On the $k$-Semispray of Nonlinear Connections in $k$-Tangent Bundle Geometry

0.14 MB·

by  Florian Munteanu

k
On the -Semispray of Nonlinear Connections in
k
6 -Tangent Bundle Geometry
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2 Florian Munteanu

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a Department of Applied Mathematics,
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University of Craiova, Romania
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[email protected]

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Abstract
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In this paper we present a method by which is obtained a sequence
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h of k-semisprays and two sequences of nonlinear connections on the k-
t tangent bundle TkM, starting from a given one. Interesting particular
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m cases appear for Lagrange and Finsler spaces of order k.
[ AMS Subject Classification: 53C05, 53C60.
Keywords: k-tangentbundle,k-semispray,nonlinearconnection,Lagrange
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space of order k, Finsler space of order k.
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1 Introduction
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0 Classical Mechanics have been entirely geometrized in terms of symplectic ge-
. ometry and in this approach there exists certain dynamical vector field on the
1
0 tangent bundle TM of a manifold M whose integral curves are the solutions of
6 theEuler-Lagrangeequations. Thisvectorfieldisusuallycalledspray orsecond-
1 order differential equation (SODE). Sometimes it is called semispray and the
:
v term spray is reserved to homogeneous second-order differential equations ([7],
i [15]). Let us remember that a SODE on TM is a vector field on TM such
X
that JC = C, where J is the almost tangent structure and C is the canonical
r
a Liouville field ([5], [6]).
In [2], [3], [4] J. Grifone studies the relationship among SODEs, nonlin-
ear connections and the autonomous Lagrangian formalism. In paper [12] Gh.
Munteanu and Gh. Piti¸s also studied the relation between sprays and nonlin-
ear connectiosn on TM. This study was extended to the non-autonomous case
by M. de Le´on and P. Rodgrigues ([5]). Also, important results for singular
non-autonomous case was obtained in [13]. In this paper, following the ideas
of papers [10], [11], [12] and [13] we will extend the study of the relationship
betweenspraysandnonlinearconnectionstothek-tangentbundleofamanifold
M. The study of the geometry of this k-tangent bundle was by introduced by
R. Miron ([7], [8], [9]). For this case the k-sprayrepresent a system of ordinary
differential equations of k+1 order.
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2 The k-Semispray of a Nonlinear Connection
Let M be a real n-dimensional manifold of class C∞ and (TkM,πk,M) the
bundle of accelerations of order k. It can be identified with the k-osculator
bundle or k-tangent bundle ([7], [9]).
A point u ∈ TkM will be written by u = (x,y(1),...,y(k)), πk(u) = x,
x ∈ M. The canonical coordinates of u are (xi,y(1)i,...,y(k)i), i = 1,n, where
1 dxi 1 dkxi
y(1)i = , ..., y(2)i = . A transformation of local coordinates
1! dt k! dtk
(xi,y(1)i,...,y(k)i)→(xi,y(1)i,...,y(k)i) on (k+1)n-dimensional manifold TkM
is given by
e e e ∂xi
xi =xi(x1,...,xn), rang =n,
∂xj
 (cid:18) (cid:19)
(1)  y2eey(1()2i)i=e=∂∂∂xxe∂yjix(y1j()1i)yj(,1)j +2∂∂yy((11))ejiy(2)j,
........................................................................
e e
A localcookreyed(ikn)ait=es∂chye∂a(kxn−jg1e)i(y1()1)tjra+n2sf∂o∂yremy(k(s−1)1tj)hiey(n2a)jtu+ra·l··b+askis∂∂yye((kk−−11))jiy(k)j.
∂ ∂ ∂
, ,··· , of the tangent space T TkM by the rule:
∂xi ∂y(1)i ∂y(k)i u
(cid:26) (cid:27)u
(2)
∂ ∂xj ∂ ∂y(1)j ∂ ∂y(k)j ∂
= + + ··· + ,
∂xi ∂xi ∂xj ∂xi ∂y(1)j ∂xi ∂y(k)j

 ∂y∂...(1)i = e e ∂∂yeey((11))ji ∂ye∂(1)j + ··· + ∂∂yeey((k1))ji ∂ye∂(k)j,
∂ e ∂y(k)j e∂
The d∂isytr(kib)iutio=n V1 : u ∈ TkM → V1,u ⊂ TuTkM generated∂eyb(yk)ith∂ey(tka)jn.gent
∂ ∂ e
vectors ,··· , is a vertical distribution on the bundle TkM.
∂y(1)i ∂y(k)i
(cid:26) (cid:27)u
Its local dimension is kn. Similarly, the distribution V : u ∈ TkM → V ⊂
2 2,u
∂ ∂
T TkM generated by ,··· , is a subdistribution of V of local
u ∂y(2)i ∂y(k)i 1
(cid:26) (cid:27)u
dimension (k−1)n. So, by this procedure on obtains a sequence of integrable
distributions V ⊃ V ⊃ ··· ⊃ V . The last distribution V is generated by
1 2 k k
∂
and dimV =n ([7]).
∂y(k)i k
(cid:26) (cid:27)u
Hereafter, we consider the open submanifold
]
TkM =TkM \{0}= (x,y(1),...,y(k))∈TkM|rank ||y(1)i||=1 ,
n o
2
where 0 is the null section of the projection πk :TkM →M.
The following operators in algebra of functions F(TkM)
1 ∂
Γ=y(1)i ,
∂y(k)i
2 ∂ ∂
Γ=y(1)i +2y(2)i ,
(3) ∂y(k−1)i ∂y(k)i
....................................................................
k ∂ ∂ ∂
Γ=y(1)i +2y(2)i +···+ky(k)i
∂y(1)i ∂y(2)i ∂y(k)i
are k vector fields, globally defined on TkM and linearly independent on the
] 1 2
manifold TkM = TkM \{0}, Γ belongs of distribution V , Γ belongs of distri-
k
k 1 2 k
bution V , ..., Γ belongs of distribution V (see [7]). Γ , Γ, ..., Γ are called
k−1 1
Liouville vector fields.
In applications we shall use also the following nonlinear operator, which is
not a vector field,
∂ ∂ ∂
(4) Γ=y(1)i +2y(2)i +···+ky(k)i .
∂xi ∂y(1)i ∂y(k−1)i
Under a coordinates transformation (1) on TkM, Γ changes as follows:
∂y(k)j ∂y(k)j ∂
(5) Γ=Γ+ y(1)i +···+ky(k)i .
∂xi ∂y(k−1)i ∂y(k)j
(cid:26) (cid:27)
A k-tangent struceture J on TekM is defined as useually ([7]) by the following
F(TkM)-linear mapping J :X(TkM)→X(TkM): e
∂ ∂ ∂ ∂
J = ,J = ,...,
∂xi ∂y(1)i ∂y(1)i ∂y(2)i
(6) (cid:18) (cid:19) (cid:18) (cid:19)
∂ ∂ ∂
J = ,J =0.
∂y(k−1)i ∂y(k)i ∂y(k)i
(cid:18) (cid:19) (cid:18) (cid:19)
J is a tensor field of type (1,1), globally defined on TkM.
Definition 2.1 ([7]) A k-semispray on TkM is a vector field S ∈ X(TkM)
with the property
k
(7) JS =Γ.
Obviously, there not always exists a k-semispray, globally defined on TkM.
Therefore the notion of local k-semispray is necessary. For example, if M is a
paracompactmanifold then on TkM there exists local k-semisprays ([7]).
Theorem 2.1 ([7]) i) A k-semispray S can be uniquely written in local coordi-
nates in the form:
∂ ∂ ∂
S = y(1)i +2y(2)i +···+ky(k)i −
∂xi ∂y(1)i ∂y(k−1)i
(8)
∂
− (k+1)Gi(x,y(1),...,y(k)) .
∂y(k)i
3
ii) With respect to (1) the coefficients Gi(x,y(1),...,y(k)) change as follows:
∂xi
(k+1)Gi = (k+1)Gj −
∂xj
(9)
∂y(k)i ∂y(k)i
e − y(1)j e+···+ky(k)j .
∂xj ∂y(k−1)j
(cid:18) (cid:19)
e e
iii) If the functions Gi(x,y(1),...,y(k)) are given on every domain of local chart
of TkM, so that (9) holds, then the vector field S from (8) is a k-semispray.
Let us consider a curve c : I → M, represented in a local chart (U,ϕ) by
xi =xi(t), t∈I. Thus, the mapping c:I →TkM, given on (πk)−1(U), by
1 dxi 1 dkxi
(10) xi =xi(t),y(1)i(t)= (te),...,y(k)i(t)= (t),t∈I
1! dt k! dtk
is a curve in TkM, called the k-extension to TkM of the curve c.
A curve c : I → M is called k-path of a k-semispray S (from (8)) if its
k-extension c is an integral curve for S, that is
(11)
dxi dy(1)i dy(k−1)i dy(k)i
=ye(1)i, =2y(2)i, ..., =ky(k)i, =−(k+1)Gi.
dt dt dt dt
(cid:26)
Definition 2.2 Thek-semisprayS iscalledk-sprayifthefunctions Gi(x,y(1),...,y(k))
are (k+1)-homogeneous, that is
(cid:0) (cid:1)
Gi(x,λy(1),...,λky(k))=λk+1Gi(x,y(1),...,y(k)), ∀λ>0.
Likeinthe caseoftangentbundle,anEulerTheoremholds. Thatis,afunction
]
f ∈F(TkM) is r-homogeneous if and only if
L f =rf.
k
Γ
Then a k-semispray S is a k-spray if and only if
∂Gi ∂Gi ∂Gi
(12) y(1)h +2y(2)h +···+ky(k)h =(k+1)Gi.
∂y(1)h ∂y(2)h ∂y(k)h
Definition 2.3 AvectorsubbundleNTkM ofthetangentbundle(TTkM,dπk,M)
which is supplementary to the vertical subbundle V TkM,
1
(13) TTkM =NTkM ⊕V TkM
1
is called a nonlinear connection on TkM.
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The fibres of NTkM determine a horizontal distribution N : u ∈ TkM →
N TkM ⊂T TkM supplementary to the vertical distribution V , that is
u u 1
(14) T TkM =N TkM ⊕V TkM, ∀u∈TkM.
u u 1,u
The dimension of horizontal distribution N is n.
If the base manifold M is paracompact then on TkM there exists the non-
linear connections ([7]).
There exists a unique local basis, adapted to the horizontal distribution N,
δ δ ∂
δxi , such that dπk δxi|u = ∂xi|πk(u), i = 1, ..., n. More over, on
(cid:26) (cid:27)i=1,n (cid:18) (cid:19)
each domain of local chart of TkM there exists the functions Ni, Ni, ..., Ni
j j j
(1) (2) (k)
such that
δ ∂ ∂ ∂
(15) = −Nj −···−Nj .
δxi ∂xi i ∂y(1)j i ∂y(k)j
(1) (k)
The functions Ni, Ni, ...,Ni are called the primal coefficients of the nonlinear
j j j
(1) (2) (k)
connection N and under a coordinates transformation (1) on TkM this coeffi-
cients are changing by the rule:
(16)
∂xm ∂xi ∂y(1)i
Ni = Nm− ,
m ∂xj ∂xm j ∂xj
 (1) (1)
 Ne(2mi) ∂∂xexmj = ∂∂xxemi N(2jm) + ∂∂yex(1m)iN(1jm) − ∂∂yx(2j)i,...,
Nei ∂xem = ∂xei Nm+ ∂ye(1)i Nm +·e··+ ∂y(k−1)iNm− ∂y(k)i.
Convers(ekem)ly,∂eixfjon each∂loxecmal(kcjh)art o∂efxTmkM(k−ja1)set of funce∂tixomns N(1ij,)..., N∂eixijs given
j j
(1) (k)
so that, according to (1), the equalities (16) hold, then there exists on TkM a
unique nonlinear connection N which has as coefficients just the given set of
function ([7]).
δ δ δ
The local adapted basis , ,··· , is given by (15) and
δxi δy(1)i δy(k)i
(cid:26) (cid:27)i=1,n
δ ∂ ∂ ∂
= −Nj −···− Nj ,...,
δy(1)i ∂y(1)i i ∂y(2)j i ∂y(k)j
(1) (k−1)
(17)
δ ∂ ∂ δ ∂
= −Nj , =
δy(k−1)i ∂y(k−1)i i ∂y(k)j δy(k)i ∂y(k)j
(1)
and the dual basis (or the adapted cobasis) of adapted basis is
5
δxi,δy(1)i,...,δy(k)i , where δxi =dxi and
i=1,n
(cid:8) (cid:9)
δy(1)i = dy(1)i+Midxj,
j
(1)
 δy(2)i = dy(2)i+Midy(1)j +Midxj,...,
(18)  δy(k)i = dy(k)i+M(1)jidy(k−1)j +(2)j···+Midxj
j j
and  (1) (k)
Mi = Ni,
j j
(1) (1)
 Mi = Ni+Ni Mm, ...,
(19)  M(2)ji = N(2j)i+ (N1m)i (M1j)m+···+Ni Mm.
j j m j m j
(k) (k) (k−1) (1) (1)(k−1)
Conversely, if the adapted cobasis δxi,δy(1)i,...,δy(k)i is given in the
i=1,n
δ δ δ
form (18), then the adapted basis (cid:8) , ,··· , (cid:9) is expressed
δxi δy(1)i δy(k)i
(cid:26) (cid:27)i=1,n
in the form (17), where
Ni = Mi,
j j
(1) (1)
 Ni = Mi−Ni Mm, ...,
(20)  N(2j)i = M(2)ji− N(1m)i (M1j)m−···−Ni Mm. .
j j m j m j
(k) (k) (k−1) (1) (1)(k−1)
The functions Mi, Mi, ..., Mi are called the dual coefficients of the nonlinear
j j j
(1) (2) (k)
connection N.
A nonlinear connection N is complete determined by a system of functions
Mi, ..., Mi which is given on each domain of local chart on TkM, so that,
j j
(1) (k)
according to (1), the relations hold:
(21)
∂xi ∂xm ∂y(1)i
Mm = Mi + ,
j ∂xm ∂xj m ∂xj
 (1) (1)
 M(2jm) ∂∂xxemi = ∂∂xexmj Mf(2m)i + ∂ye∂(x1j)mM(1m)i + ∂∂yx(2j)i,...,
Mm ∂xei = ∂xemMfi + ∂ye(1)mfMi +·e··+ ∂y(k−1)mMi + ∂y(k)i.
Let c (:kj)I∂→xemM be a∂expjarfa(kmm)etrizee∂dxjcur(vfk−em1o)n the basee∂mxjanifof(l1dm) M, g∂eixvjen by
xi =xi(t), t∈I. If we consider its k-extensionc to TkM, then we say that c is
e
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an autoparallel curve for the nonlinear connection N if its k-extension c is an
dc
horizontal curve, that is belongs to the horizontal distribution.
dt
e
From (18) and
e
dc dxi δ δy(1)i δ δy(k)i δ
(22) = + +···+
dt dt δxi dt δy(1)i dt δy(k)i
e
it result that the autoparallels curves of the nonlinear connection N with the
dualcoefficientsMi,...,Mi arecharacterizedbythesystemofdifferentialequa-
j j
(1) (k)
tions ([7]):
dxi 1 d2xi 1 dkxi
y(1)i = , y(2)i = , ..., y(k)i = ,
dt 2! dt2 k! dtk
 δy(1)i dy(1)i dxj
(23)  δydd(tt2)i == dydd(tt2)i ++MM((11))jjiiddydt(t1)=j +0,M(2)jiddxtj =0,
.....................................................................
 δyd1(tk)i = dyd(tk)i +M(1)jidy(dk−t1)j +···+M(k)j1iddxtj =0.
Now, let be S = S a k-semispray with the coefficients Gi = Gi(x,y(1),...,y(k))
like in (8). Then the set of functions
∂Gi
Mi = ,
j ∂y(k)j
 (1)
(24)  M.(.2..)ji...=....12... ...S..M.(.1.).ji..+....M.(.1.m)i...M.(.1.j.m)..!.. ,
1
gives the dual coefficientsM(okf)jia=nkon liSne(kMa−rj1ic)o+nnM(e1cm)iti(Mokn−jm1N)!determined only by the
k-semispray S (see the book [7] of Radu Miron).
Other result, obtained by Ioan Buca˘taru ([1]), give a second nonlinear con-
nection N∗ on TkM determined only by the k-semispray S. That is, the fol-
lowing set of functions
∂Gi ∂Gi ∂Gi
(25) M∗i = , M∗i = , ..., M∗i =
j ∂y(k)j j ∂y(k−1)j j ∂y(1)j
(1) (2) (k)
is the set of dual coefficients of a nonlinear connection N∗.
7
2
Let us consider the set of functions (Gi(x,y(1),...,y(k))), given on every do-
main of local chart by
(26)
1 1 1
2 1 k 1 1 ∂Gi 2 ∂Gi k ∂Gi
Gi = ΓGi = y(1)h + y(2)h +···+ y(k)h .
k+1 k+1 ∂y(1)h k+1 ∂y(2)h k+1 ∂y(k)h
2 2
Using (5) we obtain that the functions Gi verifies (9). So, the functions Gi
2
represent the coefficients of a k-semispray S,
2 ∂ ∂ ∂
S = y(1)i +2y(2)i +···+ky(k)i −
∂xi ∂y(1)i ∂y(k−1)i
(27)
2 ∂
− (k+1)Gi(x,y(1),...,y(k)) .
∂y(k)i
Obviously, there exists two nonlinear connections on TkM, which depend only
2
by the k-semispray S:
2
N with the dual coefficients
2 ∂Gi
Mi = ,
j ∂y(k)j
 (1)
(28)  M.(.22..)ji...=....21... ...S..M.(.21.).ji..+....M.(.21.m)i...M.(.21.j.m)..!.. ,
2 1 2 2 2
2  M(k)ji = k S(kM−j1i)+M(1m)i (Mk−jm1)!
and N∗ with the dual coefficients
2 2
2 ∂Gi 2 ∂Gi 2 ∂Gi
(29) M∗i = , M∗i = , ..., M∗i = .
j ∂y(k)j j ∂y(k−1)j j ∂y(1)j
(1) (2) (k)
m
By this method is obtained a sequence of k-semisprays S and two se-
(cid:18) (cid:19)m≥1
m m
quence of nonlinear connections, N , N∗ .
(cid:18) (cid:19)m≥1 (cid:18) (cid:19)m≥1
From (11), (23) and (26) we have the following results:
1
Proposition 2.1 If c is an autoparallel curve for nonlinear connection N∗,
2
then c is a k-path of k-semispray S.
8
Theorem 2.2 The following assertions are equivalent:
1
i) the k-semispray S is a k-spray;
1 2
ii) the k-paths of S and S coincide.
1 1 1 1 1
Theorem 2.3 If S is a k-spray then Mi, ..., Mi (or M∗i, ..., M∗i) are ho-
j j j j
(1) (k) (1) (k)
mogeneous functions of degree 1, 2, ..., k, respectively. The same property have
1 1 1 1
the primal coefficients Ni, ..., Ni (or N∗i, ..., N∗i).
j j j j
(1) (k) (1) (k)
We remark that the converse of this proposition is generally not valid and we
have the result:
1 m
Theorem 2.4 If S is a k-spray then the sequence S is constant and the
(cid:18) (cid:19)m≥1
m m
sequences N , N∗ are constant.
(cid:18) (cid:19)m≥1 (cid:18) (cid:19)m≥1
3 The k-Semispray of a Nonlinear Connection
in a Lagrange Space of Order k
A Lagrangian of order k is a mapping L:TkM →R. L is called differentiable
]
ifitisofC∞-classonTkM andcontinuousonthe nullsectionofthe projection
πk :TkM →M.
The Hessian of a differentiable Lagrangian L, with respect to the variables
]
y(k)i on TkM is the matrix ||2g ||, where
ij
1 ∂2L
(30) g = .
ij 2∂y(k)i∂y(k)j
]
We have that g is a d-tensor field on the manifold TkM, covariantof order 2,
ij
symmetric (see [7]).
If
]
(31) rank ||g ||=n, on TkM
ij
we say that L(x,y(1),...,y(k)) is a regular (or nondegenerate) Lagrangian.
The existence of the regularLagrangiansoforder k is provedfor the case of
paracompacts manifold M in the book [7] of Radu Miron.
Definition 3.1 ([7])WecallaLagrangespaceoforderkapairL(k)n =(M,L),
formed by a real n-dimensional manifold M and a regular differentiable La-
grangian of order k, L : (x,y(1),...,y(k)) ∈ TkM → L(x,y(1),...,y(k)) ∈ R, for
]
which the quadratic form Ψ=g ξiξj on TkM has a constant signature.
ij
9
Liscalledthe fundamental function andg the fundamental (ormetric)tensor
ij
field of the space L(k)n.
It is known that for any regular Lagrangian of order k, L(x,y(1),...,y(k)),
there exists a k-semispray S determined only by the Lagrangian L (see [7]).
L
The coefficients of S are given by
L
1 ∂L ∂L
(32) (k+1)Gi = gij Γ − .
2 ∂y(k)j ∂y(k−1)j
(cid:26) (cid:18) (cid:19) (cid:27)
This k-semispray S depending only by L will be called canonical. If L is
L
]
globally defined on TkM, then S has the same property on TkM.
L
From (24) and (25) it result that there exists two nonlinear connections:
Miron’s connection N and Buca˘taru’s connection N∗ which depending only by
the LagrangianL. For this reason, both are called canonical.
m
So, the coefficients of k-semisprays S and the coefficients of nonlinear con-
m m
nections N, N∗ depend only by the Lagrangian L, for any m ≥ 1, but their
expressions is not attractive for us.
Interesting results appear for Finsler spaces of order k.
Definition 3.2 ([8])AFinslerspaceoforderkisapairF(k)n =(M,F)formed
byarealdifferentiablemanifoldM ofdimensionnandafunctionF :TkM →R
having the following properties:
]
i) F is differentiable on TkM andcontinuousonnullsection 0:M →TkM;
ii) F is positive;
iii) F is k-homogeneous;
iv) the Hessian of F2 with elements
1 ∂2F2
(33) g =
ij 2∂y(k)i∂y(k)j
]
is positively defined on TkM.
The function F is called the fundamental function and the d-tensor field g is
ij
calledfundamental(ormetric)tensorfield oftheFinslerspaceoforderk,F(k)n.
The class of spaces F(k)n is a subclass of spaces L(k)n.
Taking into account the k-homogeneity of the fundamental function F and
2k-homogeneity of F2 we get:
1. the coefficients Gi of the canonical k-semispray SF2, determined only by
the fundamental function F,
1 ∂F2 ∂F2
(34) (k+1)Gi = gij Γ − ,
2 ∂y(k)j ∂y(k−1)j
(cid:26) (cid:18) (cid:19) (cid:27)
is (k+1)-homogeneous functions, that is SF2 is a k-spray;
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