E p · g , wher e r g denotes the r ig ht m ultiplication of g ∈ G . Equiv alently , we can consider a princip al c onne ct ion form , which is a g -v alued one- from sa tisfying the tw o conditions r ∗ g ω = Ad( g − 1 ) ω , and ω ( σ ( A ) | p ) = A for any A ∈ g , p ∈ P. There is a o ne-to-one corre s po ndence b etw een these tw o structur es, in the sense that ker ω is a principal Ehr esmann connection when ω is a principal co nnection form, and for any principal E hresmann connectio n E , we can define a principal connection for m by formula ω ( v ) = A for v ∈ T p P if v − σ ( A ) | p ∈ E p . Related to a choice o f Ehresmann co nnection E we also hav e horizontal lifts, since the mapping τ ∗ | E p : E p → T τ ( p ) P is a linear iso metry . F or a vector v ∈ T m M , we define the ho r izontal lift h p v of v at p ∈ τ − 1 ( m ) as the unique element in E p which is pro jected to v by τ ∗ . 4 E. GRONG Let us co nsider a particula r principal bundle ov er a manifold M . F or t wo vector spaces V a nd b V , let GL( V , b V ) b e the space of all linear isomorphisms from V to b V . F or any m ∈ M , we say that a frame at m is a ch oice of bas is f 1 , . . . , f n for T m M . Eq uiv alently , we can consider a frame as a linear map f ∈ GL( R n , T m M ). The cor resp ondence b etw ee n the t wo p oint of views, is given b y f (0 , . . . , 0 , 1 , 0 , . . . , 0) | {z } 1 in th e j th c o ordinate = f j . W e wr ite F m ( M ) = GL( R n , T m M ) for the space o f fra mes in T m M . There is a natural action of GL( n ) o n these spaces by comp osition on the r ight. Using this action, we define the fr ame bund le e τ : F ( M ) → M as the principal GL( n )-bundle with fib er ov er m b eing F m ( M ). Similarly , if M is an oriented Riemannian manifold, we can define the oriente d orthonormal fr ame bund le τ : F ( M ) → M as the pr incipal SO( n )-bundle whose fiber is F m ( M ) := SO( R n , T m M ). Here, R n is furnished with the standar d or ie n- tation and the E ucliean metric. Let ∇ be an affine connection on M , s een as a n op erato r on vector fields ( X, Y ) 7→ ∇ X Y . Then we can as so ciate a pr incipal E hresmann connectio n E ∇ on e τ : F ( M ) → M to ∇ , by defining E ∇ to be the distribution on F ( M ) consisting of tangent vectors of smo oth curves f ( t ) such that the vector fields f 1 ( t ) , . . . , f n ( t ) are all pa rallel alo ng m ( t ) := e τ ( f ( t )). If M is Riemannian and o riented, and if ∇ is c ompatible with a metric, then E ∇ can be defined a s a distribution on F ( M ) instead, since b o th p os itive or ientation and orthonor mality are preserved under parallel tra nsp ort. W e no w go to the co ncrete case wher e ∇ is the Levi-Civita co nnection. Define the tautolo gic al one-fr om θ = ( θ 1 , . . . , θ n ) o n F ( M ), as the R n -v alued o ne-form θ j | f = τ ∗ ( ♭f j ) , τ : F ( M ) → M , where ♭ : T M → T ∗ M is the isomorphism induced by the Riemannian metric. In other words, if v ∈ T f F ( M ), then θ j ( v ) = h f j , τ ∗ v i . Denote the so ( n )-v alued principal connection form cor resp onding to the Levi-Civita connection b y ω . The formulas for the differentials of θ and ω ar e given by the well-kno w n Cartan equa- tions. W e expres s them in no tations, tha t will b e helpful for later purp ose s . Let R b e the Riema nn curv a tur e tensor , defined by R ( Y 1 , Y 2 , Y 3 , Y 4 ) = h R ( Y 1 , Y 2 ) Y 3 , Y 4 i , wher e R ( Y 1 , Y 2 ) = ∇ Y 1 ∇ Y 2 −∇ Y 2 ∇ Y 1 −∇ [ Y 1 ,Y 2 ] . F urthermore, use it to define the cur v ature for m Ω = (Ω ij ), as the so ( n )-v alued t wo-form (2) Ω ij ( v 1 , v 2 ) = R ( τ ∗ v 1 , τ ∗ v 2 , f j , f i ) , v 1 , v 2 ∈ T f F ( M ) . Then the following relations hold (3) dθ j + P n i =1 ω j i ∧ θ i = 0 , dω ij + P n k =1 ω ik ∧ ω kj = Ω ij , where ω ij are the matrix en trie s of ω . These equa tions are g oing to b e importa nt in o rder to understand the connections be tw een geo metry a nd the control system of ro lling manifolds. CONTR OLLABILITY OF ROLLING IN HIGHER DIM. 5 R emark 1 . The Cartan equa tions can a lso be defined on the frame bundle corre- sp onding to a general affine connection. See, e.g ., [14 , 20] for details . F rom no w o n, we will adopt the con v ention that whenev e r we mention R n , it will alwa y s come furnished with the standard orientation and the Euclidean metr ic . 2.3. The rolling distribu ti on and the corresp onding sub-R iemannian struc- ture. The tange n t vectors of all p oss ible ro lling s form an n -dimensional distribu- tion D on Q . W e will call this distribution D the ro lling distribution. A curve q ( t ) is a ro lling, if and only if, it is hor izontal with resp ect to D , i.e. it is abs o lutely contin uous a nd ˙ q ( t ) ∈ D q ( t ) for a lmo st any t . W e present the following lo ca l des cription of D (see [8 ] for more details). Let us write the pro jection of Q to M × c M , as π : Q → M × c M . Giv en any sufficien tly small neighborho o d U on M , let e b e a lo cal section of the oriented or tho gonal frame bundle F ( M ) with do main U . W e write this lo cal section as ( e, U ). Let ( b e, b U ) b e a similar lo cal s ection o f F ( c M ). W e can use these lo ca l sections to trivia lize the fib er bundle Q ov er U × b U , with the map q ∈ SO( T m M , T b m c M ) 7→ ( m, b m, ( q ij )) ∈ U × b U × SO( n ) , q ij := h b e j , q e j i . Then the distribution D | π − 1 ( U × b U ) , in the ab ove co or dinates, is s pa nned by the vector fields (4) e j := e j + q e j + X 1 ≤ α ≤ β ≤ n  e α , ∇ e j e β  −  q e α , ∇ qe j q e β  W ℓ αβ . where j = 1 , . . . , n . Her e , e j is seen as a vector field on Q | U × b U = π − 1 ( U × b U ) a nd q e j stands for the v ector field q 7→ q ( e j | π ( q ) ) . The vector fields W ℓ αβ are defined by (5) W ℓ αβ = n X s =1  q sα ∂ ∂ q sβ − q sβ ∂ ∂ q sα  . The symbol ℓ here is not a para meter ; it simply stands fo r “left” (a n explana tion of this will follow in Remark 2). Let us consider the optimal cont rol problem finding of a r olling q ( t ) connecting t wo configur a tions q 0 and q 1 , such that the cur ve m ( t ) = π ( q ( t )) has minimal length. W e do this b y introducing a metr ic h· , ·i on D , defined b y (6) h v 1 , v 2 i = h π ∗ v 1 , π ∗ v 2 i , v 1 , v 2 ∈ D q . With r esp ect to this metric, the e j in (4) are a lo cal orthono rmal basis. Also, from the definition o f D , we ha ve h v 1 , v 2 i = h b π ∗ v 1 , b π ∗ v 2 i . Hence, minimizing the length of m ( t ) is equiv alent to minimizing the le ng th of b m ( t ) = b π ( q ( t )) (this is can also be seen fro m the no -slip co nditio n). Definition 2. A triple ( Q, D , h· , ·i ) , wher e Q is a c onne cte d manifold, D is a distribution on Q and h· , ·i is a m et ric on D , is c al le d a su b-Riemannia n manifold . The sub-Riemannian distance function d ( q 0 , q 1 ) b et ween t wo points is defined a s the infimum of the length of all curves which ar e horizo ntal to D . W e can view rollings b etw een c o nfiguration fro m q 0 to q 1 along a curve of minimal leng th a s a sub-Riemannian le ngth minimizer in Q . F or the distance function to b e finite, we need tha t every pa ir of co nfigurations q 0 and q 1 can b e connected by a curve horizontal to D . In other words, we need to 6 E. GRONG determine when rolling without twisting or slipping is a controllable s y stem. O ur goal is to give sufficient conditions for this to hold, in terms of geometric inv ariants on M and c M . W e will leave the question of finding optimal curves for later r esearch. R emark 2 . The vector fields W ℓ αβ in (5) can b e considered as a “lo ca lly left inv ari- ant” bas is of ker π ∗ . It will b e prac tical to a ls o int ro duce a “lo cally right in v a riant” analogue. Relative to tw o chosen lo cal s e ctions ( e, U ) a nd ( b e , b U ), define (7) W r αβ = n X s =1  q β s ∂ ∂ q αs − q αs ∂ ∂ q β s  , q ij := h b e i , q e j i . Notice that W r αβ = P n l,s q αl q β s W ℓ ls . 2.4. Co n trol lability and brac k ets. Given an initial configur ation q 0 ∈ Q , write O q 0 for a ll p oints in Q that ar e reachable by a r olling starting from q 0 . This will be the o rbit of D at q 0 , which coincides with the reachable set of D , since D is a dis tr ibution (see, e.g ., [2, 1 5] for details). The Or bit Theor em [9, 22] tells us that O q 0 is a co nnected, immersed submanifold of Q , but also that the size can be approximated by the brackets of D . Define the C ∞ ( Q )-mo dule Lie D as the limit of the pro cess D 1 = Γ( D ) , D k +1 = D k + [ D , D k ] . Γ( D ) denotes the s ections o f D . Let D k q and Lie q D be the subspaces o f T q Q obtained by ev aluating resp ectively D k and Lie D at q . Then, for any q ∈ O q 0 , (8) Lie q D ⊆ T q O q 0 . In pa r ticular, it follows from (8), that if D is br acket gener ating a t q , i.e., if Lie q D = T q Q , then O q 0 is an op en submanifold of Q , a nd we say that we have lo c al c ont r ol lability at q 0 . If O q 0 = Q for one (and hence all) q 0 ∈ Q , the sys tem is called c ompletely c ontr ol lable . The lea st amount of control happens when O q 0 is n -dimensional submanifold. As a consequence of the Orbit theore m and F rob enius theor e m, this happ ens if and only if D | O q 0 is in v olutive, that is, if Lie q D = D q , for every q ∈ O q 0 . The fo cus of this pap er will b e to provide re s ults of controllability , by in vesti- gating when the dis tribution D will b e bracket genera ting at a given p oint q . R emark 3 . When D is not brack et gener ating at q , Lie q D will in general only g ive us a low er bo und for the size of O q . How e ver, if Lie D is lo cally finitely gener ated as a C ∞ ( Q )-mo dule, i.e., has a finite basis of vector field when restr ic ted to a sufficiently sma ll neighbor ho o d, then the equality ho lds in (8). R emark 4 . W e will use the no tation introduced her e for distributions in genera l, not just for the rolling distribution. 2.5. R elationshi p b etw een frame bundl es and rolling . Consider the linear Lie a lgebra s o ( n ). F o r in tegers α and β betw een 1 a nd n a nd not equa l, let w αβ be the matrix with 1 at entry αβ , -1 at en try β α , a nd zero at all o ther entries. Clearly w αβ = − w β α . Define w αα = 0 . The commutator br ack et b etw een these matrices are given by (9) [ w αβ , w κλ ] = δ β ,κ w αλ + δ α,λ w β κ − δ α,κ w β λ − δ β ,λ w ακ , The collectio n of all w αβ with α < β for m a basis for so ( n ). CONTR OLLABILITY OF ROLLING IN HIGHER DIM. 7 Consider the principal bundle F ( M ) → M . W e introduce the following notation. W rite σ αβ for the vector fields on F ( M ) cor resp onding to w αβ in the se nse of (1). Let E b e the principal Ehresmann connection corr esp onding to the a ffine Levi- Civita connection on M , with corresp onding principal connection form ω . As before, we denote the tautolo gical o ne-form b y θ . By using horizontal lifts with re s pe ct to E , we ca n define vector fields X j on F ( M ) b y (10) X j | f := h f f j , which s atisfy θ i ( X j ) = δ i,j . Hence, the tangent bundle of F ( M ) is trivial, since it is spanned by { X j } n j =1 and { σ αβ } α<β . Finally , w e define b σ αβ , b X j , b ω and b θ similarly on F ( c M ). The co nfiguration space Q may be identified with F ( M ) × F ( c M ) quotiented out by the diagona l action o f SO( n ). L et  denote the principal SO( n )-bundle  : F ( M ) × F ( c M ) → F ( M ) × F ( c M ) / SO( n ) ∼ = Q. Then  ( f , b f ) = q , if b f = q ◦ f . By viewing ω , b ω , θ a nd b θ as forms o n F ( M ) × F ( c M ), we a re able to obta in the following r esult. Theorem 1. Le t D = ker ω ∩ ker b ω ∩ ker( θ − b θ ) , and let D b e the r ol ling distribution. Then  ∗ D = D , and the map is a line ar isomorphi sm on every fib er. Pr o of. F r o m its definition, it is clear that { X j + b X j } n j =1 is a basis for D . Cho ose any pair of lo cal section ( e, U ) and ( b e, b U ) of r esp ectively F ( M ) and F ( c M ). Give F ( M ) × F ( c M ) | U × b U lo cal co ordina tes b y asso c ia ting the pa ir of frames ( f , b f ) to the ele ment (11)  m, b m,  f ij  ,  b f ij   ∈ U × b U × SO( n ) × SO( n ) , if f j = P n i =1 f ij e i | m and b f ij = P n i =1 b f ij b e i | b m holds. Relative to this trivializ a tion, we ca n define left and r ight vector field on each of the SO( n )-factor s. O n the first, define (12) Ψ ℓ αβ = n X s =1  f sα ∂ ∂ f sβ − f sβ ∂ ∂ f sα  , Ψ r αβ = n X s =1  f β s ∂ ∂ f αs − f αs ∂ ∂ f β s  . Notice that Ψ r αβ = P n l,s =1 f αl f β s Ψ ℓ ls . Remark also that Ψ ℓ αβ is just the r e striction of σ αβ to F ( M ) × F ( c M ) | U × b U , while Ψ r αβ depe nds o n the chosen lo c a l section e . Define b Ψ ℓ αβ and b Ψ r αβ analogo usly o n the second SO ( n )-factor. Restricted to F ( M ) × F ( c M ) | U × b U and in the lo cal coo rdinates (11), the vector fields X j and b X j can b e written as (13) X j = n X s =1 f sj  e s − X α<β Γ α sβ Ψ r αβ  , b X j = n X s =1 b f sj  b e s − X α<β b Γ α sβ b Ψ r αβ  , where Γ α iβ = h e α , ∇ e i e β i a nd b Γ α iβ = h b e α , ∇ b e i b e β i . 8 E. GRONG W e no w turn to the image o f T ( F ( M ) × F ( c M ) | U × b U ) under  ∗ . Define q ij , e j , W ℓ αβ and W r αβ on Q | U × b U as in Sectio n 2.3. Remark 2 allows us to rewrite e j on the form e j = e j + q e j + X α<β  Γ α j β W ℓ αβ − n X s =1 q sj b Γ α sβ W r αβ  Lo cally the mapping  can be describ ed as  :  m, b m, ( f ij ) , ( b f ij )  7→ ( m, b m, ( q ij )) , q ij = n X s =1 b f is f j s . and the action o n the tangent vectors is given by formulas  ∗ : e i 7→ e i b e i 7→ b e i Ψ r αβ 7→ − W ℓ αβ b Ψ r αβ 7→ W r αβ . F rom this it is clear that  ∗ D = D and tha t the map is injective of e ach fiber, since e j =  ∗ n X s =1 f j s  X s + b X s  .  F rom the form of the distribution D , w e obtain the following interpretation of rolling without twisting or slipping . Corollary 1. L et q ( t ) b e a r ol ling without twisting or slipping. L et ( f ( t ) , b f ( t )) b e any lifting of q ( t ) to a curve in F ( M ) × F ( c M ) t hat is horizontal t o D , and define m ( t ) and b m ( t ) as the r esp e ctive pr oje ctions to M and c M . Then ( f ( t ) , b f ( t )) satisfy the fol lowing (i) (No twist c ondition) Every ve ctor field f j ( t ) is p ar al lel along m ( t ) . Every ve ctor field b f j ( t ) is p ar al lel along b m ( t ) . (ii) (No slip c ondition) F or almost every t , f − 1 ( t )  ˙ m ( t )  = b f − 1 ( t )  ˙ b m ( t )  . F urthermor e, if ( f ( t ) , b f ( t )) is any absolutely c ont inuous curve in F ( M ) × F ( c M ) , satisfying (i) and (ii), then  ( f ( t ) , b f ( t )) is a r ol ling without twisting or slippi ng. Pr o of. (i) fo llows from the definition o f the principal E hresmann connec tio ns ker ω and ker b ω . (ii) is exactly the requirement for a curve to b e in ker ( θ − b θ ).  The main adv a nt age of the viewp oint g iven in Theo rem 1, is that it will help us to compute Lie q D . Corollary 2.  ∗ D k q = D k q for any q ∈ Q, k ∈ N . Pr o of. W e o nly need to show this lo cally . Intro duce lo ca l co o rdinates a s in the pro of o f Theorem 1 and let e j be as in (4). The n [ e i , e j ] =  ∗ " n X s =1 f is  X s + b X s  , n X s =1 f j s  X s + b X s  # , CONTR OLLABILITY OF ROLLING IN HIGHER DIM. 9 and since P n s =1 f is  X s + b X s  is a lo cal basis for D , it fo llows that D 2 =  ∗ D 2 lo cally . The rest follows by induction.  Since [ X i , b X j ] = 0, computations of brack ets of D , and hence of brack ets D , can be reduced to mostly computing brack ets of sections in the Ehres mann connections corres p o nding to the Levi-Civita connec tio ns of the manifolds inv o lved. 2.6. R emark on previous descriptions of rol ling us ing frame bundles . The description o f rolling g iven in Theor em 1, lo o k s very simila r to the definitio n of rolling without twisting or slipping found in [4] for dimension 2 . Her e, the descrip- tion of a rolling is in terms of the distribution e D := D ⊕ ker  ∗ , which can also b e describ ed a s (14) e D = ker( ω − b ω ) ∩ ker( θ − b θ ) . In [4], a rolling of a pair of 2-dimensio nal manifold is defined as a cur ve in Q , that is hor izontal to  ∗ e D , where e D is defined in terms o f (14 ). W e could hav e used e D fo r our co mputatio n, since clearly  ∗ e D k q = D k q also, and D is brack et g enerating at a point q ∈ Q if and only if e D is brack et ge nerating at any (and hence every) ( f , b f ) ∈  − 1 ( q ). How ever, since [ D k , k er  ∗ ] ⊂ D k + ker  ∗ , the a dditional bra ck ets are not o f any interest. The definition o f rolling o r “ rolling without slipping” in pro bability theor y is defined o n frame bundles [10], and is equiv alent to c o nsidering curves in F ( M ) × F ( c M ) that are horizontal to D for the special ca se when c M is R n with the Euclidean metric and standar d orientation. 3. Brackets of D 3.1. T ensors on M and asso ciated v ector fields. W e introduce a general t ype of v ector fields asso ciated to tensors. By giving a general formula for the brack ets of these, we essentially deter mine a ll the brack ets of D . Let T b e a tensor on M . W e will only co nsider tensors defined on vectors. T o an y tenso r k -tenso r on M , we can as so ciate the functions E i 1 ,...i k ( T ) : F ( M ) 7→ R f 7→ T ( f i 1 , . . . , f i k ) , 1 ≤ i s ≤ n. If k = 2 + l , and T is antisymmetric in the first tw o arg ument s, we can define vector fields on F ( M ) by W i 1 ,...,i l ( T ) = X 1 ≤ α<β ≤ n E α,β ,i 1 ,...,i l ( T ) σ αβ , If b T is a tensor on c M , we define E i 1 ,...i k ( b T ) and W i 1 ,...,i l ( b T ) similarly a s r esp ec- tively functions and v ector fie lds on F ( c M ). Lemma 1. L et X k b e define d as in (10) . (a) F or any l -tensor T , X k E i 1 ,...,i l ( T ) = E i 1 ,...,i l ,k ( ∇ T ) . (b) F or any 2 + l - tensor T , that is antisymmetric in the first two ar gumen t s [ X k , W i 1 ,...,i l ( T )] = W i 1 ,...,i l ,k ( ∇ T ) − n X s =1 E s,k,i 1 ,...,i l ( T ) X s . 10 E. GRONG Pr o of. (a) W e use a lo cal representation of X j . Consider the form ula given in (13), and write P n s =1 f sj e s as just f j . Since Ψ ℓ αβ is just the repres entation of σ αβ in lo cal co o rdinates, we can wr ite X j as X j = f j − X 1 ≤ α ≤ n h f α , ∇ f j f β i σ αβ . In this notation, remark fir s t that σ αβ f i = δ β ,i f α − δ α,i f β , which gives us X 1 ≤ α<β ≤ n h f α , ∇ f k f β i σ αβ f i j = 1 2 n X α,β =1 h f α , ∇ f k f β i ( δ β ,i j f α − δ α,i j f β ) = n X α =1  f α , ∇ f k f i j  f α = ∇ f k f i j . Then the result follows from rea lizing that X k E i 1 ,...,i l ( T ) = f k T ( f i 1 , . . . , f i l ) − X 1 ≤ α<β ≤ n h f α , ∇ f k f β i σ αβ T ( f i 1 , . . . , f i l ) = f k T ( f i 1 , . . . , f i l ) − T ( ∇ f k f i 1 , . . . , f i l ) − T ( f i 1 , ∇ f k f i 2 , . . . , f i l ) − · · · − T ( f i 1 , f i 2 , . . . , ∇ f k f i l ) = ∇ T ( f i l , . . . , f i l , f k ) . (b) The brack e ts [ σ αβ , σ κλ ] are, by definition, given b y the same relations a s describ ed in (9). W e will c ontin ue by the following computations. [ X k , W i 1 ,...,i l ( T )] = 1 2 n X κ,λ =1 X k ( E κ,λ,i 1 ,...,i l ( T )) σ κλ − 1 4 n X α,β ,κ, λ =1 h f α , ∇ f k f β i E κ,λ,i 1 ,...,i l ( T ) [ σ αβ , σ κλ ] − 1 2 n X κ,λ =1 E κ,λ,i 1 ,...,i l ( T ) σ κλ f k + 1 4 n X α,β ,κ, λ =1 E κ,λ,i 1 ,...,i l ( T ) σ κλ ( h f α , ∇ f k f β i ) σ αβ = 1 2 n X κ,λ =1 E κ,λ,i 1 ,...,i l ( ∇ T ) σ κλ + 1 2 n X κ,λ =1 ( T ( ∇ f k f κ , f λ , f i 1 , . . . , f i l ) + T ( f κ , ∇ f k f λ , f i 1 , . . . , f i l )) σ κλ − n X s =1 E s,k,i 1 ,...,i l ( T ) f s + 1 2 n X α,β ,s =1 E s,α,i 1 ,...,i l ( T ) ( h f s , ∇ f k f β i ) σ αβ + 1 2 n X α,β ,s =1 E s,k,i 1 ,...,i l ( T ) ( h f α , ∇ f s f β i ) σ αβ + 1 2 n X α,β ,s =1 E s,β ,i 1 ,...,i l ( T ) σ w κλ ( h f α , ∇ f k f s i ) σ αβ = W i 1 ,...,i l ,k ( ∇ T ) − n X s =1 E s,k,i 1 ,...,i l ( T ) X s  CONTR OLLABILITY OF ROLLING IN HIGHER DIM. 11 The next lemma gives a n explanation for the int ro duction of the ab ove notation. Lemma 2. [ X i , X j ] = − W ij ( R ) for i, j = 1 . . . n. Pr o of. This lemma is an easy cons equence of the Ca r tan equations. Since ker θ ∩ ker ω only contains the zero section of T F ( M ), we can show equa lit y in the ab ove equation by ev aluating the left and rig ht hand side by θ and ω a nd see that it pro duces the same result. E v aluating the left hand side, we get θ ([ X i , X j ]) = − dθ ( X i , X j ) = 0 , ω ([ X i , X j ]) = − dω ( X i , X j ) = X 1 ≤ α<β ≤ n Ω( X i , X j ) = − X 1 ≤ α<β ≤ n R ( f α , f β , f i , f j ) w α,β . which is obviously what we g et from ev aluating the right hand side.  R emark 5 . Com bining these t wo lemmas, we get a way to express the comm uta- tors o f the E hresmann connection. Let E = ker ω b e the Ehresmann connection corres p o nding to the Levi-Civita connec tio n. Then E k +2 = E k +1 + span  W i 1 ,...,i k +2 ( ∇ k R )  n i 1 ,...,i k +2 =1 , for k ≥ 0 . As a co nsequence o f this, we get the well know Am bro se-Singer theor e m, see [3],[18, App C], that the subalgebra spanned by elements ω | f  W i 1 ,...,i k +2 ( ∇ k R )  is contained in the ho mology alg ebra at f . W e adopt the conv ention that if the elemen ts in the collection are vector fields, “span” means the spa n ov er s mo oth functions (so in Remar k 5, it mea ns ov er C ∞ ( F ( M ))). If the elements are vectors, the span is over the real num b er s . 3.2. O btaining the brac kets for D . Computing the brack ets on D is a bit more complicated than each individual Ehresmann connectio n, since it is ha rder to know whether or not tw o vectors fields are equal mo d span { X j + b X j } n j =1 rather than just mo d span { X j } n j =1 . W e illustrate this by co mputing the tw o next bra ck ets. Lemma 3. (a) [ X k , [ X i , X j ]] = − W ij k ( ∇ R ) + P n s =1 E skij ( R ) X s . (b) L et R 2 b e the 6-tensor on M define d by R 2 ( Y α , Y β , Y i 1 , Y i 2 , Y i 3 , Y i 4 ) = R ( R ( Y α , Y β ) Y i 1 , Y i 2 , Y i 3 , Y i 4 ) . Then [ X l , [ X k , [ X i , X j ]] = − W ij k l ( ∇ 2 R ) + W lkij ( R 2 ) + n X s =1 ( E ij slk ( ∇ R ) − E ij sk l ( ∇ R )) X s . The r eason for the notation R 2 will b e clearer in Section 5.1. 12 E. GRONG Pr o of. Statement (a) follows dir ectly fro m Lemma 1. By Lemma 1 we also ha ve [ X l , [ X k , [ X i , X j ]]] = − W ij k l ( ∇ 2 R ) + n X s =1 E slijk ( ∇ R ) X s + n X s =1 E skij l ( ∇ R ) X s + n X s =1 E skij ( R ) W ls ( R ) = − W ij k l ( ∇ 2 R ) + W lkij ( R 2 ) + n X s =1 ( E ij slk ( ∇ R ) − E ij sk l ( ∇ R )) X s .  W e c a n con tin ue this pro cedure, computing mo re of the br ack ets us ing Lemma 1. How ever, these will b eco me more and mor e complicated. Also , for a gene r al pair o f manifolds, it is ha rd to determine which brack ets actually give us something new, that is, something that c o uld not be ex pressed a s linear combinations of previo usly obtained vectors. Ra ther than giving the total pictur e , we will therefore fo cus on giving some sufficient conditions, which a re usually more simple to chec k. 4. Sufficient conditions f or contr ollability Let R a nd b R be the curv ature tensor on M and c M resp ectively . Define a new 4-tensor o f elements in D , by R = π ∗ ( R ) − b π ∗ ( b R ) . Remark that R may also b e seen as a bilinear map o f tw o elements in V 2 D . Use ∇ R to denote the 5-tensor on D , defined b y π ∗ ( ∇ R ) − b π ∗ ( ∇ b R ) . Finally , intro duce a bundle morphism R : V 2 D → V 2 D ∗ , so that (15) R ( ξ 1 )( ξ 2 ) = R  ξ 2 , ξ 2  ξ 1 , ξ 2 ∈ 2 ^ D q . 4.1. Pro jection of the results on D . F r om the discussio n in prev ious section, we have the following formulations for the brack ets o f D . Lemma 4. L et ( e , U ) and ( b e, b U ) b e two lo c al se ctions of F ( M ) and F ( c M ) , r esp e c- tively. Then on Q | U × b U , in t erms of the notation int r o duc e d in Se ction 2.3, D 2 = D 1 ⊕ s pa n  X 1 ≤ α<β ≤ n R ( e α , e β , e i , e j ) W ℓ αβ  n i,j =1 D 3 = D 2 + s pa n  X 1 ≤ α<β ≤ n ∇ R ( e α , e β , e i , e j , e k ) W ℓ αβ + q R ( e i , e j ) e k − b R ( q e i , q e j ) q e k + X 1 ≤ α<β ≤ n D q e α , ∇ qR ( e i ,e j ) e k − b R ( qe i ,qe j ) qe k q e β E  n i,j,k =1 . Pr o of. The formula for D 2 follows directly from Lemma 2 a nd the lo cal fo rmulation of  ∗ given in the pro of of Theorem 1. T o s ee how the express ion for D 3 follows CONTR OLLABILITY OF ROLLING IN HIGHER DIM. 13 from Lemma 3 (a), observe first that n X s =1  E skij ( R ) X s + E skij ( b R ) b X s  = − n X s =1  E ij k s ( R ) X s + E ij k s ( b R ) b X s  = n X s =1  E ij k s ( R ) − E ij k s ( b R )  b X s (mo d D ) . F urthermore  ∗ n X s,µ,λ,κ, =1 f iµ f j λ f kκ  E µλκs ( R ) − E µλκs ( b R )  b X s = q R ( e i , e j ) e k − b R ( q e i , q e j ) q e k − n X 1 ≤ α<β ≤ n  q e α , ∇ qR ( e i ,e j ) e k q e β  W ℓ αβ − n X 1 ≤ α<β ≤ n  q e α , ∇ R ( qe i ,qe j ) qe k q e β  W ℓ αβ .  Corollary 3. Define a bund le morphi sm Ξ : D ⊕ V 2 D → D ∗ by Ξ( v , ξ ) = ι v ◦ R ( ξ ) , v ∈ D q , ξ ∈ 2 ^ D q , wher e ι v : η 7→ η ( v , · ) for any v ∈ D q and two form η . Th en dim D 3 q ≥ n + rank Ω | q + r ank Ξ | q . Pr o of. Given point q ∈ Q , introduce lo cal co or dinates in a neighborho o d of q , in the wa y demonstr ated in Section 2.3. W e will also keep the same notation fro m the previous mentioned s e ction. Then, a ll w e need to show is that for a given q ∈ Q , the dimensio n of (16) span { q R ( e i , e j ) e k ( π ( q )) − R ( q e i , q e j ) q e k ( b π ( q )) } n i,j,k =1 ⊂ T b π ( q ) c M , is equal to rank Ξ q . Int ro duce iso mo rphisms ♭ : D → D ∗ and ♯ : D ∗ → D , r elative to the metric defined in (6). Use the sa me symbols for the is omorphisms b etw e en the tangent bundle a nd the cotang ent bundle on M and c M , induced by their resp ective metrics. Observe that Ξ( e k , e i ∧ e j ) = π ∗  ♭ ( R ( e i , e j ) e k )  − b π ∗  ♭ ( b R ( q e i , q e j ) q e k  = ♭ Y − ♭Z , where Y = R ( e i , e j ) e k + q R ( e i , e j ) e k + X 1 ≤ α<β ≤ n  e α , ∇ R ( e i ,e j ) e k e β  −  q e α , ∇ qR ( e i ,e j ) e k q e β  W ℓ αβ , Z = q − 1 b R ( q e i , q e j ) q e k + b R ( q e i , q e j ) q e k + X 1 ≤ α<β ≤ n D e α , ∇ q − 1 b R ( q e i ,qe j ) qe k e β E − D q e α , ∇ b R ( q e i ,qe j ) qe k q e β E W ℓ αβ . 14 E. GRONG F rom this, it beco mes clear that b π ∗ ♯ is a bijectiv e linear ma p from the image of Ξ q to (16).  4.2. Suffi cien t condi tion in terms of the curv ature tens or and sectional curv ature. As men tioned before, there is a strong connection b e tween controlla- bilit y and geometry in the t wo dimensional ca se. Theorem 2 ([2, 4]) . F or q ∈ Q , let κ q denote t he Gaussian curvatur e of M at π ( q ) , and let b κ q denote the Gaussian curvatur e of c M at b π ( q ) . Then dim O q = 5 , if and only if κ − b κ 6≡ 0 on O q . If κ − b κ ≡ 0 on O q , t hen dim O q = 2 . The “if and only if” in the ab ov e theo rem follows from the fact that in t wo dimensions, the rolling distribution D at a p oint q , is either bra ck et generating or inv olutive. This do es not hold in higher dimensions, how e ver, but we are able to present the following genera lization. Definition 3. The smal lest inte ger k such that D k q = Lie q D is c al le d t he step of D at q . Theorem 3. Le t R b e as define d in (15 ) . Then, for any element q , q 0 ∈ Q , the fol lowing holds. (a) dim O q 0 = n if an only if R| O q 0 ≡ 0 . (b) If R q is an isomorphism, then D is br acket gener ating of s t ep 3 at q . Henc e, if O q 0 c ontains a p oint q , so t hat R| q is an isomorphism, then O q 0 is an op en submanifold. R emark 6 . The sta temen t in Theorem 3 (a) was also presen ted in [6, Cor . 5.28]. By combining [6, Cor. 5.2 6] with [6, Prop. 5.17], and doing some simple calcula tions, we can also obtain the result of Theorem 3(b), how ever, this is not stated. The pro of is presented here, since the a pproach in [6 ] differ from our s, and since the results were obtained indep endently . Pr o of. Statement (a) b eco mes o bvious from L e mma 4 . T o prove (b), let π ( q ) = ( m, b m ), where π : Q → M × c M is the pro jection. Pick resp ective lo cal sections ( e, U ) and ( b e , b U ) o f F ( M ) and F ( c M ) around m and b m and use them to introduce the lo cal co ordinates defined in Sec tio n 2.3. Since R| q is an isomorphism, we know that span  X 1 ≤ α<β ≤ n R ( e α , e β , e i , e j ) W ℓ αβ    q  n i,j =1 = span  W ℓ ij ( q )  n i,j =1 , span   q R ( e i , e j ) e k − b R ( q e i , q e j ) q e k     q  n i,j,k =1 = span  q e i ( m )  n i =1 . Lemma 4 then tells us that D 3 q = span { e j ( q ) , q e i ( m ) , W ℓ αβ ( q ) } n i,j,αβ =1 = span { e j ( m ) , b e i ( b m ) , W ℓ αβ ( q ) } n i,j,αβ =1 = T q Q.  CONTR OLLABILITY OF ROLLING IN HIGHER DIM. 15 T o state that R q is an iso morphism is equiv a lent to claiming tha t R q induces a pseudo-inner pro duct on V 2 D q , i.e. a nondeg enerate bilinear map. Ther e fore, we hav e the following wa y w e can c hec k if R is an isomorphism at q . Corollary 4. Cho ose any orthonormal b asis { v j } of T m M , m = π ( q ) . Compute the determinant of the n ( n − 1) 2 × n ( n − 1) 2 matrix det  R ( v α , v β , v i , v j ) − b R ( q v α , q v β , q v i , q v j )  , 1 ≤ α < β ≤ n ar e ro w indic es, 1 ≤ i < j ≤ n ar e c olumn indic es, If this determinant is nonzer o, then we have lo c al c ont r ol lability at q . F rom this we obtain the following coro llary . Corollary 5. Define a function κ q on 2-dimensional planes L in D q by the formula κ q ( L ) = κ π ( q ) ( π ∗ L ) − b κ b π ( q ) ( b π ∗ L ) , wher e κ m and b κ b m denotes the r esp e ctive se ctional curvature s of M and c M at the indic ate d p oints. Then (a) dim O q 0 = n if an only if κ q ≡ 0 for any q ∈ O q 0 . (b) If κ q > 0 or κ q < 0 , then D is br acket gener ating of st ep 3 at q . Pr o of. If κ q ≡ 0, then R q is 0 also. Similarly , if κ p > 0 (r esp. κ p < 0) for every L , then − R q (resp. R q ) will b e an inner pro duct on V 2 D q . T o see this, fo r the case κ q > 0, we o nly need to show that R q ( ξ , ξ ) < 0, whenever ξ ∈ V 2 D q is nonzer o. Pick an orthono rmal basis ba sis v 1 , . . . , v n of D q . W rite L ij := span { v i , v j } . In this basis, we hav e that if ξ = P 1 ≤ i 0 . The case κ q < 0 is treated similarly .  R emark 7 . All the co nditions stated here, are sufficient conditions for lo cal co n- trollability . How ever, if they hold in a ll p oints, they will natura lly b e sufficient conditions for complete con trollability . 4.3. E xamples. Example 1 . W e start with t wo known examples, to verify our results and demon- strate their effectiveness of obtaining informa tion on controllabilit y . (a) If M is a sphere o f r adius r and c M = R n is the n dimensional Euclidean space, then M has co nstant sectio na l curv ature 1 r 2 , w hile c M has consta nt sectional curv ature 0 . It follows that κ q ≡ 1 r 2 for an y q ∈ Q . Hence D is brack et gener ating at all p oints, and the system is co mpletely controllable. (b) If M and c M a re the spheres with r esp ective radii r and b r , then κ q ≡ 1 r 2 − 1 b r 2 , for an y q ∈ Q . Hence the system is co mpletely con trollable if a nd o nly if r 6 = b r . When r = b r , D b e comes an inv olutive distribution. 16 E. GRONG T o co mpare, see [8 , 23] for a former pro of of the controllability o f (a), a nd [1 6] for a trea tmen t of the ex ample in (b). Example 2 . Mor e genera lly , if M is any manifold with only strictly p ositive or strictly negative sectional curv a tur e, ro lling on n -dimensional Euclidean spa ce, then this system is completely controllable (w e will later show that this only needs to hold in one p oint o f M ). Example 3 . Let M = S 2 × R b e the subs e t the Euclidian spa ce R 4 ,  ( x 0 , x 1 , x 2 , x 4 ) ∈ R 4 : x 2 0 + x 2 1 + x 2 2 = 1  . Define a loca l section o n the subset U = { ( x 0 , x 1 , x 2 , x 3 ) ∈ M : x 2 > 0 } , with the orthonor mal vector fields e 1 = − q x 2 1 + x 2 2  − ∂ x 0 + x 0 x 2 1 + x 2 2 ( x 1 ∂ x 1 + x 2 ∂ x 2 )  , e 2 = x 2 p x 2 1 + x 2 2  − ∂ x 1 + x 1 x 2 ∂ x 2  , e 3 = ∂ x 3 . (a) Le t us first consider M ro lling on R 4 . The r olling distribution can lo cally be descr ib es by D 1 = span { e 1 , e 2 , e 3 } . e 1 = e 1 + q e 1 , e 2 = e 2 + q e 2 + x 0 p x 2 1 + x 2 2 W 12 , e 3 = e 3 + q e 3 . D is then of step 3 for any q ∈ U a nd D 2 = D 1 ⊕ s pa n { W 12 } , D 3 = D 2 ⊕ s pa n { q e 1 , q e 2 } Since D 3 is lo cally finitely genera ted, we know that dim O q = 6 for a ny q ∈ U (a nd for symmetry reasons, every q ∈ Q ). (b) Let M roll on a co py of itself. Consider the rotation ma tr ix ( q ij ) = ( h e i , q e j i ) . Give ( q ij ) the co ordina tes ( q ij ) =  cos θ cos ϕ sin θ cos ψ − cos θ sin ϕ sin ψ sin θ sin ψ + cos θ sin ϕ cos ψ − sin θ cos ϕ cos θ cos ψ + si n θ sin ϕ cos ψ cos θ sin ψ − sin θ si n ϕ cos ψ − sin ϕ − sin ψ cos ϕ cos ϕ cos ψ  . The vector fields spanning D ar e loca lly given by e 1 = e 1 + q e 1 − x 0 (sin θ cos ψ + cos θ sin ϕ sin ψ ) p x 2 1 + x 2 2 V , e 2 = e 2 + q e 2 + x 0 p x 2 1 + x 2 2 W 12 − x 0 (cos θ co s ψ + sin θ sin ϕ s in ψ ) p x 2 1 + x 2 2 V , e 3 = e 2 + q e 2 + x 0 cos ϕ sin ψ p x 2 1 + x 2 2 V , V := cos ϕ co s ψ W 12 − co s ϕ sin ψ W 13 − s in ϕW 23 . The ma trix −  R ( e α , e β , e i , e j )  , i < j, α < β is then given by  1 − cos 2 ϕ cos 2 ψ − cos 2 ϕ sin ψ cos ψ cos ϕ sin ϕ cos ψ − cos 2 ϕ sin ψ cos ψ − cos 2 ϕ sin 2 ψ cos ϕ sin ϕ sin ψ cos ϕ sin ϕ cos ψ cos ϕ sin ϕ sin ψ − sin 2 ϕ  . CONTR OLLABILITY OF ROLLING IN HIGHER DIM. 17 It is easily c hec ked that this matrix has rank 2 , ex cept when sin ϕ = sin ψ = 0. Res tricted to the s ubset of Q where the latter equation hold, that is, the configuratio ns where the tw o copies o f the line λ = { (0 , 0 , 0 , x 3 ) ∈ M } , lie tang ent to ea ch o ther, D is inv o lutiove and the o rbits are 3 dimensional. On the other points, we hav e that D 2 = D 1 ⊕ s pa n { W 12 , V } , D 3 = D 2 ⊕ s pa n { q e 1 , q e 2 , q e 3 } . so the orbits hav e dimension 8, or co dimensio n 1. This example illustrates that if we are rolling manifolds of dimension higher than t wo, the dimensio n of O q do es not only depend on the co nnect- ing pair of points π ( q ) = ( m, b m ). 5. P ar ticular cases W e present so me sp ecial results for when the manifolds inv olved in the rolling are par ticular nice. W e will first deal with lo cally symmetr ic space s, then presen t some re sults for ro lling on complete spa ces. Remar k that all of these results are applicable to the case of ro lling on R n , since this is bo th lo ca lly s ymmetric and complete. 5.1. Lo cally symmetric spaces. Recall the definitio n of Ω from (2). Prop ositi o n 1. L et M b e lo c al ly symmetric and let c M b e flat ( b R ≡ 0 ) . Then D is at most of step 3. D is br acket gener at ing at q ∈ Q if and only if Ω | π ( q ) : 2 ^ D q → so ( n ) , is a line ar isomorp hism. Pr o of. Cons ider the bundle D , and let X j and b X j be defined as in (10) . Since c M is flat [ X i + b X i , X i + b X j ] = [ X i , X j ] . Then [ X i , X j ] = − W ij ( R ) , [ X k , [ X i , X j ]] = n X s =1 E skij ( R ) X s , [ X l , [ X k , [ X i , X j ]]] = − n X s =1 E skij ( R ) W sl ( R ) ∈ D 2 , Hence, D 2 ( f , b f ) + ker  ∗ ( f , b f ) = T ( f , b f )  F ( M ) × b F ( M )  only if span n W ij ( R )   ( f , b f ) o n i,j =1 = span n σ ij   ( f , b f ) o n i,j =1 . and this also implies that span { σ ij , P n s =1 E skij ( R ) X s } = span { σ ij , f j } , whic h gives us the desired r esult.  When c M is not flat, the results become a little bit more complicated, and require us to in tro duce so me notation. Let R l denote the 2 l + 2 -tensor defined by R 1 = R and R l ( Y α , Y β , Y i 1 , Y i 2 , . . . , Y i 2 l − 1 , Y 2 l ) := R l − 1 ( R ( Y α , Y β ) Y i 1 , Y i 2 , . . . , Y i 2 l − 1 , Y 2 l ) . Lemma 5. If ∇ R = 0 , then ∇ R l = 0 for any l ≥ 1 . 18 E. GRONG Pr o of. W e give the pro o f b y induction. Assume that ∇ R k = 0 for 1 ≤ k < l. Let m ( t ) b e any smo o th curve in M . Let v 1 ( t ) , . . . , v n ( t ) b e par allel v ector fields alo ng m ( t ). Then d dt R l ( v α , v β , v i 1 , v i 2 , . . . , v 2 l − 1 , v 2 l ) = d dt n X s =1 R ( v α , v β , v i l , v s ) R l − 1 ( v s , v i 2 , . . . , v 2 l − 1 , v 2 l ) = 0 . Hence ∇ R l = 0 als o.  W e introduce a notation re la ted to R l , similar to what we did for R . Use b R l for the analog ues tensor on c M , and write R l for the tenso r on D defined by R l = π ∗ ( R l ) − b π ∗ ( b R l ) . Prop ositi o n 2. L et M a nd c M b oth b e lo c al ly symmetric. Then D is br acket gener ating at q if and only if (17) [ l ≥ 1 span  X 1 ≤ α<β ≤ n R l ( e α , e β , e i 1 , . . . , e i 2 l ) W αβ   q  = k e r π ∗ | q . Pr o of. W e will lo ok a t the brack ets of D . F rom Lemma 5, we know that for an y l ≥ 1,  X i 1 ,  X i 2 ,  · · ·  X i 2 l − 1 , X i 2 l  · · ·  = ( − 1) l W i 1 ,...,i 2 l ( R l ) ,  X i 1 ,  X i 2 ,  · · ·  X i 2 l , X i 2 l +1  · · ·  = ( − 1) l +1 n X s =1 E s,i 1 ,...,i 2 l +1 ( R l ) X s . Analogues rela tions hold for the brack ets of b X j . Pro jecting the even brack ets to T q Q , we get the left hand side of (17), which has to b e equa l to all of ker π ∗ | q in order for D to b e bra ck et gener ating. Conv ersely , if (17) holds , then the pro jection of the o dd brack ets will span T q Q tog e ther with ker π ∗ | q and D q .  5.2. R olling on a complete manifold. The fa ct that one of the manifolds is complete, makes it eas ier to give statements ab out co mplete controllability . The reason for this, can b e summed up in the following Lemma. Lemma 6 . As s ume that c M is c omplete. L et t 7→ m ( t ) b e any absolutely c ontinuous curve in M with domain [0 , τ ] . L et q 0 ∈ Q b e any p oint with π ( q 0 ) = m (0) . Then ther e is a r ol ling t 7→ q ( t ) of M on c M , so that q (0) = q 0 , π ◦ q ( t ) = m ( t ) for any t ∈ [0 , τ ] . Pr o of. If we assume first that b oth M a nd c M are complete, then such a r olling q ( t ) exist. The pro o f for this can b e fo und in [2, p. 386]. This pro of is done for the case when M and c M ar e tw o dimensio nal, but can, with simple mo difications, be generalized to higher dimensions. Assume now that M is not c omplete. Let f ( t ) b e a lifting of m ( t ) to a curve in F ( M ) that is ho r izontal to the Ehres mann connection, i.e . each f j ( t ) is parallel along m ( t ). Define the curve in R n by , e m ( t ) = Z t 0 f − 1 ( s )  ˙ m ( s )  ds. CONTR OLLABILITY OF ROLLING IN HIGHER DIM. 19 Let e f ( t ) b e a lifting of e m ( t ) to a curve F ( R n ), so that each e f j ( t ) is pa rallel alo ng e m ( t ) . Then, fro m Coro llary 1, e q ( t ) =  ( f ( t ) , e f ( t )) is a r olling of M on R n . Let e q 0 := e q (0). Since bo th R n and c M are complete, we know that there is a rolling b q ( t ) o f R n on c M along e m ( t ), so that b q (0) = q 0 ◦ e q − 1 0 . W e can then o bta in our desired rolling b y defining q ( t ) = b q ( t ) ◦ e q ( t ) .  Prop ositi o n 3. L et the manifold c M b e c omplete. Assume t hat ther e is a p oint m ∈ M , so that D q is br acket gener ating for every p oint q ∈ π − 1 ( m ) . Then t he system is c ompletely c ontr ol lable. Pr o of. Let q 0 be any element in Q . F ro m Lemma 6 we know that there is a rolling q ( t ) from q 0 to some p oint q 1 ∈ π − 1 ( m ). Hence O q 0 = O q 1 . But since D is bracket generating in q 1 , O q 1 is a n op en submanifold. Since q 0 was arbitrary , we hav e lo cal controllabilit y at every p oint, so O q 0 = Q for any q 0 ∈ Q .  Corollary 6. L et c M b e a manifold t hat is b oth c omplete and flat. Assume that ther e is a p oint m ∈ M , so that for some (and henc e any) orthonormal b asis { v j } n j =1 of T m M , det ( R ( v α , v β , v i , v j )) 6 = 0 . 1 ≤ α < β ≤ 1 ar e r ow indic es , 1 ≤ i < j ≤ 1 ar e c olumn indic es . Then the system is c ompletely c ontr ol lable. Example 4 . Let M be the surface in R 3 , defined by M = { ( x 1 , x 2 , x 3 ) ∈ R 3 : q x 2 2 + x 2 3 = 1 − f ( x 1 ) , | x 1 | < 3 2 } , where f ( x 1 ) = ( 0 if | x 1 | ≤ 1 e − 1 ( | x 1 |− 1) 2 if 1 < | x 1 | < 3 2 . Define the following orthonorma l basis o n M , e 1 = 1 p 1 + f ′ ( x 1 ) 2  ∂ x 1 − f ′ ( x 1 ) 1 − f ( x 1 ) ( x 2 ∂ x 2 + x 3 ∂ x 3 )  , e 2 = 1 1 − f ( x 1 ) ( − x 3 ∂ x 2 + x 2 ∂ x 3 ) . All Chr istoffel symbols are determined by Γ 1 12 = h e 1 , ∇ e 1 e 2 i = 0 , Γ 1 22 = h e 1 , ∇ e 2 e 2 i = f ′ ( x 1 ) (1 − f ( x 1 )) p 1 + f ′ ( x 1 ) 2 . and from this we can compute the Gaussia n curv ature κ ( x ) = f ′′ ( x 1 ) (1 + f ′ ( x 1 ) 2 ) 2 (1 − f ( x 1 )) . Inserting the v a lue o f f ( x ) we obtain that κ ( x ) = 0 for | x 1 | ≤ 1, but strictly positive for 1 < | x 1 | < 3 2 . It follo ws that , if w e ro ll M on R 2 , the system is completely controllable. Observe that in this ca se Lie D = spa n  e 1 + q e 1 , e 2 + q e 2 + Γ 1 22 W 12 , f ( x 1 ) W 12 , f ( x 1 ) q e 1 , f ( x 1 ) q e 2  , fails to be lo cally finitely generated around p oints fulfilling | x 1 | = 1. 20 E. GRONG 6. Fur ther generaliza tion of r ol ling without twisting or slipping Up un til now, we ha ve o nly been concerned with rolling t wo Riemannian ma ni- folds on each other without twisting o r slipping. The definition can ea sily be g en- eralized to manifolds with an affine connectio n. W e intro duce the genera lization here. Let M and c M be tw o c o nnected ma nifo lds , with resp ective affine connections ∇ and b ∇ . Then a rolling without t wisting or s lipping can be seen a s an abs o lutely contin uous cur ve q ( t ) in to the manifold Q = n q ∈ GL( T m , T b m c M ) : m ∈ M , b m ∈ c M o . satisfying (no slip condition) and (no twist co nditio n) fro m section 2.1. Reexamining the pr o ofs, it turns o ut that the descr iption of many of the results we had for rolling rela ted to the Levi-Civita connection, generalizes to ge ner al connections. W e will descr ib e this here br iefly . Define the tautological one-for m on F ( M ) as the R n -v alued o ne-form obtained by θ ( v ) = f − 1 ( v ) , v ∈ T f F ( M ) . Define b θ similarly , while ω and b ω are defined in terms of the connection. W e will also cho ose Riemannia n structure s on M a nd c M in o rder to pres ent the results in a s imilar way and make them eas ier to co mpa re, but these do not need to b e compatible with the connections. The rolling distribution D , will still be a n n dimens io nal distribution, a nd the relation in Theorem 1 is still v alid. D is loca lly spanned by vector fields e j = e j + q e j + n X α,β =1   e α , ∇ e j e β  − D q e α , b ∇ qe j q e β E E ℓ ij . where E ℓ ij = P s =1 q r i ∂ ∂ q rj . The study o f controllability b eco mes ha rder, since torsio n may app ear in the equations (3). How e ver, this is actually the o nly complication. Let ǫ αβ ∈ gl ( n, R ) be the matrix with only 1 in a t entry αβ and zer o o therwise. Define X j as in equation (1 0). If we then modify the definition o f W i 1 ,...,i l ( T ) to W i 1 ,...,i l ( T ) = n X α,β =1 E α,β ,i 1 ,...,i l ( T ) σ ( ǫ αβ ) , now defined for an y tensor T , then Lemma 1 still ho lds for any connection. Since all of our results follow from Lemma 1 and 2, it follows that o ur results also holds for a ny pair of manifolds with tor sion fr ee connectio ns. References [1] A. A grachev , R ol ling b al ls and o ctonions . 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