Symmetries of the Rolling Model

0.3 MB·English

by  Yacine Chitour

SYMMETRIES OF THE ROLLING MODEL
YACINE CHITOUR
MAURICIO GODOY MOLINA
PETRI KOKKONEN
Abstract. Inthe presentpaper,we study the infinitesimalsymmetries ofthe modelof
two Riemannian manifolds (M,g) and (Mˆ,gˆ) rolling without twisting or slipping. We
3 show that, under certain genericity hypotheses, the natural bundle projection from the
1 state space Q of the rolling model onto M is a principal bundle if and only if Mˆ has
0 constantsectionalcurvature. Additionally, we provethat when M and Mˆ havedifferent
2
constant sectional curvatures and dimension n≥3, the rolling distribution is never flat,

n
contrarytothetwodimensionalsituationofrollingtwospheresofradiiintheproportion
a
1: 3, which is a well-known system satisfying É. Cartan’s flatness condition.
J

1
1
Contents

]
G 1. Introduction 1
D 2. Notations and terminology 3
. 3. Symmetries of the Rolling Distribution 5
h
t 3.1. General Symmetries 5
a
m 3.2. Inner Symmetries 7
4. Principal Bundle Structure 9
[

5. Constant Curvature and Flatness 16
1
v 5.1. Nilpotent Approximation of the rolling distribution DR 16
9 5.2. Non-Flatness of the Rolling Distribution 17
7
References 21
5
2
.
1
0
3 1. Introduction
1
: A very old and difficult problem in differential geometry is the study of symmetries of
v
i distributions. A seminal contribution is the celebrated paper by É. Cartan [9] in which, in
X
modern terms, he studied distributions of rank two on a manifold of dimension five and,
r
a moreprecisely, theassociatedequivalenceproblem. Recallthatarankl vectordistribution
D on an n-dimensional manifold M or (l,n)-distribution (where l < n) is, by definition,
an l-dimensional subbundle of the tangent bundle TM, i.e., a smooth assignment q 7→ D|
q
defined on M where D| is an l-dimensional subspace of the tangent space T M. Two
q q
vector distributions D and D are said to be equivalent, if there exists a diffeomorphism
1 2
F : M → M such that F D | = D | for every q ∈ M. Local equivalence of two
∗ 1 q 2 F(q)
2000 Mathematics Subject Classification. 53C07,53A45, 53A55, 53C17.
Keywordsandphrases. rollingmodel,non-holonomicdistributions,symmetriesofdistributions,nilpo-
tent approximation.
The work of the first author is supported by the ANR project GCM, program “Blanche”, (project
number NT09_504490) and the DIGITEO-Région Ile-de-France project CONGEO. The work of the
second author is partially supported by the ERC Starting Grant 2009 GeCoMethods. The work of the
thirdauthorissupportedbyFinnishAcademyofScienceandLetters,KAUTEFoundationandl’Institut
français de Finlande.
1
2 Y.CHITOUR,M. GODOY M., P. KOKKONEN
distributions is defined analogously. The equivalence problem consists in constructing
invariants of distributions with respect to the equivalence relation defined above. The
main contribution of Cartan in [9] was the introduction of the “reduction-prolongation”
procedure for building invariants and the characterization for (2,5)-distributions via a
functional invariant (Cartan’s tensor) which vanishes precisely when the distribution is
flat, that is, when it is locally equivalent to the (unique) graded nilpotent Lie algebra
of step 3 with growth vector (2,3,5). In this case, the Lie algebra of symmetries of
the distribution corresponds to the 14-dimensional Lie algebra g and this situation is
2
maximal, that is, in the non-flat case the dimension of the Lie algebra of symmetries is
strictly less than 14. In fact, Cartan gave a geometric description of the flat G -structure
2
as the differential system that describes space curves of constant torsion 2 or 1/2 in the
standard unit 3-sphere (see Section 53 in Paragraph XI in [9].) It has been a folkloric
fact among the control theory community that the flat situation described above occurs
in the problem of two 2-dimensional spheres rolling one against the other without slipping
or spinning, assuming that the ratio of their radii is 1: 3, see [6] for some historical notes
and a thorough attempt of an explanation for this ratio. In fact, whenever the ratio of
their radii is different from 1: 3, the Lie algebra of symmetries becomes so(3)×so(3), thus
dropping its dimension to 6. A complete answer to this strange phenomenon as well as a
geometric reason for Cartan’s tensor was finally given in two remarkable papers [32, 33]
(cf. also [4]), where ageometric methodforconstruction of functionalinvariants ofgeneric
germs of (2,n)-distributionfor arbitrary n ≥ 5 is developed. It has been recently observed
in [26] that the Lie algebra of symmetries of a system of rolling surfaces can be g in the
2
case of non-constant Gaussian curvature.
As for the rolling model, its two dimensional version has been intensively studied by
the control community for quite a while, see for example [1, 2, 8, 12, 20, 23, 25]. Indeed,
the mechanical problem of a sphere rolling can be traced back to the 19th century, in
two seminal papers by S. A. Chaplygin [10, 11], recently translated. It was not until the
publication of the book [31] that the higher dimensional problem became better known to
the control theorists, though it had been introduced several years before in [27]. A major
disadvantage of Sharpe’s definition was the use of submanifolds of Euclidean space, with a
strongdependence ontheirconcreterealizations, nevertheless itstillyieldsomeinteresting
results, for example [21]. Trying to deal with this inconvenience was the starting point
of the studies [13, 18] in which a coordinate-free model for the rolling dynamics was
introduced, where the restrictions of no-twist and no-slip were encoded in terms of the so-
called rolling distribution D . Recently non-trivial extensions to manifolds with different
R
dimensions [15], semi-Riemannian manifolds [24], and Cartan geometries [17] have been
presented. Besides geometric issues that are associated to the intrinsic definition for the
rolling model (e.g., the question of existence of such dynamics [19]), one can address the
problemoffindingconditionsonthepairofmanifoldsM andMˆ sothattherollingmodelis
completely controllable, i.e., if Q denotes the state space of the model of two Riemannian
manifolds M and Mˆ rolling without slipping or spinning, one says that the associated
rolling model is completely controllable if, given arbitrary q ,q ∈ Q, one can roll Mˆ on
0 1
M without slipping or spinning from the initial position q to the final position q . That
0 1
typical issue of control theory is usually solved by evaluating, at every point q ∈ Q, the
Lie algebra generated by the distribution D . It turns out that this approach is almost
R
impossible to carry over for the general n-dimensional rolling model (cf. [15]) except for
n = 3. On the other hand, when one of the manifolds has constant sectional curvature,
the distribution D is a principal bundle connection for the canonical projection map
R
π : Q → M and that key feature enables one to successfully address the controllability
Q,M
SYMMETRIES OF THE ROLLING MODEL 3
issue “without Lie brackets computations” because the latter reduces to the determination
of a certain holonomy group associated to an appropriate linear connection [14, 16].
In this paper, we study the Lie algebra of symmetries of the rolling distribution D
R
over the state space Q. We obtain as consequences of this analysis the answers of two
problems arising from the issues above mentioned. The first of these says that, under
ˆ
certain genericity assumptions on M and M, the distribution D is a principal bundle
R
ˆ
connection for π if and only if M has constant sectional curvature. Our second main
Q,M
result refers to the question of flatness of the rolling distribution for the case of spaces
of constant curvature. In this context, a regular distribution of rank k on a manifold of
dimension n is said to be flat if it is locally equivalent to its nilpotent approximation. We
prove that, as long as the curvatures of M and Mˆ are different, the rolling distribution is
never flat in dimensions ≥ 3, contrary to what happens for the 1 : 3 phenomenon in two
dimensions described previously.
The paper is organized as follows. In Section 2 we introduce the basic terminology con-
cerning the higher dimensional rolling problem that will be used throughout the paper.
Section 3 starts the study of the symmetries of the rolling model, addressing later the
restricted case of inner symmetries, that is, symmetries induced by vector fields in the
rolling distribution. Section 4 presents the first of our main results mentioned above. A
key tool in this section is the set Sym (D ) of symmetries that lie in the kernel of the
0 R
differential (π ) . In fact, the aforementioned result follows from a complete charac-
Q,M ∗
terization of Sym (D ) as the symmetries induced by the Killing vector fields of Mˆ. In
0 R
Section 5 we present the second main result mentioned in the previous paragraph. We
begin by studying the nilpotent approximation of the rolling distribution, from which we
can deduce its non-flatness if the dimension is greater than 3.
2. Notations and terminology
If D is a smooth constant rank distribution on M, we write VF for the set of X ∈
D
VF(M) such that X| ∈ D| for all x ∈ M. If N is a submanifold of M, then we say that
x x
D is tangent to N, if D| ⊂ T| N for all x ∈ N.
x x
Definition 2.1 Let D be a smooth distribution of constant rank on M. Then X ∈ VF(M)
is called an infinitesimal symmetry of D if [X,VF ] ⊂ VF . The vector space of all the
D D
infinitesimal symmetries of D is denoted by Sym(D).
An infinitesimal symmetry X ∈ Sym(D) is called an inner infinitesimal symmetry if X ∈
VF . The set of all inner infinitesimal symmetries is denoted by InnSym(D).
D
Remark 2.2 The set InnSym(D) is a vector subspace of VF , given by InnSym(D) =
D
Sym(D)∩VF .
D
Definition 2.3 For a distribution D on M, we define the D-orbit of x ∈ M, denoted by
O (x) as the set of all points in M that can be connected to x by an absolutely continuous
D
curve with velocity almost everywhere contained in D.
Remark 2.4 By Nagano-Sussman’s theorem, see [3], the orbit O (x) is an immersed
D
submanifold of M.
As an abbreviation, we usually refer to infinitesimal symmetries (resp. infinitesimal
inner symmetries) of D simply as symmetries (resp. inner symmetries) of D.
Forthesakeofcompleteness, werecallsomeoftheterminologyforthemodeloftwoRie-
mannian manifolds, one rolling against the other without twisting or slipping, introduced
4 Y.CHITOUR,M. GODOY M., P. KOKKONEN
in [13, 14]. For a more detailed discussion, we refer the reader to [13]. Let (M,g) and
(Mˆ,gˆ) be two oriented connected Riemannian manifolds. The state space Q = Q(M,Mˆ)
of the rolling model is the manifold
Q = Q(M,Mˆ) = A : T| M → T| Mˆ | x ∈ M, xˆ ∈ Mˆ,
x xˆ
A linear isometry, det(A) > 0 .
(cid:8)
Given a point q = (x,xˆ;A) ∈ Q, a vector X = (X,Xˆ) ∈ T| (M × Mˆ(cid:9)) and any
(x,xˆ)
smooth curve t 7→ γ(t) = (γ(t),γˆ(t)) in M ×Mˆ defined on an open interval I ∋ 0 such
that γ(0) = (x,xˆ), and γ˙(0) = X, the no-spinning lift of X at q if defined by
d
(1) L (X)| := Pt(γˆ)◦A◦P0(γ) ∈ T| Q,
NS q dt 0 0 t q
where Pb(γ) (resp. Pb(γˆ)) denotes th(cid:12)e(cid:0)parallel transport (cid:1)map along γ from γ(a) to γ(b)
a a (cid:12)
(resp. along γˆ from γˆ(a) to γˆ(b)). It is readily seen that the definition of L (X)| does
NS q
not depend on the choice of the smooth curve γ as long as it satisfies γ(0) = (x,xˆ) and
γ˙(0) = X.
Similarly, we define the rolling lift of X ∈ T| M to q = (x,xˆ;A) ∈ Q as
x
(2) L (X)| := L (X,AX)| .
R q NS q
Notice that L also defines a natural map L : VF(M) → VF(Q) such that L (X) :=
R R R
q 7→ L (X)| .
R q
(cid:0) (cid:1)
Definition 2.5 The rolling distribution D on Q is the n-dimensional smooth distribution
R
defined, for (x,xˆ;A) ∈ Q, by
(3) D | = L (T| M)| .
R (x,xˆ;A) R x (x,xˆ;A)
An absolutely continuous curve t 7→ q(t) = (γ(t),γˆ(t);A(t)) in Q that is almost ev-
erywhere tangent to D is called a rolling curve. This condition can be rewritten as
R
q˙(t) = L (γ˙(t))| , a.e. t. It was shown in [13, 18] that such curves are exactly those
R q(t)
ˆ
that describe the dynamics of rolling M against M without twisting or spinning.
As it can be noticed already, there are several fiber and vector bundles that will play an
important role in the main results of this article. As an abuse of notation, we will often
denote the bundles only by its projection maps. The fiber bundles π : Q → M ×Mˆ and
Q
π : Q → M are the projections π (x,xˆ;A) = (x,xˆ) and π (x,xˆ;A) = x. Observe
Q,M Q Q,M
that π is a fiber subbundle of the vector bundle π : T∗M ⊗TMˆ → M ×Mˆ. For
Q T∗M⊗TMˆ
anymanifoldN, the mapπ : TkN → N denotes thevector bundle of(k,m)-tensorson
TmkN m
N, and the special case (k,m) = (1,0) for the tangent bundle is simply denoted by π .
TN
Given two fiber bundles ξ and η over the same manifold M, we denote by C∞(ξ,η) the
space of smooth bundle maps from ξ to η. Assuming that ξ and η are vector bundles, for
x ∈ M and f ∈ C∞(ξ,η), one defines the vertical derivative ν(w)| (f) of f at u ∈ ξ−1(x)
u
in the direction of w ∈ ξ−1(x) as
d
ν(w)| (f) = f(u+tw),
u
dt
(cid:12)t=0
(cid:12)
which can be identified with an element of the(cid:12)fiber η−1(x). This notion then immediately
(cid:12)
extends to the situation where there is a (possibly non vector) fiber subbundle λ of ξ, and
f ∈ C∞(λ,η) if, moreover, ν(w)| is tangent to the total space of λ.
u
We still need to extend the notion of the vector L (X)| , q ∈ Q, to an operator acting
R q
on tensor valued maps. Suppose N is a submanifold of Q such that D is tangent to N,
R
i.e., D | ⊂ T| N for all q ∈ N. Suppose that F : N → Tk(M × Mˆ) is smooth and
R q q n
SYMMETRIES OF THE ROLLING MODEL 5
pr ◦π ◦F = π where pr : M ×Mˆ → M; (x,xˆ) 7→ x. For q = (x,xˆ;A) ∈ N
1 Tk(M×Mˆ) Q,M 1
m
and X ∈ T| M one defines L (X)| F as the element of Tk(M ×Mˆ) given by
x R q m
L (X)| F := ∇ F(q(t)),
R q (X,AX)
where q(t) is any smooth curve in Q such that q˙(0) = L (X)| (as vectors) and ∇ is the
R q
connection induced by the Levi-Civita connections ∇ and ∇ˆ on the bundle π .
Tk(M×Mˆ)
m
Note that above F(q(t)) is a tensor field along the curve (γ(t),γˆ(t)) := π (q(t)), whose
Q
˙
initial velocity is (γ˙(0),γˆ(0)) = (X,AX), so that the expression ∇ F(q(t)) makes
(X,AX)
sense. Moreover, this expression is independent of the choice of the smooth curve q(t) as
long as q˙(0) = L (X)| (e.g. q(t) could be taken as a rolling curve).
R q
Foraninnerproductspace(V,h·,·i),denotebyso(V)theLiealgebraofskew-symmetric
endomorphisms of V with respect to h·,·i. For the rest of the paper, given x ∈ M, we
identify the vector space 2T| M with so(T| M) as follows: If X,Y,Z ∈ T| M, then
x x x
(X ∧Y)Z = g(Z,Y)X −g(Z,X)Y.
V
To conclude this section, we present a convenient result that allows us to compute Lie
brackets of vector fields on Q. Its proof follows after a careful calculation, and the details
can be found in [13].
Proposition 2.6 Let ∇ and ∇ˆ be the Levi-Civita connections of M and Mˆ respectively.
Let T = (T,Tˆ),S = (S,Sˆ) ∈ C∞(π ,π ) and U,V ∈ C∞(π ,π ) be such
Q T(M×Mˆ) Q T∗M⊗TMˆ
that U(q),V(q) ∈ Aso(T| M) for all q = (x,xˆ;A) ∈ Q. Then if
x
X| := L (T(q))| +ν(U(q))| , Y| := L (S(q))| +ν(V(q))| ,
q NS q q q NS q q
one has
[X,Y]| =L (X| S −Y| T)| +ν(X| V −Y| U)|
q NS q q q q q q
+ν(AR(T(q),S(q))−Rˆ(Tˆ(q),Sˆ(q))A)| +R∇(T(q),S(q)),
q
where R and Rˆ are the Riemannian curvatures of (M,g) and (Mˆ,gˆ) respectively, and R∇ is
the curvature of the connection ∇.
Remark 2.7 The above proposition holds true if one replaces everywhere Q by any sub-
manifold N ⊂ Q such that D is tangent to it, replacing the condition that U(q),V(q) ∈
R
Aso(T| M) for q ∈ Q by the assumption that X,Y be tangent to N.
x
3. Symmetries of the Rolling Distribution
3.1. General Symmetries. We begin our study of the symmetries of the rolling model
byfinding a condition, equivalent tothe oneinDefinition 2.1, fora vector field S ∈ VF(Q)
to be a symmetry.
Proposition3.1 LetZ ∈ C∞(π ,π ),Zˆ ∈ C∞(π ,π ),U ∈ C∞(π ,π )
Q,M TM Q,Mˆ TMˆ Q T∗M⊗TMˆ
be such that U(q) ∈ Aso(T| M) for all q = (x,xˆ;A) ∈ Q. Defining
x
S| := L (Z(q),Zˆ(q))| +ν(U(q))| ,
q NS q q
then S ∈ Sym(D ) if and only if for all q = (x,xˆ;A) ∈ Q and all X ∈ T| M, one has
R x
(4) U(q)X = −AL (X)| Z +L (X)| Zˆ
R q R q
(5) L (X)| U = −AR(X ∧Z(q))+Rˆ(AX ∧Zˆ(q))A
R q
6 Y.CHITOUR,M. GODOY M., P. KOKKONEN
Proof. If X ∈ VF(M), then
[S,L (X)]| =L (∇ X)| −L (L (X)| Z,L (X)| Zˆ)|
R q R Z(q) q NS R q R q q
+ν(AR(Z(q),X)−Rˆ(Zˆ(q),AX)A)|
q
+L (0,U(q)X)| −ν(L (X)| U)|
NS q R q q
=L (∇ X −L (X)| Z)|
R Z(q) R q q
(6) +L (0,U(q)X +AL (X)| Z −L (X)| Zˆ(q))|
NS R q R q q
(7) +ν(−L (X)| U +AR(Z(q),X)−Rˆ(Zˆ(q),AX)A)| .
R q q
Note that S ∈ Sym(D ) if and only if [S,L (X)] ∈ D , for all X ∈ VF(M). Hence,
R R R
S ∈ Sym(D ) if and only if the terms (6),(7) above vanish for every X ∈ VF(M). (cid:3)
R
We will often use the notation
S | := L (Z(q),Zˆ(q))| +ν(U(q))| .
(Z,Zˆ,U) q NS q q
In [14] a notion of curvature especially adapted to the rolling model was introduced. This
idea will play a fundamental role in the subsequent developments, so we briefly recall it
here for the sake of completeness. For q = (x,xˆ;A) ∈ Q, the rolling curvature is the linear
map
2
Rol : T| M → T∗| M ⊗T| Mˆ; Rol (X ∧Y) := AR(X,Y)−Rˆ(X,Y)A.
q x x xˆ q
For convenie^nce, we also define
2 2
Rol : T| M → T| M; Rol (X ∧Y) = R(X,Y)−A−1Rˆ(X,Y)A,
q x x q
i.e. Rol = A^Rol . The f^act that the values of Rol are in 2T| M instead of just
qf q f q x
T∗M ⊗TM, follows from well-known properties of the curvature tensors R,Rˆ.
V
As usual, forfa smooth distribution D, we denfote by D(k) the kth element in the
canonical flag of D, that is the kth step in the iterative definition
D(1) = D, D(k+1) = D(k) +[D(k),D].
Proposition 3.2 Let S ∈ Sym(D ). Then for all q = (x,xˆ;A) ∈ Q, X,Y ∈ T| M,
(Z,Zˆ,U) R x
Aν(Rol (X ∧Y))| Z = ν(Rol (X ∧Y))| Zˆ.
q q q q
Proof. Notice that if S ∈ Sym(D ), then S ∈ Sym(D(k)) for all k ∈ N. Note
(Z,Zˆ,U) R (Z,Zˆ,U) R
that ν(Rol(X ∧Y)) = q 7→ ν(Rol (X ∧Y))| belongs to D(2) and
q q R
[S ,ν(Rol(X ∧Y)(cid:0))] =−L ν(Rol (X ∧(cid:1)Y))| Z,ν(Rol (X ∧Y))| Zˆ
(Z,Zˆ,U) NS q q q q q
+ν L(cid:0) (Z(q),Zˆ(q))| Rol(X ∧Y)+ν(U(q))| (cid:1)R(cid:12)ol(X ∧Y)|
NS q q (cid:12) q q
−ν(cid:0)ν(Rol (X ∧Y))| U (cid:1)(cid:12)
q q
(cid:12)
=−L(cid:0)NS 0,−Aν(Rolq(X ∧(cid:1) Y))|qZ +ν(Rolq(X ∧Y))|qZˆ q
−LR(ν(cid:0)(Rolq(X ∧Y))|qZ)|q +ν(···)|q. (cid:1)(cid:12)
(cid:12)
SinceD(2) isspanned byvectorsoftheformL (X)andν(Rol(X∧Y))andsince S ∈
R R (Z,Zˆ,U)
Sym(D(2)), it follows that the L term above vanishes, i.e.
R NS
−Aν(Rol (X ∧Y))| Z +ν(Rol (X ∧Y))| Zˆ = 0. (cid:3)
q q q q
SYMMETRIES OF THE ROLLING MODEL 7
Remark 3.3 The above propositions holds true if one replaces everywhere Q by a subman-
ifold N ⊂ Q to which D is tangent, one replaces the set Sym(D ) by Sym(D | ), and
R R R N
if the condition that U(q) ∈ Aso(T| M) for q ∈ Q is replaced by the assumption that S
x
be tangent to N. The notation π (resp. π , π ) would mean in this context π |
N,M N,Mˆ N Q,M N
(resp. π | , π | ).
Q,Mˆ N Q N
3.2. Inner Symmetries. The aim of this subsection is to briefly study some basic prop-
erties or inner symmetries of D as well as of D | for q ∈ Q. In particular, we
R R ODR(q0) 0
will unveil a connection between the existence of inner symmetries of the type L (Z),
R
Z ∈ VF(M), and one of the manifolds having constant sectional curvature. We will begin
by characterizing the inner symmetries.
Proposition 3.4 The following properties hold.
(i) If S ∈ InnSym(D ), then for all q = (x,xˆ;A) ∈ Q,
(Z,Zˆ,U) R
Zˆ(q) = AZ(q), U(q) = 0,
Rol (X ∧Z(q)) = 0, ∀X ∈ T| M.
q x
(ii) If there exists Z ∈ C∞(π ,π ) such that Rol (X ∧ Z(q)) = 0 for all q =
Q,M TM q
(x,xˆ;A) ∈ Q and X ∈ T| M, then defining Zˆ(q) := AZ(q), we have that
x
S ∈ InnSym(D ).
(Z,Zˆ,0) R
Proof. (i) Since
S = L (Z(q))| +L (0,−AZ(q)+Zˆ(q))| +ν(U(q))| ,
(Z,Zˆ,U) R q NS q q
we see that if S ∈ InnSym(D ), then −AZ(q) + Zˆ(q) = 0 and U(q) = 0 for all
(Z,Zˆ,U) R
q ∈ Q. Then by (5),
0 = L (X)| U = −AR(X ∧Z(q))+Rˆ(X ∧AZ(q))A = −Rol (X ∧Z(q)).
R q q
(ii) Setting U(q) = 0, we see that
−AL (X)| Z +L (X)| Zˆ = −L (X)| (·)Z(·)−Zˆ(·) = 0 = U(q)X
R q R q R q
and (cid:0) (cid:1)
−AR(X ∧Z(q))+Rˆ(AX ∧Zˆ(q))A = −AR(X ∧Z(q))+Rˆ(AX ∧AZ(q))A
= −Rol (X ∧Z(q)) = 0 = L (X)| U.
q R q
Hence the vector field S = L (Z(·)) satisfies equations (4)-(5), in other words, we
(Z,Zˆ,0) R
have S ∈ InnSym(D ). (cid:3)
(Z,Zˆ,0) R
Remark 3.5 Notice that if S ∈ InnSym(D ), then fS ∈ InnSym(D ) for all f ∈ C∞(Q).
R R
Therefore, if InnSym(D ) is a non-trivial space, it has to be infinite dimensional as a vector
R
space over R.
Remark 3.6 As before, one can replace in the above proposition the space Q by any
of its submanifolds N such that D is tangent to N, if one also replaces InnSym(D ) by
R R
InnSym(D | ).
R N
Example 3.7 Suppose that (M,g),(Mˆ,gˆ) both have constant, equal curvature. Then any
vector field Y ∈ VF(M) gives rise to an inner symmetry. Indeed, defining Z(q) := Y| , for
x
8 Y.CHITOUR,M. GODOY M., P. KOKKONEN
q = (x,xˆ;A), one has Rol (X ∧Z(q)) = 0 for all X ∈ T| M. In other words, in this setting
q x
L (VF(M)) ⊂ InnSym(D ).
R R
Next we present the result announced at the beginning of this subsection. As a no-
tational remark, if X,Y ∈ T| M is an orthonormal pair of vector, then σ denotes
x (X,Y)
the sectional curvature of M at x with respect of the plane spanned by X and Y. For
ˆ
convenience, we use σˆ for the analogous concept on M.
Proposition 3.8 Suppose that there exists Z ∈ VF(M), Z 6= 0, such that L (Z) ∈
R
InnSym(D ). Then (Mˆ,gˆ) has constant curvature cˆ ∈ R and for every x ∈ M such that
R
Z| 6= 0, the sectional curvatures of (M,g) along all the planes containing Z| are equal to cˆ.
x x
Proof. Fix x ∈ M such that Z| 6= 0. Then for all q ∈ (π )−1(x ), say q = (x ,xˆ;A),
1 x1 Q,M 1 1
we have
0 = Rol (X ∧Z| ) = AR(X ∧Z| )−Rˆ(AX ∧AZ| )A, ∀X ∈ T| M,
q x1 x1 x1 x1
from which we get, whenever X ∈ T| M is a unit vector orthogonal to Z| ,
x1 x1
σ = σˆ .
(cid:16)X,Z|x1/kZ|x1kg(cid:17) (cid:16)AX,AZ|x1/kZ|x1kg(cid:17)
Let us thus fix a unit vector X ∈ T| M orthogonal to Z| . Given any xˆ ∈ Mˆ and
x1 x1
ˆ ˆ ˆ ˆ
orthonormal pair of vectors X,Y ∈ T| M, there exists a A ∈ Q| such that AX = X,
xˆ (x1,xˆ)
AZ| /kZ| k = Yˆ, and therefore,
x1 x1 g
σˆ = σ .
(Xˆ,Yˆ) (cid:16)X,Z|x1/kZ|x1kg(cid:17)
Since xˆ ∈ Mˆ and Xˆ,Yˆ ∈ T| Mˆ were an arbitrary point and an arbitrary orthonormal
xˆ
pair of vectors, it follows that the sectional curvatures of (Mˆ,gˆ) are all equal to cˆ :=
σ .
(X,Z|x1/kZ|x1kg)
Now if x ∈ M is such that Z| 6= 0, then again, for any unit vector X ∈ T| M
x x
orthogonal to Z| ,
x
σ = σˆ = cˆ,
(X,Z|x/kZ|xkg) (AX,AZ|x/kZ|xkg)
that is, the sectional curvatures of all the two dimensional planes of T| M that contain
x
Z| are equal to cˆ. (cid:3)
x
The following examples show that there do exist Riemannian manifolds, not both of
constant curvature, for which D in Q has non-trivial (even nowhere vanishing) inner
R
symmetries of the type as described by the previous proposition.
Example 3.9 Suppose that (M,g) is a Sasakian manifold of dimension n with a charac-
teristic unit vector field ξ (cf. [7]), and suppose that (Mˆ,gˆ) is the n-dimensional unit sphere.
We show that L (ξ) is an inner symmetry of D on Q.
R R
Indeed, we have for any X,Y ∈ T| M,
x
AR(X ∧ξ)Y =A(g(ξ,Y)X −g(X,Y)ξ) = gˆ(Aξ,AY)AX −gˆ(AX,AY)Aξ
=(AX ∧Aξ)AY = Rˆ(AX ∧Aξ)AY,
where the last equality follows from the fact that (Mˆ,gˆ) has constant curvature = 1. Thus
for all q = (x,xˆ;A) ∈ Q and X ∈ T| M,
x
Rol (X ∧ξ) = 0.
q
SYMMETRIES OF THE ROLLING MODEL 9
Setting Z(q) := ξ| , Zˆ(q) := AZ(q) and U(q) := 0 for all q = (x,xˆ;A) ∈ Q, it follows from
x
Proposition 3.4 that L (ξ) = S ∈ InnSym(D ).
R (Z,Zˆ,0) R
Example 3.10 Let K ∈ R and suppose that (Mˆ,gˆ) is a space of constant curvature K.
Takeas (M,g) a warped product (I×N,dr2+f(r)2h) where (N,h) is a Riemannian manifold
of dimension n−1, I is a real interval, and f : I → R is a strictly positive smooth function
that satisfies
f′′ = −Kf.
Denote by ∂ the canonical coordinate vector field on I. We claim that defining for q =
r
(x,xˆ;A) ∈ Q, Z(q) = ∂ | if x = (s,y), Zˆ(q) = A∂ | and U(q) = 0, then S ∈
r s r s (Z,Zˆ,0)
InnSym(D ).
R
Indeed, if Y,Z ∈ T| N, we have (see [28], Chapter 7, Prop. 42)
y
f′′
AR(Y ∧∂ )∂ = − AY = KAY = K(AY ∧A∂ )A∂ ,
r r r r
f
AR(Y ∧∂ )Z = f′′fh(Y,Z)∂ = −Kf2h(Y,Z)∂ = −Kg(Y,Z)∂ = K(AY ∧A∂ )AZ,
r r r r r
which proves that Rol (X∧∂ ) = 0 for all q = (x,xˆ;A) ∈ Q and X ∈ T| M. Thus the claim
q r x
follows again from Proposition 3.4.
4. Principal Bundle Structure
In this section, we state and prove one of the main results of the present paper: we
give a necessary condition for the fiber bundle π : Q → M to be a principal bundle.
Q,M
This characterization follows from a fundamental relation between the existence of certain
symmetries and the group of Riemannian isometries. We use freely some classical results
in Riemannian geometry, which can be found for example in [22, 29].
We introduce the convenient notation S := S . Moreover, we define
(Zˆ,U) (0,Zˆ,U)
Sym (D ) := {S ∈ Sym(D ) | (π ) S = 0},
0 R R Q,M ∗
i.e., S ∈ Sym (D ) if and only if Z = 0 and S ∈ Sym(D ).
(Z,Zˆ,U) 0 R (0,Zˆ,U) R
Observe thatthereisanequivalent characterization oftheelements inSym (D ), which
0 R
follows easily from Proposition 3.1. We simply state it as a fact.
Proposition 4.1 S ∈ Sym (D ) if and only if
(Zˆ,U) 0 R
(8) U(q)X = L (X)| Zˆ
R q
(9) L (X)| U = Rˆ(AX ∧Zˆ(q))A,
R q
for all q = (x,xˆ;A) ∈ Q and X ∈ T| M.
x
The following theorem gives a precise bound for the dimension of the vector space
Sym (D )| of restrictions of elements of Sym (D ) onto an orbit O (q ). Note
0 R ODR(q0) 0 R DR 0
the contrast with the space of inner symmetries since, as observed in Remark 3.5, if it is
non trivial, then it is infinite dimensional.
Theorem 4.2 Let q = (x ,xˆ ;A ) ∈ Q. Then the linear space Sym (D )| has
0 0 0 0 0 R ODR(q0)
dimension at most n(n+1).
2
10 Y.CHITOUR,M. GODOY M., P. KOKKONEN
Proof. We claim that the map
Sym (D )| → T| Mˆ ×so(T| Mˆ); S | 7→ (Zˆ(q ),A−1U(q ))
0 R ODR(q0) xˆ0 xˆ0 (Zˆ,U) ODR(q0) 0 0 0
is injective. This will then imply
n(n+1)
dim Sym (D )| ≤ dim T| Mˆ ×so(T| Mˆ) = ,
0 R ODR(q0) xˆ0 xˆ0 2
which is what w(cid:0)e set out to prove.(cid:1) (cid:0) (cid:1)
Indeed, let q = (x ,xˆ ;A ) ∈ O (q ) and suppose that γ : [0,1] → M is a geodesic
1 1 1 1 DR 0
such that γ(0) = x and γ˙(0) = X ∈ T| M. Write q(t) = (γ(t),γˆ(t);A(t)) for the
1 x1
unique rolling curve in Q starting at q and satisfying π (q(t)) = γ(t). Also write
1 Q,M
ˆ ˆ
Y(t) := Z(q(t)). Notice that
L (γ˙(t))| Zˆ = ∇ (Zˆ(q(·)))−ν(∇ A(·))| Zˆ = ∇ˆ Yˆ(·).
R q(t) (γ˙(t),γˆ˙(t)) (γ˙(t),γˆ˙(t)) q(t) γˆ˙(t)
Then by (8)-(9) one has
Rˆ(γˆ˙(t)∧Yˆ(t))γˆ˙(t) =Rˆ(A(t)γ˙(t)∧Zˆ(q(t)))A(t)γ˙(t) = L (γ˙(t))| U(·) γ˙(t)
R q(t)
=LR(γ˙(t))|q(t) U(·)γ˙(·) −U(q(t))∇(cid:0) γ˙γ˙ (cid:1)
=LR(γ˙(t))|q(t)(cid:0)LR(γ˙(·)(cid:1))|q(·)Zˆ(q(·))
=∇ˆγˆ(t)∇ˆγˆ(·)Yˆ(·(cid:0)), (cid:1)
where in the second to last equality we used that γ is a geodesic.
Therefore, Yˆ is a Jacobi field along the geodesic γˆ(t) = γˆ (t) and hence is uniquely
A1X
determined by the initial values Yˆ(0) = Zˆ(q ), ∇ˆ Yˆ = L (X)| Zˆ = U(q )X. More-
1 A1X R q1 1
over, Yˆ uniquely determines U(q(t)) for all t since
L (γ˙(t))| U = ∇ U(q(·))−ν(∇ A(·))| U = ∇ U(q(·)),
R q(t) (γ˙(t),γˆ˙(t)) (γ˙(t),γˆ˙(t)) q(t) (γ˙(t),γˆ˙(t))
and hence
t
U(q(t)) = Pt(γˆ) U(q )+ P0(γˆ)Rˆ(γˆ˙(s)∧Yˆ(s))Ps(γˆ)dsA P0(γ).
0 1 s 0 1 t
(cid:16) Z0 (cid:17)
This implies that if Zˆ(q ) = Yˆ(q ) and U(q ) = V(q ), then for all X ∈ T| M and all
1 1 1 1 x1
t one has S | = S | , where q(t) = q (γ ,q ) and γ is the geodesic with
(Zˆ,U) q(t) (Yˆ,V) q(t) DR X 1 X
γ (0) = x , γ˙ (0) = X.
X 1 X
To finish the proof, suppose (Zˆ(q ),A−1U(q )) = (Yˆ(q ),A−1V(q )). Given a point
0 0 0 0 0 0
q = (x,xˆ;A) ∈ O (q ), there exists geodesics γ : [0,1] → M, i = 1,...,N, on M such
DR 0 i
that γ (0) = x , γ (1) = x, γ (1) = γ (0), i = 2,...,N and q (γ ...γ .γ ,q )(1) =
1 0 N i−1 i DR N 2 1 0
q. Since Zˆ(q ) = Yˆ(q ), U(q ) = V(q ), we have S | = S | ,
0 0 0 0 (Zˆ,Uˆ) qDR(γ1,q0)(t) (Yˆ,Vˆ) qDR(γ1,q0)(t)
t ∈ [0,1], by what we just proved above. In particular, Zˆ(q ) = Yˆ(q ), U(q ) = V(q ),
1 1 1 1
where q := q (γ ...γ .γ ,q ) = q (γ ,q ). Inductively we obtain Zˆ(q ) = Yˆ(q ),
i DR i 2 1 0 DR i i−1 N N
U(q ) = V(q ), where q = q, and so S | = S | . Since q ∈ O (q ) was
N N N (Zˆ,U) q (Yˆ,V) q DR 0
arbitrary, we have proven the claim. (cid:3)
Remark 4.3 The proof of the previous theorem shows in fact that for any q ∈ Q the space
0
Sym (D | ) of all S ∈ Sym(D | ) such that (π | ) S = 0 has at most
0 R ODR(q0) R ODR(q0) Q,M ODR(q0) ∗
dimension n(n+1)/2.
The theorem above has a very natural consequence in the case of a completely control-
lable rolling dynamics. Recall that the rolling distribution D is said to be completely
R
controllable if O (q ) = Q, q ∈ Q.
DR 0 0