A vertical Liouville subfoliation on the cotangent bundle of a Cartan space and some related structures

0.21 MB·English

by  Cristian Ida

A vertical Liouville subfoliation on the cotangent bundle of a
Cartan space and some related structures
Cristian Ida and Adelina Manea
3
1
0 Abstract
2
In this paper we study some problems related to a vertical Liouville distribution
n
onthe cotangentbundle ofa Cartanspace. We study the cohomologyofc–indicatrix
a
J cotangentbundle and the existence ofsome linear conections ofVr˘anceanuand Vais-
man type on Cartan spaces related to foliated structures. Also we identify a certain
2
∗
2 (n,2n−1)–codimensionalsubfoliation(FV,FC∗) onT M0 givenby verticalfoliation
FV andthe line foliationFC∗ spannedby the verticalLiouville-Hamiltonvectorfield
] C∗ and we give a triplet of basic connections adapted to this subfoliation.
G
D
.
h 2010 Mathematics Subject Classification: 53B40, 53C12, 53C60.
t
a Key Words: Cartan space, foliation, basic connection.
m
[
1 Introduction and preliminaries

1
v
6 1.1 Introduction
1
3 The study of interrelations between the geometry of foliations on the tangent manifold of
5 a Finsler space and the geometry of the Finsler space itself was initiated and intensively
.
1 studied by Bejancu and Farran [9]. The main idea of their paper is to emphasize the
0
importanceofsomefoliations whichexistonthetangentbundleofaFinslerspace(M,F),
3
1 in studying the differential geometry of (M,F) itself. On the other hand in a very recent
: paper [3] a similar study of some natural foliations in cotangent bundle T∗M of a Cartan
v
i space (M,K) is given. It is shown that geometry of these foliations is closely related to
X
the geometry of the Cartan space (M,K) itself. This approach is used to obtain new
r
a characterizations of Cartan spaces with negative constant curvature.
The aim of our paper is to continue the study of these foliations on the cotangent
bundle of a Cartan space (M,K) from more points of view as: the cohomology of c–
indicatrix cotangent bundle, the study of some linear conections of Vra˘nceanu and Vais-
mantypeonCartanspacesrelatedtofoliatedstructures,thestudyofacertain(n,2n−1)–
∗
codimensional subfoliation (FV,FC∗) on T M0 given by vertical foliation FV and the line
∗
foliation FC∗ spannedby thevertical Liouville-Hamilton vector field C andthe existence
1
of a triplet basic connections adapted to this subfoliation. The notions are introduced by
analogy with similar notions on the tangent manifold of a Finsler space.
The paper is organized as follows: In the preliminary subsection we briefly recall
some basic facts on the geometry of a Cartan space (M,K) and we present the almost
∗ ∗ ∗
Ka¨hlerian model (T M ,G,J) of the cotangent manifold T M = T M −{zero section}
0 0
together with the Riemannian metric G given by Sasaki type lift of the fundamental
metric tensor gij = 1 ∂2K2 and with the natural almost complex structure J. In the
2∂pi∂pj
second section, following an argument inspired from [7], we define a vertical Liouville
∗
distribution LC∗ on T M0 as the complementary orthogonal distribution in the vertical
∗
distribution V(T M ) to the line distribution spanned by the vertical Liouville-Hamilton
0
vector field C∗ = pi∂∂pi and we prove that the distribution LC∗ is an integrable one. Also,
we find geometric properties of both leaves of vertical Liouville distribution LC∗ and the
∗
vertical distribution V(T M0). We notice that the vertical Liouville distribution LC∗ will
bean important tool in studyof futureproblems in this paper. In the third section, using
∗
the vertical Liouville-Hamilton vector field C and the natural almost complex structure
∗ ∗
J on T M , we give an adapted basis in T(T M ). Next we prove that the c–indicatrix
0 0
cotangent bundle I(M,K)(c) of (M,K) is a CR–submanifold of the almost Ka¨hlerian
∗
manifold(T M ,G,J)andwestudysomecomologicalpropertiesofI(M,K)(c)inrelation
0
with classical cohomology of CR-submanifolds, [13]. In the fourth section, following
some ideas from [8], we investigate the existence of some linear connections of Vra˘nceanu
and Vaisman type on a Cartan space, related with the vertical and Liouville-Hamilton
foliations on it. In the last section, following [15], we briefly recall the notion of a (q ,q )-
1 2
codimensional subfoliation on a manifold and we identify a (n,2n − 1)-codimensional
∗
subfoliation (FV,FC∗) on the cotangent manifold T M0 of a Cartan space (M,K), where
FV is the vertical foliation and FC∗ is the line foliation spanned by the vertical Liouville-
∗
Hamilton vector field C . Firstly we make a general approach about basic connections
on the normal bundles related to this subfoliation and next a triple of adapted basic
connections with respect to this subfoliation is given. A similar study on the tangent
manifold of a Finsler space (M,F) is given in [18].
1.2 Preliminaries and notations
In this subsection we briefly recall some basic facts from the geometry of a Cartan space
(M,K). For more see [19, 20, 21, 23].
∞ ∗ ∗
LetM bean n-dimensional C manifold and π : T M → M its cotangent bundle. If
(xi), i = 1,...,n are local coordinates in a local chart U on M, then (xi,p ), i= 1,...,n
i
will betaken as local coordinates in the local chart (π∗)−1(U) on T∗M with the momenta
(p )provided by p = p dxi wherep ∈ T∗M, x ∈ M and {dxi}is thenaturalbasis of T∗M.
i i x x
The indices i,j,k,... will run from 1 to n and the Einstein convention on summation will
be used. A change of coordinates on U ∩U 6= ∅ given by (xi,p )7→ (xi,p ) has the form
i i
∂xj ∂xi
xi = xi(xi), pi = ∂xeipj, rank ∂xj = n, e e (1.1)
(cid:18) (cid:19)
e
e e e
e2
where ∂xj is the inverse of the Jacobian matrix ∂xei .
∂xei ∂xj
Let(cid:16)usco(cid:17)nsider ∂ , ∂ ,i= 1,...,nthenatu(cid:16)ralba(cid:17)sisinT (T∗M)and{dxi,dp }
∂xi ∂pi (x,p) i
the dual basis of it.nThe chaonge of coordinates (1.1) produces the following changes:
∂ ∂xj ∂ ∂p ∂ ∂ ∂xi ∂
j
= + , = (1.2)
∂xi ∂xi ∂xj ∂xi ∂p ∂p ∂xj ∂p
j i j
e e e
and
∂xi e ∂xje e ∂2xj
dxi = dxj, dp = dp + p dxk. (1.3)
∂xj i ∂xi j ∂xi∂xk j
The kernel V (T∗M) of theedifferential dπ∗ : T (T∗M) → T M is called the vertical
(x,p) e e (x,p) x
∗ ∗
subspace of T (T M) and the mapping (xe,p) 7→ V e (Te M) is a regular distribution
(x,p) (x,p)
∗
onT M calledthevertical distribution. Thisisintegrableanddefinestheverticalfoliation
F with the leaves characterized by xk = constant and it is locally spanned by ∂ .
V ∂pi
The vector field C∗ = p ∂ is called the vertical Liouville-Hamilton vector fienld anod
i∂pi
ω = p dxi s called the Liouville 1-form on T∗M. Then Ω = dω = dp ∧ dxi is the
i i
∗
canonical symplectic structure on T M. For an easer handling of the geometrical objects
∗
on T M it is usual to consider a supplementary distribution to the vertical distribution,
∗
(x,p) 7→ H (T M), called the horizontal distribution and to report all geometrical
(x,p)
∗
objects on T M to the decomposition
∗ ∗ ∗
T (T M)= H (T M)⊕V (T M). (1.4)
(x,p) (x,p) (x,p)
The horizontal distribution is taken as being locally spanned by the local vector fields
δ ∂ ∂
:= +N (x,p) . (1.5)
δxi ∂xi ij ∂p
j
∗
Thehorizontaldistributioniscalledalsoanonlinear connection onT M andthefunctions
N are called the local coefficients of this nonlinear connection. It is important to note
ij
∗
that any regular Hamiltonian on T M determines a nonlinear connection whose local
coefficients verify N = N . The basis δ , ∂ is adapted to the decomposition (1.4).
ij ji δxi ∂pi
The dual of it is {dxi,δp := dp −N dxnj}. o
i i ji
∗
According to [23] a Cartan structure on M is a function K :T M → [0,∞) which has
the following properties:
∞ ∗ ∗
i) K is C on T M := T M −{zero section};
0
ii) K(x,λp) = λK(x,p) for all λ > 0;
iii) then×nmatrix (gij), wheregij = 1 ∂2K2 , ispositivedefiniteatall pointsofT∗M .
2∂pi∂pj 0
We notice that in fact K(x,p) > 0, whenever p 6= 0.
3
Definition 1.1. The pair (M,K) is called a Cartan space.
Let us put
1∂K2 1 ∂3K2
pi = , Cijk = − . (1.6)
2 ∂p 4∂p ∂p ∂p
i i j k
The properties of K imply that
pi = gijp , p =g pj, K2 = gijp p = p pi, Cijkp = Cikjp = Ckijp = 0, (1.7)
j i ij i j i k k k
where (g ) is the inverse matrix of (gji).
ij
One considers the formal Christoffel symbols by
1 ∂g ∂g ∂g
γi (x,p) := gis js + sk − jk , (1.8)
jk 2 ∂xk ∂xj ∂xs
(cid:18) (cid:19)
and the contractions γ0 (x,p) := γi (x,p)p , γ0 := γi p pk. Then the functions
jk jk i j0 jk i
1 ∂g
N (x,p) = γ0(x,p)− γ0 (x,p) ij(x,p), (1.9)
ij ij 2 h0 ∂p
h
∗
define a nonlinear connection on T M. This nonlinear connection was discovered by R.
Miron[21]andiscalledthecanonical nonlinear connection of(M,K). Wealsonoticethat
the coefficients from (1.9) satisfies N (x,p) = N (x,p) and are positively homogeneous
ij ji
of degree 1 in momenta. For details we refer to Ch. 6 in [23]. Thus a decomposition (1.4)
holds. From now on we shall use only the canonical nonlinear connection given by (1.9).
To a Cartan space (M,K) we can associate some important geometrical object fields
on the manifold T∗M . Namely, the N-lift G of the fundamental tensor gij, the almost
0
complexstructureJ, etc. IfN isthecanonicalnonlinearof(M,K), thus(G,J)determine
an almost Hermitian structure, which is derived only from the fundamental function K
of the Cartan space.
The N-lift of the fundamental tensor field gij of the space (M,K) is defined by
G = g dxi ⊗dxj +gijδp ⊗δp (1.10)
ij i j
∗
and there is a natural almost complex structure J on T M which is locally given by
0
∂ δ δ ∂ ∂ δ
J =−g ⊗dxj +gij ⊗δp , J = −g , J = gij . (1.11)
ij∂p δxi j δxi ij∂p ∂p δxj
i j i
(cid:18) (cid:19) (cid:18) (cid:19)
∗
Thus, according to [23], (T M ,G,J) has a model of almost Ka¨hlerian manifold with the
0
fundamental form Ω given by Ω(X,Y) = G(JX,Y) and locally expressed by
Ω = δp ∧dxi = dp ∧dxi. (1.12)
i i
4
2 A vertical Liouville distribution on T∗M
0
∗
Followinganargumentinspiredfrom[7]wedefineaverticalLiouvilledistributiononT M
0
∗
asthecomplementaryorthogonaldistributioninV(T M )tothelinedistributionspanned
0
∗
by the vertical Liouville-Hamilton vector field C and we prove that this distribution is
an integrable one.
By (1.7) we have
G(C∗,C∗)= K2. (2.1)
∗
By means of G and C , we define the vertical one form ζ by
1
∗ ∗
ζ(X) = G(X,C ) ∀X ∈ Γ(V(T M )). (2.2)
0
K
∗ ∗ ∗
Denote by {C } the line vector bundleover T M spanned by C and definethe Liouville
0
∗ ∗
distribution as the complementary orthogonal distribution LC∗ to {C } in V(T M0) with
respect to G. Hence, LC∗ is defined by ζ, that is we have
∗
Γ(LC∗)= {X ∈ Γ(V(T M0)) : ζ(X) = 0}. (2.3)
Thus, any vertica vector field X = X ∂ can be expressed as follows:
i∂pi
1
∗
X = PX + ζ(X)C , (2.4)
K
∗
where P is the projection morphism of V(T M0) on LC∗. By direct calculations, we
obtain
∗
G(X,PY) = G(PX,PY) = G(X,Y)−ζ(X)ζ(Y), ∀X,Y ∈ Γ(V(T M )). (2.5)
0
Then the local components of ζ and P with respect to the basis {δp } and δp ⊗ ∂ ,
i i ∂pi
respectively, are given by n o
pi ζip
ζi = , Pi = δi − j, (2.6)
K j j K
where δi are the components of the Kronecker delta.
j
Theorem 2.1. The Liouville distribution LC∗ is integrable and it defines a foliation on
∗
T M0 denoted by FC∗.
∗ ∗
Proof. Let X,Y ∈ Γ(LC∗). As V(T M0) is an integrable distribution on T M0, it is
∗
sufficient to prove that [X,Y] has no component with respect to C .
It is easy to see that a vertical vector field X = Xi∂∂pi is in Γ(LC∗) if and only if
gijX p = 0. (2.7)
i j
5
Differentiate (2.7) with respect to p we get
k
∂gij ∂X
X p +gikX +gij ip = 0, ∀k = 1,...,n (2.8)
i j i j
∂p ∂p
k k
∂gij
and taking into account the relation p = 0 (see (1.7)), one gets
∂p j
k
∂X
gikX +gijp i = 0, ∀k = 1,...,n. (2.9)
i j
∂p
k
Then, by direct calculations using (2.9), we have
∂Y ∂X
G([X,Y],C∗) = gijp iX − iY
j k k
∂p ∂p
(cid:18) k k (cid:19)
= −gikY X +gikX Y
i k i k
= 0
which completes the proof.
Based on the above results, we may say that the geometry of the leaves of F should
V
∗
be derived from the geometry of the leaves of FC∗ and of integral curves of C . In order
to get this interplay, we consider a leaf F of F given locally by xi = ai, i = 1,...,n,
V V
where the ai’s are constants. Then, gij(a,p) are the components of a Riemannian metric
G on F . Denote by ∇ the Levi-Civita connection on F with respect to G and
FV V FV
jk jk
consider the Christoffel symbols C of ∇. Then we obtain the usual formula for C ,
i i
namely
1 ∂gsk
Cjk(a,p) = − g (a,p) (a,p) = g (a,p)Csjk(a,p), (2.10)
i 2 is ∂p is
j
where g (a,p) are the entries of the inverse matrix of the n×n matrix gsi(a,p) . Con-
is
tracting (2.10) by p , we deduce that
j
(cid:0) (cid:1)
jk
C (a,p)p = 0. (2.11)
i j
By straightforward calculations using (2.11), (2.5) and (2.6), we obtain the covariant
∗
derivatives of C , ζ and P in the following lemma:
Lemma 2.1. Let (M,K) be a Cartan space. Then, on any leaf F of F , we have
V V
1 1
∗
∇ C = PX, (2.12)
X
K K
(cid:18) (cid:19)
1
(∇ ζ)Y = G(PX,PY), (2.13)
X
K
and
1
∗
(∇ P)Y = − [G(PX,PY)C +Kζ(Y)PX] (2.14)
X
K2
for any X,Y ∈ Γ(TF ).
V
6
Proof. Indeed, if we take X = X ∂ , Y = Y ∂ ∈ Γ(TF ) the relation (2.12) it follows
i∂pi j∂pj V
by:
1 X ∂K ∂ X p ∂
∇ C∗ = i δiK −p + i jCij
X K K2 j j∂p ∂p K k ∂p
(cid:18) (cid:19) (cid:18) i(cid:19) j k
X p ζi ∂
= i δi − j +0
K j K ∂p
j
(cid:18) (cid:19)
1
= PX.
K
For the relation (2.13) we have
∂ζj X Y pjpi
(∇ ζ)Y = X(ζ(Y))−ζ(∇ Y)= X Y = i j gij −
X X i j
∂p K K2
i
(cid:18) (cid:19)
and
1 X Y pjpi
G(PX,PY)= i j gij − .
K K K2
(cid:18) (cid:19)
The relation (2.14) it folows using (2.12) and (2.13). Indeed, we have
(∇ P)Y = ∇ (PY)−P(∇ Y)
X X X
1 1
∗ ∗
= ∇ Y −ζ(Y) C −∇ Y + ζ(∇ Y)C
X X X
K K
(cid:18) (cid:19)
1 1 1
∗ ∗
= −X(ζ(Y)) C −ζ(Y) PX + ζ(∇ Y)C
X
K K K
1 1
∗
= −[X(ζ(Y))−ζ(∇ Y)] C −ζ(Y) PX
X
K K
1
∗
= − [G(PX,PY)C +Kζ(Y)PX].
K2
Now, in similar manner with [7], we obtain:
Theorem 2.2. Let (M,K) be an n-dimensional Cartan space and FV, FC∗ and γ be a
leaf of FV, a leaf of FC∗ that lies in FV, and an integral curve of K1C∗, respectively. Then
we have the following assertions:
i) γ is a geodesic of F with respect to ∇.
V
ii) FC∗ is totally umbilical immersed in FV.
∗
iii) FC∗ lies in the indicatrix Ia(M,K) ={p ∈ TaM0 : K(a,p) =1} of (M,K) and has
constant mean curvature equal to −1.
7
Theorem 2.3. Let (M,K) be an n-dimensional Cartan space and F be a leaf of the
V
vertical foliation F . Then the sectional curvature of any nondegenerate plane section on
V
F containing the vertical Liouville-Hamilton vector field is equal to zero.
V
Corollary 2.1. Let (M,K) be an n-dimensional Cartan space. Then there exist no leaves
of F which are positively or negatively curved.
V
3 An adapted basis in T(T∗M ) and cohomology of the c-
0
indicatrix cotangent bundle
∗
In this section, using the vertical Liouville-Hamilton vector field C and the natural
∗ ∗
almost complex structure J on T M , we give an adapted basis in T(T M ). Next we
0 0
prove that the c-indicatrix cotangent bundle I(M,K)(c) of (M,K) is a CR-submanifold
∗
of thealmost Ka¨hlerian manifold (T M ,G,J) and we studysome comological properties
0
of I(M,K)(c) in relation with classical cohomology of CR-submanifolds, [13].
3.1 An adapted basis in T(T∗M )
0
Aswealreadysaw,thevertical bundleV(T∗M )islocally spannedby ∂ , i= 1,...,n
0 ∂pi
and it admits decomposition n o
∗ ∗
V(T M0) = LC∗ ⊕{C }. (3.1)
∗
In the sequel, following [17], we give another basis on V(T M0), adapted to FC∗, and
next we extend this basis to an adapted basis in T(T∗M0), by the same reasons as in [2],
[11].
We consider the following vertical vector fields:
∂ ∂
= −tjC∗, j = 1,...,n, (3.2)
∂pj ∂pj
where functions tj are defined by the conditions
∂
∗
G ,C = 0,∀j = 1,...,n. (3.3)
∂p
(cid:18) j (cid:19)
The above conditions become
∂ ∂
G ,p −tjG(C∗,C∗) =0
i
∂p ∂p
j i
(cid:18) (cid:19)
so, taking into account also (1.10) and (2.1), we obtain the local expression of functions
tj in a local chart (U,(xi,p )):
i
p gji pj 1 ∂K ζj
tj = i = = = , ∀j = 1,...,n. (3.4)
K2 K2 K ∂p K
j
8
If (U,(xi,p )) is another local chart on T∗M , in U ∩U 6= ∅, then we have:
i 0
p gki 1 ∂xj ∂xk ∂xi ∂xk
e e e tk = Ki 2 = K2 ∂xipj ∂xl ∂xhgelh = ∂xltl.
So, we obtain the following echeanging rule for thee veector fieldse(3.2):
e
e
∂ ∂xi ∂
= , ∀i = 1,...,n. (3.5)
∂p ∂xk ∂p
i k
e
By a straightforward calculation, using (1.7), it results:
e
Proposition 3.1. The functions {tj}, j = 1,...,n defined by (3.4) are satisfying:
∂ ∂ti gij
p ti = 1, p = 0, = −2titj + , C∗tj = −tj, ∀i,j = 1,...,n. (3.6)
i i
∂pi ∂pj K2
Proposition 3.2. The following relations hold:
∂ ∂ ∂ ∂ ∂ ∂
, = ti −tj , ,C∗ = , (3.7)
∂p ∂p ∂p ∂p ∂p ∂p
(cid:20) i j(cid:21) j i (cid:20) i (cid:21) i
for all i,j =1,...,n.
By conditions (3.3), the vector fields ∂ ,..., ∂ are orthogonal to C∗, so they be-
∂p1 ∂pn
longto the (n−1)-dimensionaldistributinon LC∗. Itresoults that they are linear dependent
and, from the first relation of (3.6), we have
∂ p ∂
a
= − , (3.8)
∂pn pn∂pa
since the local coordinate p is nonzero everywhere.
n
We also have
Proposition 3.3. The system ∂ ,..., ∂ ,C∗ of vertical vector fields is a locally
∂p1 ∂pn−1
∗
adapted basis to the vertical Liounville foliation FC∗,oon V(T M0).
Thus, we can denote
∂ ∂
=Ea , a = 1,...,n−1,
∂pa i ∂pi
where rankEa = n−1 and Eap gij = 0.
i i j
∗
Now, using the natural almost complex structure J on T(T M ), the new local vector
0
∗
field frame in T(T M ) is
0
a ∗ ∂ ∗
X ,ξ , ,C , (3.9)
∂p
(cid:26) a (cid:27)
where
ξ∗ = J(C∗)= pi δ , Xa = J ∂ = Eagij δ .
δxi ∂p i δxj
(cid:18) a(cid:19)
9
Remark 3.1. In some future calculations we shall replace the local basis ∂ , a =
∂pa
1,...,n−1 by the system ∂ , i= 1,...,n taking into account relation (3n.8) foor some
∂pi
easier calculations. n o
3.2 Cohomology of c-indicatrix cotangent bundle
∗
Since the vertical Liouville-Hamilton vector field C is orthogonal to the level hypersur-
faces of the fundamental function K, the vector fields Xa,ξ∗, ∂ are tangent to these
∂pa
hypersurfaces in TM0, so they generate the distributinon {C∗}⊥ wohich is the orthogonal
∗ ∗
complement of {C } in T(T M0). The vertical indicatrix distribution LC∗ is locally gen-
erated by ∂ , a = 1,...,n−1, and the vertical foliation has the structural bundle
∂pa
locally gennerateod by ∂ ,C∗ , a = 1,...,n−1.
∂pa
∗
Also, if we considner the lione distribution {ξ } spanned by the horizontal Liouville-
∗ ∗
Hamilton vector field ξ and its complement in H(T M0) denoted by Lξ∗, then it is easy
a
to see that Lξ∗ is locally spanned by the vector fields X , a = 1,...,n −1, and we
have the decomposition
∗ ⊥ ∗ (cid:8) (cid:9)
{C } ={ξ }⊕Lξ∗ ⊕LC∗. (3.10)
For any c > 0, we consider now the c-indicatrix cotangent bundle over M, given by
∗
I(M,K)(c) = I (M,K)(c) , I (M,K)(c) = {p ∈ T M : K(x,p)= c}.
x x x 0
x∈M
[
According to [5, 8], if (N,g,J) is an (almost) Ka¨hler manifold, whereg is the Rieman-
nian metric and J is the (almost) complex structure on N, then N is a CR-submanifold
of N if N admits two compeleemeentary orthogonal distributions D and D⊥esuch that
e e
i) D is J-invariant, i.e., J(D) ⊂ D;
e
⊥ ⊥ ⊥
ii) D is J-anti-invariant, i.e., J D ⊂ (TN) .
e e
⊥
D is called maximal complex (holom(cid:0)orph(cid:1)ic) distribution of N and D is called totally real
e e
distribution of N.
We have
Proposition 3.4. Let i : I(M,K)(c) ֒→ T∗M be the imersion of I(M,K)(c) in T∗M .
0 0
∗
Then I(M,K)(c) is a CR-submanifold of T M with holomorphic distribution given by
0
⊥ ∗
D =Lξ∗ ⊕LC∗ and the totally real distribution given by D = {ξ }.
∗ ⊥ ∗
Proof. We have that {C } = {ξ } ⊕ Lξ∗ ⊕ LC∗ is the tangent bundle of I(M,K)(c).
∗
Taking into account the behaviour of the almost complex structure J of (T M ,G) we
0
have
⊥
∗ ∗ ∗ ⊥
J(Lξ∗ ⊕LC∗) ⊂ LC∗ ⊕Lξ∗ =Lξ∗ ⊕LC∗, J({ξ }) ⊂ {C } = {C }
(cid:16) (cid:17)
10