Table Of ContentCONTROLLABILITY OF ROLLING WITHOUT TWISTING OR
SLIPPING IN HIGHER DIMENSIONS
2
1 ERLENDGRONG
0
2
n
a
J Abstract. Wedescribehowthedynamicalsystemofrollingtwon-dimensional
connected,orientedRiemannianmanifoldsM andMcwithouttwistingorslip-
1
ping,canbeliftedtoanonholonomicsystemofelementsintheproductofthe
3
orientedorthonormalframebundlesbelongingtothemanifolds. Byconsider-
ing the lifted problem and using properties of the elements in the respective
]
C principalEhresmannconnections, weobtainsufficientconditionsforthelocal
controllability of the system in terms of the curvature tensors and the sec-
O
tional curvatures ofthemanifoldsinvolved. Wealsogivesomeresultsforthe
. particularcaseswhenM andMcarelocallysymmetricorcomplete.
h
t
a
1. Introduction
m
[ The rollingoftwosurfaces,without twistingorslipping, isa goodillustrationof
anonholonomicmechanicalsystem,whosepropertiesareintimatelyconnectedwith
3
geometry. It has therefore receivedmuch interest, and we can mention [1, 4, 5, 17]
v
8 and [2, Chapter 24] as examples of research produced in this area. In particular,
5 thetreatmentofrollingin[2,4]wasdonebyformulatingitasanintrinsicproblem,
2 independent of the imbedding of the surfaces into Euclidean space.
5
The generalization of this concept to that of an n-dimensional manifold rolling
.
3 without twisting or slipping on the n-dimensional Euclidean space, is well known
0
(see e.g. [13, p. 268], [10, Chapter 2.1]). It is usually formulated intrinsically, in
1
terms of frame bundles, and is an important tool in stochastic calculus on mani-
1
folds. A definition for two arbitrary n-dimensional manifolds rolling on each other
:
v
withouttwistingorslipping,firstappearedin[20,App. B],however,thisonlydealt
i
X withmanifoldsimbeddedintoEuclideanspace. Anintrinsicdefinitionforrollingof
r higher dimensional manifolds, that connected the definition in [20] with the intrin-
a sic approach in [2, 4], was presented in [8]. Apart from appearing as mechanical
systems, rolling of higher dimensional manifolds can be also used as a tool in in-
terpolationtheory. Fordemonstrationof the “rollingandwrapping”-technique,we
refer to [11]. See also [21] for an example where this is applied in robot motion
planning.
For the rolling of two 2-dimensional manifolds, there is a beautiful correspon-
dence between the degree of control and the geometry of the manifolds [2, 4].
Essentially, we have complete control over our dynamical system if the respective
Gaussian curvatures M and M do not coincide. Controllability in higher dimen-
sions has been addressed in some special cases [16, 23]. The first general result
c
2000 Mathematics Subject Classification. 37J60,53A55,53A17.
Keywordsandphrases. Rollingmaps,controllability,bracket-generating,framebundles,non-
holonomicconstraints.
1
2 E.GRONG
on controllability in higher dimension is presented in [6], where it is shown that
the curvature tensors determine the brackets of the distribution obtained from the
constraintsgivenbyneither twisting,norslipping. Itisalsoprovedthatbeing able
tofindarollingtoanarbitrarycloseconfiguration,connectingthesametwopoints,
is a sufficient condition for complete controllability.
The objective of this paper will be to describe the connection between geome-
try and controllability for rolling of higher dimensional manifolds. We do this by
connecting the earlier mentioned viewpoint from stochastic calculus with the one
presented in [8].
This paper is organized as follows. In Section 2, we state the intrinsic defini-
tion of rolling. We present some of the theory of frame bundles, and develop our
notation there. We end this section by showing how we can lift our problem to
the oriented orthonormalframe bundles of the involvedmanifolds. We continue in
sectionSection3,bydoingcomputationsontheliftedproblem. Byusingproperties
of the sections in the Ehresmann connections, we obtain formulas for computation
of the brackets of the rolling distribution. In Section 4 we project the results back
to our configuration space of relative positions of the manifolds. Section 4.2 con-
sist of conditions for controllability in terms of the Riemann curvature tensor and
the sectional curvature of the manifolds involved. We end this section with some
examples. Section 5 focuses on results concerning the rolling of locally symmetric
and complete manifolds. Section 6 contains a brief comment on how to general-
ize the concept of rolling without twisting or slipping to manifolds with an affine
connection,andwhy the results presentedhere also holds for a rollingof manifolds
with a torsion free affine connection.
The author would like to express his gratitude to Mauricio Godoy Molina for
many fruitful discussions concerning this subject.
2. Intrinsic definition of rolling and its relations to frame bundles
2.1. Intrinsicdefinitionofrollingwithouttwistingorslipping. Throughout
this paper, M and M will denote connected, oriented, n-dimensional Riemannian
manifolds. Sinceinthespecialcasen=1,theconditionsofrollingwithouttwisting
orslippingbecomehcolonomic(see[8]),wewillalwaysassumethatn≥2. Weadopt
the convention to equip objects (points, projection, etc.) related to M with a hat
(ˆ). Objectsrelatedtobothofthemareusuallydenotedbyabar(¯),whileobjects
connected to M are not given any special distinction. The exceptioncto this rule is
theRiemannianmetricandtheaffineLevi-Civitaconnectionwhicharerespectively
denoted by h·,·i and ∇ on both M and M. The context will make it clear which
manifoldtheseobjectsarerelatedto. ForavectorfieldX onM,wewillwriteX|
m
rather than X(m), and we will use similacr notation for other sections of bundles.
For any pair of oriented inner product spaces V and V, we let SO(V,V) denote
the space of all orientation preserving linear isometries from V to V. This allows
us to define the SO(n)-fiber bundle Q over M ×M by b b
b
Q= q ∈SO TmM,TmbM : mc∈M,m∈M .
n (cid:16) (cid:17) o
We can be sure that this fiber bundle is principal in the case when n=2, but not
c b c
in general. The space Q represents all configurations or relative positions of M
and M, so that the two manifolds lie tangent to each other at some pair of points.
c
CONTROLLABILITY OF ROLLING IN HIGHER DIM. 3
The isometry q : TmM → TmbM, represents a configuration where M at m lies
tangent to M at m. The relative positioning of their tangent spaces is given by
how q maps TmM into TmbM. Acrollingthen becomes a curve in the space ofthese
configuratiocns. b
c
Definition 1. Let π and π denote the respective natural projections from Q to M
and M. A rolling without twisting or slipping is an absolutely continuous curve
q :[0,τ]→Q,b with m(t):=π(q(t)), m(t):=π(q(t)),
c
satisfying the following conditions:
No slip condition: m˙ (t)=q(t)m˙ (t) for almost everby t, b
No twist condition: an arbitrary vector field X(t) is parallel along m(t)
if and only if q(t)X(t) is parallel along m(t).
b
From now on we will mostly refer to a rolling q(t) without twisting or slipping
b
as simply a rolling.
These two conditions can be described in terms of a distribution D of rank
n on Q. Consider the problem of finding the rolling q(t) between two different
configurationsq andq suchthatm(t)(andtherebyalsom(t))hasminimallength.
0 1
Here,byadistributiononQ,wemeanasub-bundleofTQ. Thistypeofcurvescan
beviewedasoptimalcurvesinaninput-lineardriftfreeoptimalcontrolproblemor
b
length minimizers in a sub-Riemannian manifold. We will describe this structure
in more detail in Section 2.3. First, however, we will review some facts about
connections and frame bundles.
2.2. Fromprincipalbundlestoorientedorthonormalframebundles. Con-
sider a general principal G-bundle τ : P →M, where the Lie group G acts on the
right. We call the sub-bundle V :=kerτ of TP the vertical space of P. If g is the
∗
Lie algebra of G, then for any element A ∈ g, we have a vector field σ(A) defined
by
d
(1) σ(A)| φ= φ(p·exp (tA)), for any p∈P,φ∈C∞(P).
p dt G
(cid:12)t=0
(cid:12)
Here, expG :g→G is(cid:12)the group exponential. We remark that σ(A) is a section of
(cid:12)
V and for any p∈P, the map g→V ,A7→σ(A)| is a linear isomorphism.
p p
A sub-bundle E of TP is called a principal Ehresmann connection if TP =
E⊕V and satisfy r E =E , where r denotes the right multiplication of g ∈G.
g p p·g g
Equivalently,we canconsidera principal connection form, whichisa g-valuedone-
from satisfying the two conditions
r∗ω =Ad(g−1)ω, and ω(σ(A)| )=A for any A∈g,p∈P.
g p
There is a one-to-one correspondence between these two structures, in the sense
that kerω is a principal Ehresmann connection when ω is a principal connection
form, and for any principal Ehresmann connection E, we can define a principal
connection form by formula
ω(v)=A for v ∈T P if v−σ(A)| ∈E .
p p p
Related to a choice of Ehresmann connection E we also have horizontal lifts,
since the mapping τ | :E →T P is a linear isometry. For a vectorv ∈T M,
∗ Ep p τ(p) m
we define the horizontal lift h v of v at p ∈ τ−1(m) as the unique element in E
p p
which is projected to v by τ .
∗
4 E.GRONG
Let us considera particularprincipalbundle overa manifold M. Fortwo vector
spaces V and V, let GL(V,V) be the space of all linear isomorphisms from V to
V. For any m ∈ M, we say that a frame at m is a choice of basis f ,...,f for
1 n
T M. Equivalbently, we canbconsider a frame as a linear map f ∈ GL(Rn,T M).
m m
Tbhe correspondence between the two point of views, is given by
f(0,...,0,1,0,...,0)=f .
j
1inthejthcoordinate
We write F (M) = GL(Rn,|T M) fo{rz the spa}ce of frames in T M. There is a
m m m
natural action of GL(n) on these spaces by composition on the right. Using this
action, we define the frame bundle τ : F(M) → M as the principal GL(n)-bundle
with fiber over m being F (M).
m
Similarly, if M is an oriented Riemannian manifold, we can define the oriented
e
orthonormal frame bundle τ : F(M) → M as the principal SO(n)-bundle whose
fiber is F (M) := SO(Rn,T M). Here, Rn is furnished with the standard orien-
m m
tation and the Eucliean metric.
Let ∇ be an affine connection on M, seen as an operator on vector fields
(X,Y) 7→ ∇ Y. Then we can associate a principal Ehresmann connection E
X ∇
onτ :F(M)→ M to∇,bydefiningE tobethedistributiononF(M)consisting
∇
of tangent vectors of smooth curves f(t) such that the vector fields f (t),...,f (t)
1 n
are all parallel along m(t) := τ(f(t)). If M is Riemannian and oriented, and if ∇
e
is compatible with a metric, then E can be defined as a distribution on F(M)
∇
instead, since both positive oerientation and orthonormality are preserved under
parallel transport.
We now go to the concrete case where ∇ is the Levi-Civita connection. Define
the tautological one-from θ =(θ ,...,θ ) on F(M), as the Rn-valued one-form
1 n
θ | =τ∗(♭f ), τ :F(M)→M,
j f j
where ♭ : TM → T∗M is the isomorphism induced by the Riemannian metric.
In other words, if v ∈ T F(M), then θ (v) = hf ,τ vi. Denote the so(n)-valued
f j j ∗
principal connection form corresponding to the Levi-Civita connection by ω. The
formulas for the differentials of θ and ω are given by the well-known Cartan equa-
tions. We express them in notations, that will be helpful for later purposes.
Let R be the Riemann curvature tensor, defined by
R(Y ,Y ,Y ,Y )=hR(Y ,Y )Y ,Y i,where R(Y ,Y )=∇ ∇ −∇ ∇ −∇ .
1 2 3 4 1 2 3 4 1 2 Y1 Y2 Y2 Y1 [Y1,Y2]
Furthermore, use it to define the curvature form Ω = (Ω ), as the so(n)-valued
ij
two-form
(2) Ω (v ,v )=R(τ v ,τ v ,f ,f ), v ,v ∈T F(M).
ij 1 2 ∗ 1 ∗ 2 j i 1 2 f
Then the following relations hold
dθ + n ω ∧θ = 0,
j i=1 ji i
(3)
Pn
dω + ω ∧ω = Ω ,
ij k=1 ik kj ij
where ωij are the matrix entriesPof ω. These equations are going to be important
in order to understand the connections between geometry and the control system
of rolling manifolds.
CONTROLLABILITY OF ROLLING IN HIGHER DIM. 5
Remark 1. The Cartan equations can also be defined on the frame bundle corre-
sponding to a general affine connection. See, e.g., [14, 20] for details.
Fromnowon,wewilladopttheconventionthatwheneverwementionRn,itwill
always come furnished with the standard orientation and the Euclidean metric.
2.3. Therollingdistributionandthecorrespondingsub-Riemannianstruc-
ture. The tangent vectors of all possible rollings form an n-dimensional distribu-
tion D on Q. We will call this distribution D the rolling distribution. A curve q(t)
is a rolling, if and only if, it is horizontal with respect to D, i.e. it is absolutely
continuous and q˙(t)∈D for almost any t.
q(t)
We present the following localdescriptionof D (see [8] for more details). Let us
writetheprojectionofQtoM×M,asπ :Q→M×M.Givenanysufficientlysmall
neighborhood U on M, let e be a local section of the oriented orthogonal frame
bundle F(M)with domainU. Wce write this localsecctionas(e,U). Let(e,U)be a
similarlocalsectionofF(M). We canuse theselocalsectionsto trivializethe fiber
bundle Q over U ×U, with the map b b
c
q ∈SO(TmM,TmbM)7→(m,m,(qij))∈U ×U ×SO(n), qij :=hej,qeji.
b
vTehcetnortfiheelddsistributioncD|π−1(U×bUb), in the abovbe coordinates, is spannbed by the
(4) e :=e +qe + e ,∇ e − qe ,∇ qe Wℓ .
j j j α ej β α qej β αβ
1≤α≤β≤n
X (cid:0)(cid:10) (cid:11) (cid:10) (cid:11)(cid:1)
where j =1,...,n. Here, ej is seen as a vector field on Q|U×Ub =π−1(U ×U) and
qe stands for the vector field q 7→q(e | ). The vector fields Wℓ are defined by
j j π(q) αβ
b
n
∂ ∂
(5) Wℓ = q −q .
αβ sα∂q sβ∂q
s=1(cid:18) sβ sα(cid:19)
X
The symbol ℓ here is not a parameter; it simply stands for “left” (an explanation
of this will follow in Remark 2).
Let us consider the optimal control problem finding of a rolling q(t) connecting
two configurations q and q , such that the curve m(t) = π(q(t)) has minimal
0 1
length. We do this by introducing a metric h·,·i on D, defined by
(6) hv ,v i=hπ v ,π v i, v ,v ∈D .
1 2 ∗ 1 ∗ 2 1 2 q
With respect to this metric, the e in (4) are a local orthonormalbasis. Also, from
j
the definition of D, we have hv ,v i=hπ v ,π v i. Hence, minimizing the length
1 2 ∗ 1 ∗ 2
ofm(t) is equivalentto minimizing the lengthofm(t)=π(q(t)) (this is canalsobe
seen from the no-slip condition).
b b
Definition 2. A triple (Q,D,h·,·i), where Q ibs a conbnected manifold, D is a
distribution on Q and h·,·i is a metric on D, is called a sub-Riemannian manifold.
Thesub-Riemanniandistancefunctiond(q ,q )betweentwopointsisdefinedas
0 1
the infimum of the length of all curves which are horizontal to D. We can view
rollings between configuration from q to q along a curve of minimal length as a
0 1
sub-Riemannian length minimizer in Q.
For the distance function to be finite, we need that every pair of configurations
q andq can be connected by a curve horizontalto D. In other words,we need to
0 1
6 E.GRONG
determine when rolling without twisting or slipping is a controllable system. Our
goalisto givesufficientconditions forthis tohold,interms ofgeometricinvariants
onM andM. Wewillleavethequestionoffindingoptimalcurvesforlaterresearch.
Remark 2. The vector fields Wℓ in (5) can be consideredas a “locally left invari-
c αβ
ant”basisofkerπ . Itwillbepracticaltoalsointroducea“locallyrightinvariant”
∗
analogue. Relative to two chosen local sections (e,U) and (e,U), define
n
∂ ∂
(7) Wαrβ = qβs∂q −qαs∂q , qij :=hbei,bqeji.
s=1(cid:18) αs βs(cid:19)
X
Notice that Wr = n q q Wℓ. b
αβ l,s αl βs ls
2.4. ControllabilitPy and brackets. Given an initial configuration q ∈Q, write
0
O for all points in Q that are reachable by a rolling starting from q . This will
q0 0
be the orbit of D at q , which coincides with the reachable set of D, since D is
0
a distribution (see, e.g., [2, 15] for details). The Orbit Theorem [9, 22] tells us
that O is a connected, immersed submanifold of Q, but also that the size can be
q0
approximated by the brackets of D. Define the C∞(Q)-module Lie D as the limit
of the process
D1 =Γ(D), Dk+1 =Dk+[D,Dk].
Γ(D) denotes the sections of D. Let Dk and Lie D be the subspaces of T Q
q q q
obtained by evaluating respectively Dk and LieD at q. Then, for any q ∈O ,
q0
(8) Lie D ⊆T O .
q q q0
In particular, it follows from (8), that if D is bracket generating at q, i.e., if
Lie D = T Q, then O is an open submanifold of Q, and we say that we have
q q q0
local controllability at q . If O =Q for one (and hence all) q ∈Q, the system is
0 q0 0
called completely controllable.
The least amount of control happens when O is n-dimensional submanifold.
q0
Asaconsequenceofthe OrbittheoremandFrobeniustheorem,thishappensifand
only if D| is involutive, that is, if Lie D =D , for every q ∈O .
Oq0 q q q0
The focus of this paper will be to provide results of controllability, by investi-
gating when the distribution D will be bracket generating at a given point q.
Remark 3. When D is notbracketgeneratingat q, Lie D will in generalonly give
q
us a lower bound for the size of O . However, if LieD is locally finitely generated
q
as a C∞(Q)-module, i.e., has a finite basis of vector field when restricted to a
sufficiently small neighborhood, then the equality holds in (8).
Remark 4. We will use the notation introduced here for distributions in general,
not just for the rolling distribution.
2.5. Relationship between frame bundles and rolling. Consider the linear
Lie algebraso(n). For integers α andβ between 1 and n and notequal, let w be
αβ
thematrixwith1atentryαβ, -1atentryβα, andzeroatallotherentries. Clearly
w = −w . Define w = 0. The commutator bracket between these matrices
αβ βα αα
are given by
(9) [w ,w ]=δ w +δ w −δ w −δ w ,
αβ κλ β,κ αλ α,λ βκ α,κ βλ β,λ ακ
The collection of all w with α<β form a basis for so(n).
αβ
CONTROLLABILITY OF ROLLING IN HIGHER DIM. 7
ConsidertheprincipalbundleF(M)→M. Weintroducethefollowingnotation.
Write σ for the vector fields on F(M) corresponding to w in the sense of (1).
αβ αβ
LetE betheprincipalEhresmannconnectioncorrespondingtotheaffineLevi-Civita
connection on M, with corresponding principal connection form ω. As before, we
denote the tautological one-form by θ. By using horizontal lifts with respect to E,
we can define vector fields X on F(M) by
j
(10) X | :=h f ,
j f f j
whichsatisfyθ (X )=δ .Hence, thetangentbundleofF(M)istrivial,sinceitis
i j i,j
spanned by {X }n and {σ } . Finally, we define σ ,X ,ω and θ similarly
j j=1 αβ α<β αβ j
on F(M).
The configurationspaceQmaybe identifiedwithF(Mb)×Fb(Mb)quotbientedout
by thecdiagonal action of SO(n). Let ̟ denote the principal SO(n)-bundle
c
̟ :F(M)×F(M)→F(M)×F(M)/SO(n)∼=Q.
Then̟(f,f)=q,iff =q◦f.Byviewingω,ω,θ andθ asformsonF(M)×F(M),
c c
we are able to obtain the following result.
b b b b c
Theorem 1. Let
D=kerω∩kerω∩ker(θ−θ),
and let D be the rolling distribution. Then ̟ D = D, and the map is a linear
∗
isomorphism on every fiber. b b
Proof. From its definition, it is clear that {X +X }n is a basis for D.
j j j=1
Choose any pair of local section (e,U) and (e,U) of respectively F(M) and
b
F(M). GiveF(M)×F(M)| b localcoordinatesbyassociatingthepairofframes
U×U
(f,f) to the element b b
c c
(11)b m,m, fij , fij ∈U ×U ×SO(n)×SO(n),
n (cid:16) (cid:0) (cid:1) (cid:0) n(cid:1)(cid:17)
if fj = i=1fijei|m anbd fij = bi=1fijei|mb hbolds.
Relative to this trivialization,we candefine left andrightvectorfieldoneachof
P P
the SO(n)-factors. On thebfirst, defineb b
n n
∂ ∂ ∂ ∂
(12) Ψℓ = f −f , Ψr = f −f .
αβ sα∂f sβ∂f αβ βs∂f αs∂f
s=1(cid:18) sβ sα(cid:19) s=1(cid:18) αs βs(cid:19)
X X
Notice that Ψr = n f f Ψℓ . Remark also that Ψℓ is just the restriction
αβ l,s=1 αl βs ls αβ
of σαβ to F(M)×FP(M)|U×Ub, while Ψrαβ depends on the chosen local section e.
Define Ψℓ and Ψr analogously on the second SO(n)-factor.
αβ αβ
c
Restricted to F(M)×F(M)| b and in the local coordinates (11), the vector
U×U
b b
fields X and X can be written as
j j
c
n n
(13) X = b f e − Γα Ψr , X = f e − Γα Ψr ,
j sj s sβ αβ j sj s sβ αβ
Xs=1 (cid:16) αX<β (cid:17) Xs=1 (cid:16) αX<β (cid:17)
b b b b b
where Γαiβ =heα,∇eieβi and Γαiβ =heα,∇ebieβi.
b b b
8 E.GRONG
WenowturntotheimageofT(F(M)×F(M)|U×Ub)under̟∗. Defineqij,ej,Wαℓβ
andWαrβ onQ|U×Ub asinSection2.3. Remark2allowsustorewriteej onthe form
c
n
e =e +qe + Γα Wℓ − q Γα Wr
j j j jβ αβ sj sβ αβ
αX<β(cid:16) Xs=1 (cid:17)
Locally the mapping ̟ can be described as b
n
̟ : m,m,(f ),(f ) 7→(m,m,(q )), q = f f .
ij ij ij ij is js
(cid:16) (cid:17) Xs=1
and the action on thebtangentbvectors is givben by formulas b
e 7→ e
i i
e 7→ e
i i
̟ : .
∗ Ψr 7→ −Wℓ
αβ αβ
Ψrb 7→ Wb r
αβ αβ
From this it is clear that ̟ D = D and that the map is injective of each fiber,
∗
b
since
n
e =̟ f X +X .
j ∗ js s s
Xs=1 (cid:16) (cid:17)
(cid:3)
b
From the form of the distribution D, we obtain the following interpretation of
rolling without twisting or slipping.
Corollary 1. Let q(t) be a rolling without twisting or slipping. Let (f(t),f(t)) be
any lifting of q(t) to a curve in F(M)×F(M) that is horizontal to D, and define
m(t) and m(t) as the respective projections to M and M. Then (f(t),f(t))bsatisfy
the following c
(i) (Nbo twist condition) Every vector field fj(t) iscparallel along mb(t). Every
vector field f (t) is parallel along m(t).
j
(ii) (No slip condition) For almost every t,
b f−1(t) m˙ (t) =bf−1(t) m˙ (t) .
Furthermore, if (f(t),f(t)) is a(cid:0)ny ab(cid:1)solutely co(cid:0)ntinu(cid:1)ous curve in F(M)×F(M),
b b
satisfying (i) and (ii), then ̟(f(t),f(t)) is a rolling without twisting or slipping.
b c
Proof. (i) follows from the definition of the principalEhresmann connections kerω
b
and kerω. (ii) is exactly the requirement for a curve to be in ker(θ−θ). (cid:3)
The main advantage of the viewpoint given in Theorem1, is that it will help us
b b
to compute Lie D.
q
Corollary 2. ̟ Dk =Dk for any q ∈Q,k∈N.
∗ q q
Proof. We only need to show this locally. Introduce local coordinates as in the
proof of Theorem 1 and let e be as in (4). Then
j
n n
[e ,e ]=̟ f X +X , f X +X ,
i j ∗ is s s js s s
" #
Xs=1 (cid:16) (cid:17) Xs=1 (cid:16) (cid:17)
b b
CONTROLLABILITY OF ROLLING IN HIGHER DIM. 9
and since n f X +X is a local basis for D, it follows that D2 = ̟ D2
s=1 is s s ∗
locally. The rest fol(cid:16)lows by in(cid:17)duction. (cid:3)
P
b
Since [X ,X ]=0, computations of bracketsofD, andhence of bracketsD, can
i j
bereducedtomostlycomputingbracketsofsectionsinthe Ehresmannconnections
correspondingbto the Levi-Civita connections of the manifolds involved.
2.6. Remark on previous descriptionsof rollingusingframe bundles. The
description of rolling given in Theorem 1, looks very similar to the definition of
rolling without twisting or slipping found in [4] for dimension 2. Here, the descrip-
tion of a rolling is in terms of the distribution D :=D⊕ker̟ , which can also be
∗
described as
e
(14) D=ker(ω−ω)∩ker(θ−θ).
In [4], a rolling of a pair of 2-dimensional manifold is defined as a curve in Q, that
is horizontal to ̟ D, whereeD is definedbin terms ofb(14).
∗
k
We could have used D for our computation, since clearly ̟ D =Dk also, and
∗ q q
e e
D is bracket generating at a point q ∈ Q if and only if D is bracket generating at
any(and hence every)(fe,f)∈̟−1(q). However,since [Dk,ker̟e ]⊂Dk+ker̟ ,
∗ ∗
the additional brackets are not of any interest. e
The definition of rollingb or “rolling without slipping” in probability theory is
defined on frame bundles [10], and is equivalent to considering curves in F(M)×
F(M)thatarehorizontaltoDforthespecialcasewhenM isRnwiththeEuclidean
metric and standard orientation.
c c
3. Brackets of D
3.1. Tensors on M and associated vector fields. We introduce a generaltype
of vector fields associated to tensors. By giving a general formula for the brackets
of these, we essentially determine all the brackets of D. Let T be a tensor on M.
We will only consider tensors defined on vectors. To any tensor k-tensor on M, we
can associate the functions
E (T): F(M) 7→ R
i1,...ik , 1≤i ≤n.
f 7→ T(f ,...,f ) s
i1 ik
Ifk =2+l,andT isantisymmetricinthefirsttwoarguments,wecandefinevector
fields on F(M) by
W (T)= E (T)σ ,
i1,...,il α,β,i1,...,il αβ
1≤α<β≤n
X
If T is a tensor on M, we define E (T) and W (T) similarly as respec-
i1,...ik i1,...,il
tively functions and vector fields on F(M).
b c b b
Lemma 1. Let X be defined as in (10).
k
c
(a) For any l-tensor T,
X E (T)=E (∇T).
k i1,...,il i1,...,il,k
(b) For any 2+l-tensor T, that is antisymmetric in the first two arguments
n
[X ,W (T)]=W (∇T)− E (T)X .
k i1,...,il i1,...,il,k s,k,i1,...,il s
s=1
X
10 E.GRONG
Proof. (a) We use a local representation of X . Consider the formula given in
j
(13), and write n f e as just f . Since Ψℓ is just the representation
s=1 sj s j αβ
of σ in local coordinates, we can write X as
αβ j
P
X =f − hf ,∇ f iσ .
j j α fj β αβ
1≤α≤n
X
In this notation, remark first that σ f =δ f −δ f , which gives us
αβ i β,i α α,i β
n
1
hf ,∇ f iσ f = hf ,∇ f i(δ f −δ f )
α fk β αβ ij 2 α fk β β,ij α α,ij β
1≤α<β≤n α,β=1
X X
n
= f ,∇ f f =∇ f .
α fk ij α fk ij
α=1
X(cid:10) (cid:11)
Then the result follows from realizing that
X E (T)=f T(f ,...,f )− hf ,∇ f iσ T(f ,...,f )
k i1,...,il k i1 il α fk β αβ i1 il
1≤α<β≤n
X
=f T(f ,...,f )−T(∇ f ,...,f )−T(f ,∇ f ,...,f )
k i1 il fk i1 il i1 fk i2 il
−···−T(f ,f ,...,∇ f )
i1 i2 fk il
=∇T(f ,...,f ,f ).
il il k
(b) The brackets [σ ,σ ] are, by definition, given by the same relations as
αβ κλ
described in (9). We will continue by the following computations.
[X ,W (T)]
k i1,...,il
n n
1 1
= X (E (T))σ − hf ,∇ f iE (T)[σ ,σ ]
2 k κ,λ,i1,...,il κλ 4 α fk β κ,λ,i1,...,il αβ κλ
κ,λ=1 α,β,κ,λ=1
X X
n n
1 1
− E (T)σ f + E (T)σ (hf ,∇ f i)σ
2 κ,λ,i1,...,il κλ k 4 κ,λ,i1,...,il κλ α fk β αβ
κ,λ=1 α,β,κ,λ=1
X X
n
1
= E (∇T)σ
2 κ,λ,i1,...,il κλ
κ,λ=1
X
n
1
+ (T(∇ f ,f ,f ,...,f )+T(f ,∇ f ,f ,...,f ))σ
2 fk κ λ i1 il κ fk λ i1 il κλ
κ,λ=1
X
n n
1
− E (T)f + E (T)(hf ,∇ f i)σ
s,k,i1,...,il s 2 s,α,i1,...,il s fk β αβ
s=1 α,β,s=1
X X
n
1
+ E (T)(hf ,∇ f i)σ
2 s,k,i1,...,il α fs β αβ
α,β,s=1
X
n
1
+ E (T)σ (hf ,∇ f i)σ
2 s,β,i1,...,il wκλ α fk s αβ
α,β,s=1
X
n
= W (∇T)− E (T)X
i1,...,il,k s,k,i1,...,il s
s=1
X
(cid:3)
