Received 30 July 2019, Accepted 5 December 2019, Available Online 4 September 2020.
1. Introduction
The framework of differential geometry is very useful in modelling and understanding of a large class of natural phenomena. The Lie geometric methods are applied successfully in differential equations, optimal control theory or theoretical physics. In most of the cases the study starts with a variational problem formulated for a regular Lagrangian on the tangent bundle TM over the manifold M and very often the whole set of problems is transferred on the dual space T*M, endowed with a Hamiltonian function, via Legendre transformation. The problem is that in a lot of cases the proposed Lagrangian formalism yields a singular Lagrangian description, which makes the Legendre transformation ill-defined and thus no straightforward Hamiltonian formulation can be related. In the last years the investigations have led to a geometric framework which is covering these phenomena. It is precisely the underlying structure of a Lie algebroid on the phase space which allows a unified treatment. This idea was first introduced by A. Weinstein [24] in order to define a Lagrangian formalism which is very useful for the various types of such systems. One of the motivations for the present work is the study of Lagrangian systems subjected to external holonomic constraints, which come from the theory of optimal control, using the framework of Lie algebroids.
Optimal control problems belong to the class of extremum optimization theory, i.e., minimization or maximization of some functions equipped with some external constraints. This theory extends classical variational calculus that is based on control variations of a continuous trajectory. One of the most important methods in the analysis of solutions for the optimal control problems is provided by Pontryagin’s Maximum Principle. A curve c(t) = (x(t), u(t)) is an optimal trajectory if there exists a lifting of x(t) to the dual space (x(t), p(t)) satisfying the Hamilton-Jacobi-Bellman equations. This theory has very important applications in different domains because it provides a pseudo-Hamiltonian formulation of the variational problem in the case in which the standard Legendre transformation is not well defined.
However, finding a complete solution to an optimal control problem, still remains extremely difficult for several reasons. Firstly, we are dealing with the problem of integrating a Hamiltonian system, which is generally difficult to integrate, except for particular dynamics and costs. Secondly, even though all solutions are found, there remains the problem of selecting optimal solutions from them. For these reasons, it is very important to find new methods and new working space that could simplify the study. In this paper we prove that the framework of Lie algebroids is more suitable than the cotangent bundle for the study of driftless control affine systems with holonomic distribution.
Lie algebroids are also important issues in physics and mechanics since the extension of Lagrangian and Hamiltonian systems to their entity [9, 10, 12, 24] and catching the Poisson structure [17]. They are also related to optimization theory [14, 15, 18]. They have such a flexibility that holonomy of orbit foliation carried on them [6]. Thus Lie algebroids are strong assorted structures to assemble the physics and mechanics notions on them. For good details about penetration of Lie algebroids, see [23].
The notion of Lie algebroids was first introduced by Pradines [19]. Research on this field has been continued by mathematicians with various purposes up to now. Lie algebroids are studied pure or in relation with other subjects [1–3, 11, 14–16, 22]. Precisely, a Lie algebroid is a vector bundle with the property that its sections involve a real Lie algebra. Each section is anchored on a vector field, by means of a linear bundle map named as anchor map, which is further supposed to induce a Lie algebra homomorphism. Especially, when the base manifold M is a point, a Lie algebroid reduces to a Lie algebra. The simplest examples of Lie algebroids are the zero bundle over M which is denoted by M and tangent bundle over M with identity as anchor map which is denoted by TM. Then the tangent bundle is a special case of Lie algebroid structure. Therefore, a Lie algebroid is a generalization of a Lie algebra and tangent bundle.
The second motivation of this paper is the study of some geometric structures on Lie algebroids, such as (non)-linear connections, torsion and curvature. These structures can be useful in the study of optimal solutions behavior of control problems. In many situations we cannot find the exact solution of the optimal control problem, but using the geometry of the space, we can find information on their local or global behavior. Indeed, if the geodesics curves in the framework of Lie algebroids, which are also the optimal trajectories of control systems, are situated on a manifold with positive constant curvature, then the geodesics focus and the negative curvature spreads geodesics out, these never intersect again.
The Berwald connection is a well-known concept in Finsler geometry. This connection can be characterized as the unique good vertical connection with a minimal amount of metric compatibility and the most vanishing of the torsion. Also, this connection can be regarded as associated primarily with the geodesic spray of the energy metric, and its metrical properties as consequences of those of the geodesic spray. The construction of the Berwald connection from the geodesic spray is in turn a particular case of a more general construction which associates, in a unique way, a certain linear connection with an arbitrary non-linear connection on the tangent bundle of a differentiable manifold, which is called a Berwald-type connection [5]. Using Berwald-type connections we can construct other linear connections, which are called Yano-type connections. The Yano connection is a special case of a Yano-type connection that is used to construct the Douglas tensor. This tensor always vanishes in affinely connected manifolds (hence, in particular, in Riemannian manifolds). So, it has a typically ”non-Riemannian” character. The importance of the Douglas tensor can be readily realized. Those Finsler manifolds which have vanishing projective Weyl tensor and Douglas tensor are just the solutions of the Hilbert’s fourth problem [21]. The third aim of the paper is the study of the Berwald-type and Yano-type linear connections, Douglas tensor and their properties. Moreover, the study of the ρ£-covariant derivatives in π*π (in particular, Berwald and Yano derivatives), theirs torsions and curvatures and the study of their properties are the other aims of the paper.
This paper is organized as follows. In Section 2, we briefly review general definitions on Lie algebroids and we recall some geometric objects on these spaces such as vertical lifts, complete lifts, almost complex structures and sprays. In Section 3, considering two classes of distinguished connections, namely Berwald-type and Yano-type connections, we introduce Douglas tensor of Berwald endomorphism and we obtain some properties of this tensor. In Section 4, we construct ρ£-covariant derivatives in π*π as the generalization of covariant derivative in π*π to £πE. Moreover, Berwald and Yano derivatives as two important classes of ρ£-covariant derivatives in π*π are introduced in this section.
In the last section, we present an example from optimal control theory. We prove that the framework of Lie algebroids is more useful than the cotangent space in order to apply the Pontryagin Maximum Principle and find the optimal solution. Also, we calculate in this case some geometric structures as semispray, nonlinear connection, curvature, Berwald connection and Douglas tensor. We prove that the solutions of control system are integral curves of the spray. Finally, we find the optimal solution of the control system, using the framework of a Lie algebroid, which is in this case an integrable distribution of the tangent bundle.
2. Basic concepts on Lie algebroids
Let E be a vector bundle of rank n over a manifold M of dimension m and π : E → M be the vector bundle projection. Denote by Γ (E) the C∞(M)-module of sections of π : E → M. A Lie algebroid structure ([·, ·]E, ρ) on E is a Lie bracket [·, ·]E on the space Γ(E) and a bundle map ρ : E → TM, called the anchor map, such that if we also denote by ρ : Γ(E) → χ(M) the homomorphism of C∞(M)-modules induced by the anchor map then
[X,fY]E=f[X,Y]E+ρ(X)f(Y), ∀X,Y∈Γ(E), ∀f∈C∞(M).
Moreover, we have the relations
[ρ(X),ρ(Y)]=ρ([X,Y]E).
Then triple (E,[·,·]E,ρ) is called a Lie algebroid over M.
Trivial examples of Lie algebroids are real Lie algebras of finite dimension, the tangent bundle TM of an arbitrary manifold M and an integrable distribution of TM.
If we take local coordinates (xi) on M and a local basis {eα} of sections of E, then we have the corresponding local coordinates (xi, yα) on E, where xi = xi ∘ π and yα(u) is the α-th coordinate of u ∈ E in the given basis. Such coordinates determine local functions ραi, Lαβγ on M which contain the local information of the Lie algebroid structure, and accordingly they are called the structure functions of the Lie algebroid. They are given by
ρ(eα)=ραi∂∂xi, [eα,eβ]E=Lαβγeγ,
such that
ραj∂ρβi∂xj−ρβj∂ραi∂xj=ργiLαβγ, ∑(α,β,γ)[ραi∂Lβγν∂xi+LαμνLβγμ]=0,
which are usually called the structure equations.
For a function f on M, one defines its vertical lift f∨ on E by f∨(u) = f(π(u)) for u ∈ E. Now, let X be a section of E. Then, we can consider the vertical lift of X as the vector field on E given by X∨(u)=X(π(u))u∨, u ∈ E, where ∨u:Eπ(u)→Tu(Eπ(u)) is the canonical isomorphism between the vector spaces Eπ(u) and Tu(Eπ(u)). Let {eα} be a basis of sections of E, then we have eα∨=∂∂yα. Also, using it we deduce that if X = Xαeα ∈ Γ(E), then the vertical lift X∨ has the local expression X∨=(Xα∘π)∂∂yα.
The complete lift of a smooth function f ∈ C∞(M) into C∞(E) is the smooth function
fc:E→ℝ, fc(u)=dEf(u)=ρ(u)f,
where
dE is the differential of a function on
E (see, [
10,
11] for more details). In the local basis we have
fc|π−1(U)=yα((ραi∂f∂xi)∘π).
Also, the complete lift Xc of a section X on E is the unique vector field on E given by [7], [8]:
Xc={(Xαραi)∘π}∂∂xi+yβ{(ρβj∂Xα∂xj−XγLγβα)∘π}∂∂yα.(2.1)
2.1. The Prolongation of a Lie algebroid
Let £πE be the subset of E × TE defined by £πE = {(u, z) ∈ E × TE | ρ(u) = π*(z)} and denote by π£ : £πE → E the mapping given by π£(u, z) = πE(z), where πE : TE → E is the natural projection. Then (£πE, π£, E) is a vector bundle over E of rank 2n. Indeed, the total space of the prolongation is the total space of the pull-back of π* : TE → TM by the anchor map ρ.
We introduce the vertical subbundle
V£πE=Ker τ£={(u,z)∈£πE|τ£(u,z)=0},
where
τ£ :
£πE →
E is the projection onto the first factor, i.e.,
τ£(
u,
z) =
u. Therefore an element of
V£πE is of the form (0,
z) ∈
E ×
TE such that
π*(
z) = 0 which is called vertical. Since
π*(
z) = 0 and Ker
π* =
VE(
π* :
TE →
TM), then we deduce that if (0,
z) is vertical then
z is a vertical vector on
E.
For a local basis {eα} of sections of E and coordinates (xi, yα) on E, we have local coordinates (xi, yα, kα, zα) on £πE given as follows. If (u, z) is an element of £πE, then by using ρ(u) = π*(z), z has the form
z=((ραiuα)∘π)∂∂xi|v+zα∂∂yα|v, z∈TvE.
The local basis {𝒳α, 𝒱α} of sections of £πE associated to the coordinate system is given by
𝒳α(v)=(eα(π(v)),(ραi∘π)∂∂xi|v), 𝒱α(v)=(0,∂∂yα|v).(2.2)
If V is a section of £πE which in coordinates writes
V(x,y)=(xi,yα,Zα(x,y),Vα(x,y)),
then the expression of
V in terms of base {
𝒳α,
𝒱α} is [
10]
V=Zα𝒳α+Vα𝒱α.
The vertical lift XV and the complete lift XC of a section X ∈ Γ(E) as the sections of £πE → E are given by
XV(u)=(0,X∨(u)), XC(u)=(X(π(u)),Xc(u)), u∈E.
It is known that X∨ and XC have the following coordinate expressions [13]:
XV=(Xα∘π)𝒱α, XC=(Xα∘π)𝒳α+yβ[(ρβj∂Xα∂xj−XγLγβα)∘π]𝒱α,(2.3)
where
X =
Xαeα ∈ Γ(
E). In particular we have
eαV=𝒱α(2.4)
Here, we consider the anchor map ρ£ : £πE → TE defined by ρ£(u, z) = z and the bracket [·, ·]£ satisfying the relations
[XV,YV]£=0, [XV,YC]£=[X,Y]EV, [XC,YC]£=[X,Y]EC,(2.5)
for
X,
Y ∈ Γ(
E). Then this vector bundle (
£πE,
π£,
E) is a Lie algebroid with structure ([·, ·]
£,
ρ£).
2.2. A setting for semispray on £πE
A section of π along smooth map f : N → M is a smooth map σ : N → E such that π ∘ σ = f. The set of sections of π along f will be denoted by Γf(π). Then there is a canonical isomorphism between Γ(f*π) and Γf(π) (see [20]). Now we consider the pullback bundle π*π = (π*E, pr1, E) of the vector bundle (E, π, M), where
π*E:=E×ME:={(u,v)∈E×E|π(u)=π(v)},
and
pr1 is the projection map onto the first component. The fibres of
π*π are the
n-dimensional real vector spaces {
u} ×
Eπ(u) ≅
Eπ(u), and so any section in Γ(
π*π) is of the form
X¯:u∈E→X¯(u)=(u,X_(u)),
where
X_:E→E is a smooth map such that
π∘X_=π. In these terms, the map
X¯∈Γ(π*π)→X_∈Γπ(π),
is an isomorphism of
C∞(
E)-modules. Therefore we have Γ(
π*π) ≅ Γ
π(
π). In Γ(
π*π), there is a distinguished section
δ:u∈E→δ(u)=(u,u)∈π*E,(2.6)
that is called
the canonical section along π. This section corresponds to the identity map 1
E under the isomorphism Γ
π(
π) ≅ Γ(
π*π). For any section
X on
E, the map
X^:E→π*E,
defined by
X^(u)=(u,X∘π(u)) is a section of
π*π, called the lift of
X into Γ(
π*π).
X^ may be identified with the map
X ∘
π :
E →
E. It is easy to see that,
{X^|X∈Γ(E)} generates locally the
C∞(
E)-module Γ(
π*π).
Here, we consider the exact sequence
0→π*(E)→i£πE→jπ*(E)→0,(2.7)
with
j(
u,
z) = (
πE(
z),
Id(
u)) = (
v,
u),
z ∈
TvE, and
i(u,v)=(0,vu∨). Function
J =
i j :
£πE →
£πE is called the
vertical endomorphism (
almost tangent structure) of
£πE. This endomorphism has the locally expression
J=𝒱α⊗𝒳α,(2.8)
and it has the following properties
ImJ=Imi=V£πE, KerJ=Kerj=V£πE, J∘J=0.
Let δ be the canonical section along π given by (2.6). Then section C given by
C:=i∘δ,
is called
Liouville or
Euler section. The Liouville section
C has the coordinate expression
C =
yα𝒱α with respect to {
𝒳α,𝒱α}. Moreover, we have
(i) [J,C]£F−N=J, (ii) [XV,C]£F−N=XV, (iii) JC=0,(2.9)
where
[⋅,⋅]£F−N is the generalized Frölicher-Nijenhuis bracket on
£πE (see [
13], for more details).
A section X˜ of vector bundle (£πE, π£, E) is said to be homogeneous of degree r, where r is an integer, if [C,X˜]£=(r−1)X˜. Moreover, f˜∈C∞(E) is said to be homogeneous of degree r if £C£f˜=ρ£(C)(f˜)=rf˜. It is known that a real valued smooth function f˜ on E is homogeneous of degree r if and only if yα∂f˜∂yα=rf˜ [13]. A section S of the vector bundle (£πE, π£, E) is said to be a semispray if it satisfies the condition J(S) = C. Moreover, if S is homogeneous of degree 2, i.e., [C, S]£ = S, then we call it spray. It is known that S = Aα𝒳α + Sα𝒱α is a spray on £πE if and only if Aα = yα and 2Sβ=yα∂Sβ∂yα. In [13], the first author proved that if S1 is a spray on £πE and f˜:E→ℝ is a homogeneous function of degree 1 on E − {0}, then S2=S1+f˜C is a spray on £πE. This is said to be projective change of S1 by f˜.
A function h : £πE → £πE is called a horizontal endomorphism if h ∘ h = h, Kerh = V£πE and h is smooth on £π∘E=£πE−{0}. Also, v := Id − h is called the vertical projector associated to h. It is known that h has the following locally expression:
h=(𝒳β+ℬβα𝒱α)⊗𝒳β.
Setting H£πE := Imh we have the following splitting for £πE:
£πE=V£πE⊕H£πE.(2.10)
Moreover, we get
Imv=KerJ=V£πE, Kerh=ImJ=KerJ=Imv=V£πE,hJ=hv=Jv=0, v∘v=v, vh=0, Jh=J=vJ,
and
j∘h=j,(2.11)
where
j :
£πE →
E ×
M E is the map introduced in
(2.7).
Let h be a horizontal endomorphism on £πE. Then H=[h,C]£F−N:£πE→£πE is called the tension of h. If H = 0, then h is called homogeneous. Using the above definition H has the coordinate expression
H=(ℬβα−yγ∂ℬβα∂yγ)𝒱α⊗𝒳β.
It is known that
h is homogeneous if and if
ℬβα=yγ∂ℬβα∂yγ. The
weak torsion of
h is defined by
t=[J,h]£F−N∈Γ(£πE). It is known that,
t has the following coordinate expression:
t=12tαβγ𝒳α∧𝒳β⊗𝒱γ,(2.12)
where
tαβγ:=∂ℬβγ∂yα−∂ℬαγ∂yβ−(Lαβγ∘π).(2.13)
The curvature of a horizontal endomorphism h is defined by Ω = −Nh, where Nh is the Nijenhuis tensor of h given by
Nh(X˜,Y˜)=[hX˜,hY˜]£−h[hX˜,Y˜]£−h[X˜,hY˜]£+h[X˜,Y˜]£, ∀X˜,Y˜∈Γ(£πE).
For sections X˜ and Ỹ of £pE we have
Ω(X˜,Y˜)=Ω(hX˜,hY˜)=−v[hX˜,hY˜]£.
Moreover, the curvature Ω has the following coordinate expression:
Ω=−12Rαβγ𝒳α∧𝒳β⊗𝒱γ,(2.14)
where
Rαβγ=(ραi∘π)∂ℬβγ∂xi−(ρβi∘π)∂ℬαγ∂xi+ℬαλ∂ℬβγ∂yλ−ℬβλ∂ℬαγ∂yλ+(Lβαλ∘π)ℬλγ.(2.15)
If S is an arbitrary semispray of £πE, then S¯=hS is also a semispray of £πE which does not depend on the choice of S. S¯ is called the semispray associated to h. Moreover, if h is homogeneous, then the semispray associated to h is a spray [13]. Let S be the semispray associated to h. Then the almost complex structure F : £πE → £πE given by F:=h[S,h]£F−N−J is called the almost complex structure induced by h. The following relations hold
(i) F∘J=h, (ii) F∘h=−J, (iii) J∘F=v, (iv) F∘v=h∘F.
Also, F has the following coordinate expression
F=−(ℬαγ(𝒳γ+ℬγβ𝒱β)+𝒱α)⊗𝒳α+(𝒳α+ℬαβ𝒱β)⊗𝒱α.
Considering ℋ := F ∘ i : E ×M E → £πE and 𝒱 := j ∘ F : £πE → E ×M E, the following sequence is a double short exact sequence
0⇄π*E⇄𝒱i£πE⇄ℋjπ*E⇄0.
Furthermore, we have
(i) h=ℋ∘j, (ii) v=i∘𝒱.
We define the horizontal endomorphism generated by a semispray S by hS:=12(1£πE+[J,S]£F−N). hS has the coordinate expression
hS=(𝒳α+ℬαγ𝒱γ)⊗𝒳α,(2.16)
where
ℬαγ=12(∂Sγ∂yα−yβ(Lαβγ∘π)).(2.17)
It is known that the horizontal endomorphism generated by a semispray S is torsion-free (see [13], Th. 4.20). We also have HS=12[[C,S]£−S,J]£F−N, where HS is the tension of hS. Moreover, the tension HS has the following coordinate expression [13]
HS=12(∂Sα∂yβ−yγ∂2Sα∂yγ∂yβ)𝒱α⊗𝒳β.
The horizontal endomorphism generated by an spray is called Berwald endomorphism and we denote it by hSB.
Lemma 2.1
([13]). Let hS be the horizontal endomorphism generated by a semispray S. Then the semispray associated by hS is12(S+[C,S]£). Moreover the spray associated by hS is S and hS is homogeneous.
Let h be a horizontal endomorphism on £πE. We consider the map
X∈Γ(E)→Xh:=hXC∈H£πE,
and we call it
horizontal lift b h. If
X =
Xαeα, then we have
Xh=(Xα∘π)(𝒳α+ℬαβ𝒱β).
The following relations hold for any X, Y ∈ Γ(E).
(i) JXh=XV, (ii) h[Xh,Yh]£=[X,Y]Eh, (iii) [X,Y]EV=J[Xh,Yh]£.
Setting δα=eαh=𝒳α+ℬαβ𝒱β, it is easy to see that {δα} generate a basis of H£πE and {δα, 𝒱α} is a local basis of £πE adapted to splitting (2.10) which is called adapted basis (see [13], for more details). The dual adapted basis is {𝒳α, δ𝒱α}, where
δ𝒱α=𝒱α−ℬβα𝒳β.
The Lie brackets of the adapted basis {δα, 𝒱α} are
[δα,δβ]£=(Lαβγ∘π)δγ+Rαβγ𝒱γ, [δα,𝒱β]£=−∂ℬαγ∂yβ𝒱γ, [𝒱α,𝒱β]£=0,
where
Rαβγ is given by
(2.15). It is easy to check that
h and
F have the following coordinate expressions with respect to adapted basis
h=δα⊗𝒳α, F=−𝒱α⊗𝒳α+δα⊗δ𝒱α.
3. The Douglas tensor of a Berwald endomorphism
In [13], the first author introduced two distinguished connections, namely Berwald-type and Yano-type connections and he studied some properties of them. In this section, using these connections, we introduce the Douglas tensor of the Berwald endomorphism and study properties of it.
A linear connection on a Lie algebroid (E, [·, ·]E, ρ) is a map
D:Γ(E)×Γ(E)→Γ(E)
which satisfies the rules
DfX+YZ=fDXY+DYZ,DX(fY+Z)=(ρ(X)f)Y+fDXY+DXZ,
for any function
f ∈
C∞(
M) and
X,
Y,
Z ∈ Γ(
E). Let
D be a linear connection on
£πE and h be a horizontal endomorphism on
£πE. Then (
D,
h) is called a distinguished connection (or d-connection) on
£πE, if
- i)
D is reducible, i.e., Dh = 0, which gives us
DX˜hY˜=hDX˜Y˜∈H£πE, DX˜vY˜=vDX˜Y˜∈V£πE, ∀X˜,Y˜∈Γ(£πE),
- ii)
D is almost complex, i.e., DF = 0, where F is the almost complex structure associated to h. It is known that DJ = 0 [13]. Moreover, it is shown that a d-connection (D, h) has the following coordinate expressions
Dδαδβ=Dδα𝒱β=Fαβγ𝒱γ, D𝒱αδβ=D𝒱α𝒱β=Cαβγ𝒱γ.
Let (D, h) be a d-connection. Then
{Dh:Γ(£πE)×Γ(£πE)→Γ(£πE)(X˜,Y˜)↦DX˜hY˜:=DhX˜Y˜,
and
{Dv:Γ(£πE)×Γ(£πE)→Γ(£πE)(X˜,Y˜)↦DX˜vY˜:=DvX˜Y˜,
are called
h-covariant derivative and v-covariant derivative, respectively. Moreover,
{h*(DC):Γ(£πE)→Γ(£πE)X˜↦DC(hX˜):=DhX˜C,
and
{v*(DC):Γ(£πE)→Γ(£πE)X˜↦DC(vX˜):=DvX˜C,
are called
h-deflection and
v-deflection of (
D,
h), respectively. It is easy to see that
h*(
DC) and
v*(
DC) have the following coordinate expressions:
h*(DC)=(ℬαγ+yβFαβγ)𝒱γ⊗𝒳α, v*(DC)=(δαγ+yβCαβγ)𝒱γ⊗δ𝒱α,
where
δαγ is the Kronecker symbol. Also, the curvature tensor field
K of
D is completely determined by the following (see [
13], for more details)
R(X˜,Y˜)Z˜:=K(hX˜,hY˜)JZ˜,P(X˜,Y˜)Z˜:=K(hX˜,hY˜)JZ˜,Q(X˜,Y˜)Z˜:=K(JX˜,JY˜)JZ˜,
where
R,
P and
Q are horizontal, mixed and vertical curvatures, respectively.
For a d-connection (D, h) on £πE, the tensor field
{Pric:Γ(£πE)×Γ(£πE)→C∞(E),(X˜,Y˜)→tr[F∘(Z˜→P(Y˜,Z˜)X˜)],
is called the mixed Ricci tensor of d-connection (
D,
h), where
F is the almost complex structure associated to
h. It is known that the mixed Ricci tensor of (
D, h) has the coordinate expression
Pric =
Pαβ 𝒳α 𝒳β , where
Pαβ=Pαβγβ [
13].
Here we present two examples of d-connections on £πE. These connections which are called Berwald-type and Yano-type connections have studied by the first author in [13].
Let h be a horizontal endomorphism on £πE. The couple (DB,h), where
{DB:Γ(£πE)×Γ(£πE)→Γ(£πE),(X˜,Y˜)↦DX˜BY˜,
is given by
DX˜BY˜:=hF[hX˜,JY˜]£+v[hX˜,vY˜]£+h[vX˜,Y˜]£+J[vX˜,FY˜]£,
is called the
Berwald-type connection. If, in particular,
hSB is a Berwald endomorphism, then we call
(DB,hSB) a
Berwald connection. It is easy to see that
DδαBvβ=DδαBδβ=−∂ℬαγ∂yβδγ, DvαBδβ=DvαBvβ=0.
We have also
RBαβγλ=−(ραi∘π)∂2ℬβλ∂xi∂yγ−ℬαμ∂2ℬβλ∂yμ∂yγ+(ρβi∘π)∂2ℬαλ∂xi∂yγ+ℬαμ∂2ℬαλ∂yμ∂yγ+∂ℬβμ∂yγ∂ℬαλ∂yμ−∂ℬαμ∂yγ∂ℬαλ∂yμ+(Lαβμ∘π)∂ℬαλ∂yγ,(3.1)
PBαβγλ=∂2ℬαλ∂yβ∂yγ,(3.2)
SBαβγλ=0,(3.3)
where
RBαβγλ,
PBαβγλ and
SBαβγλ are the coefficients of the horizontal, mixed and vertical curvatures of d-connection
(DB,h), respectively. Here, let
h be a homogeneous and torsion-free horizontal endo-morphism on
£πE and
PricB be the mixed Ricci tensor of the Berwald-type connection
(DB,h). We consider the following d-connection
DYvX˜vY˜=DBvX˜vY˜,DYhX˜vY˜=DBhX˜vY˜+1n+1PBric(X˜,FY˜)C,DYvX˜hY˜=DBvX˜hY˜,DYhX˜hY˜=DBhX˜hY˜+1n+1PBric(X˜,Y˜)FC,
where
n = rank
E. This d-connection is said to be the
Yano-type connection induced by h. If, in particular,
hSB is a Berwald endomorphism, then we call it a
Yano connection. It is easy to see that the Yano-type connection has the following coordinate expression:
DYδα𝒱β=DYδαδβ=(1n+1∂2ℬαλ∂yλ∂yβyγ−∂ℬαγ∂yβ)δγ, DY𝒱αδβ=DY𝒱α𝒱β=0.
It is known that the coefficients of mixed curvature of Yano-type connection are
PYαβγλ=∂2ℬαλ∂yβ∂yγ−1n+1(∂2ℬαμ∂yμ∂yγδβλ+∂3ℬαμ∂yβ∂yμ∂yγyλ).(3.4)
Definition 3.1.
Let hSB be a Berwald endomorphism on the manifold £πE. If (DY,hSB) is the Yano connection induced by hSB and PY is the mixed curvature of DY, then the tensor
D=PY−12(PYric⊗J+J⊗PYric),
is said to be the Douglas tensor of the Berwald endomorphism.
Using (2.8) and (3.4), the Douglas tensor D has the following coordinate expression:
D=Dαβγλ𝒱λ⊗𝒳α⊗𝒳β⊗𝒳γ,(3.5)
where
Dαβγλ=∂2ℬαλ∂yβ∂yγ−1n+1(∂2ℬαμ∂yμ∂yγδβλ+∂3ℬβμ∂yα∂yμ∂yγyλ+∂2ℬαμ∂yμ∂yβδγλ+∂2ℬβμ∂yμ∂yγδαλ).(3.6)
Since the Berwald endomorphism is homogeneous and torsion-free, then from the above equation we deduce Dαβγλ=Dβαγλ=Dγβαλ, i.e., D is symmetric.
Proposition 3.1.
Let D be the Douglas tensor of a Berwald endomorphism. Then iSD = 0 and Dric = 0.
Proof.
Let X˜=X˜αδα+X˜α¯𝒱α and Y˜=Y˜γδγ+Y˜γ¯𝒱γ. Since D is symmetric then using (3.5) we get
(iSD)(X˜,Y˜)=D(X˜,S)Y˜=yβX˜αY˜γDαβγλ.
Moreover, since hSB is homogeneous, we have yβ∂ℬαλ∂yβ=ℬαλ. Differentiating this with respect to yγ we obtain
yβ∂2ℬαλ∂yβ∂yγ=0.(3.7)
Differentiating (3.7) with respect to yμ gives us
yβ∂3ℬαλ∂yμ∂yβ∂yγ=∂2ℬαλ∂yμ∂yγ.(3.8)
But using (3.7) and (3.8) we deduce yβDαβγλ=0. Therefore we have iSD = 0. Now we prove the second part of assertion. It is easy to see that
Dric=Dαγ𝒳α⊗𝒳γ,
where
Dαγ=Dαλγλ. But using
(3.8) and
(3.6) we deduce
Dαγ = 0 and consequently
Dric = 0.
Theorem 3.1.
The Douglas tensor of a Berwald endomorphism is invariant under the projective changes of the associated spray.
Proof.
Let hSB be a Berwald endomorphism on £πE with associated spray S and D be the Douglas tensor of hSB. Also, let S¯ be the projective change of S by f˜. Then S¯ generates a Berwald endomorphism h¯SB. Denote by D¯ the Douglas tensor of h¯SB. If S = yα 𝒳α + Sα𝒱α
and S¯=yα𝒳α+S¯α𝒱α, then S¯=S+f˜C gives us
S¯α=Sα+yαf˜.(3.9)
From (2.16) and (2.17), hSB and h¯SB have the following coordinate expressions:
hSB=(𝒳α+ℬαγ𝒱γ)⊗𝒳α, h¯SB=(𝒳α+ℬ¯αγ𝒱γ)⊗𝒳α,
where
ℬαγ=12(∂Sγ∂yα−yβ(Lαβγ∘π)), ℬ¯αγ=12(∂S¯γ∂yα−yβ(Lαβγ∘π)).(3.10)
Using (3.9) and (3.10) we get
ℬ¯αγ=ℬαγ+f˜αγ,(3.11)
where
f˜αγ=12(f˜δαγ+yγ∂f˜∂yα). If we denote by
Dαβγλ and
D¯αβγλ the coefficients of
D and
D¯, respectively, then using
(3.6) and
(3.11) we get
D¯αβγλ=Dαβγλ+∂f˜αλ∂yβ∂yγ−1n+1(∂2f˜αμ∂yμ∂yγδβλ+∂3f˜βμ∂yα∂yμ∂yγyλ+∂2f˜αμ∂yμ∂yβδγλ+∂2f˜βμ∂yμ∂yγδαλ).(3.12)
Since f˜ is homogeneous of degree 1, then we can obtain
∂3f˜∂yβ∂yγ∂yαyβ=−∂2f˜∂yγ∂yα.
The above equation and direct calculation give us
∂2f˜αλ∂yβ∂yγ=12(∂2f˜∂yβ∂yγδαλ+∂2f˜∂yγ∂yαδβλ+∂2f˜∂yβ∂yαδγλ+∂3f˜∂yβ∂yγ∂yαyλ),(3.13)
∂2f˜αμ∂yμ∂yγ=12(n+1)∂2f˜˜∂yγ∂yα,(3.14)
∂2f˜αμ∂yμ∂yβ=12(n+1)∂2f˜˜∂yβ∂yα,(3.15)
∂2f˜βμ∂yμ∂yγ=12(n+1)∂2f˜˜∂yγ∂yβ,(3.16)
∂3f˜βμ∂yα∂yα∂yγ=12(n+1)∂3f˜˜∂yα∂yγ∂yβ.(3.17)
Setting (3.13)–(3.17) in (3.12) we obtain D¯αβγ=Dαβγλ, i.e., D¯=D.
4. ρ£-covariant derivatives in π*π
In this section, we introduce ρ£-covariant derivatives in π*π and we investigate geometric properties of ρ£-covariant derivatives in π*π like torsion and partial curvature. Results are in a deep relation with the Berwald derivative.
We can deduce the following double-exact short sequence from the double-exact short sequence (2.7)
0⇄Γ(π*π)⇄𝒱¯i¯Γ(£πE)⇄ℋ¯j¯Γ(π*π)⇄0,
such that for every
X¯∈Γ(π*π) and
ξ ∈ Γ(
£πE) the following hold
i¯(X¯):=i∘X¯, j¯(ξ):=j∘ξ, ℋ¯(X¯):=ℋ∘X¯, 𝒱¯(ξ):=𝒱∘ξ.
Proposition 4.1.
Let X belongs to Γ(E). Then we have the following
(i) i¯(X^)=XV,(iii) j¯(XC)=X^,(v) 𝒱¯(XV)=X^,(ii) j¯(XV)=0,(iv) ℋ¯(X^)=Xh,(vi) 𝒱¯(Xh)=0.
Proof.
Let u ∈ E. Then we have
i¯(X^)(u)=i∘X^=i(u,X(π(π)))=(0,X(π(u))u∨)=(0,X∨(u))=XV(u),
that gives us the first one. The second one is obvious. For the third, since
J(
XC) =
XV, then we have
i∘j(XC)=XV=i(X^). Because
i is injective,
j(XC)=X^ and consequently
j¯(XC)=X^. For the forth, we can deduce
ℋ¯(X^)=ℋ∘X^=F∘i∘X^=F∘XV=F∘J(XC)=h(XC)=Xh.
Using (2.11), the fifth equation proves as follows
𝒱¯(XV)=j∘F∘XV=j∘F∘J(XC)=j∘h(XC)=j∘XC=X^.
The last one is obvious.
Definition 4.1.
Operator ∇v
with properties
- (i)
- (ii)
- (iii)
(∇X¯vα¯)(Y¯):=ρ£(iX¯)(α¯(Y¯))−α¯(∇X¯vY¯),
is called the
canonical v-covariant differential, where
f˜∈C∞(E),
X¯,
Y¯∈Γ(π*π),
α¯∈Ω1(π).
Definition 4.2.
Let f˜ be a smooth function on E. Then the tensor field
∇v∇vf˜:=∇v(∇vf˜)∈𝒯20(π),
is said to be hessian of
f˜.
Proposition 4.2.
f˜∈C∞(E) is homogeneous of degree 1 if and only if∇δvf˜=f˜.
Proof.
Let f˜ be a homogeneous function of degree 1 on E. Then we have ρ£(C)f˜=f˜. Thus
∇δvf˜=ρ£(i¯δ)f˜=ρ£(i∘δ)f˜=ρ£(C)f˜=f˜.
From the above equation, also we can deduce the converse of assertion.
Proposition 4.3.
Let X and Y be sections of E andf˜∈C∞(E). Then
∇v∇vf˜(X^,Y^)=ρ£(XV)(ρ£(YV)f˜).
Moreover, the hessian of f˜ is symmetric.
Proof.
Using the definition of hessian of f˜, (i) of proposition 4.1 and (iii) of definition 4.1 we get
∇v∇vf˜(X^,Y^)=(∇X^v(∇vf˜))(Y^)=ρ£(iX^)((∇vf˜)(Y^))−∇vf˜(∇X^Y^)=ρ£(i¯X^)(ρ£(i¯Y^)f˜)−ρ£(i¯(∇X^Y^))f˜=ρ£(XV)(ρ£(YV)f˜)−ρ£(i¯(∇X^Y^))f˜.(4.1)
But using (i), (ii) and (iv) of proposition 4.1 we deduce
∇X^vY^=j¯[i¯X^,ℋ¯Y^]£=j¯[XV,Yh]£=0,
because [
XV,
Yh]
£ ∈ Γ(
V£πE). Plugging the above equation into
(4.1) implies the first part of assertion. Now, we prove the second part of the assertion. Since [
XV,
YV]
£ = 0, then using the first part of assertion we get
∇v∇vf˜(X^,Y^)=ρ£(XV)(ρ£(YV)f˜)=ρ£([XV,YV]£)(f˜)+ρ£(YV)(ρ£(XV)f˜)=ρ£(YV)(ρ£(XV)f˜)=∇v∇vf˜(Y^,X^).
Proposition 4.4.
Let f˜∈C∞(E) be a homogeneous function of degree 1. Then
∇δv(∇v∇vf˜)=−∇v∇vf˜.
Proof.
Setting A¯=∇v∇vf˜, we must show ∇δvA¯=−A¯. Let X and Y be sections of E. Then we have
(∇δvA¯)(X^,Y^)=ρ£(i¯δ)A¯(X^,Y^)−A¯(∇δvX^,Y^)−A¯(X^,∇δvY^).(4.2)
But using (ii) of Definition 4.1, we deduce
∇δvX^=j¯[i¯δ,ℋ¯Y^]£=j¯[C,Yh]£=0,
because [
C,
Yh]
£ ∈ Γ(
V£πE). Similarly we have
∇δvY^=0. Therefore
(4.2) reduces to the following
(∇δvA¯)(X^,Y^)=ρ£A¯(X^,Y^)=ρ£(C)(ρ£(XV)(ρ£(YV)f˜)).(4.3)
In other hand, using (ii) of (2.9) we get
A¯(X^,Y^)=ρ£(XV)(ρ£(YV)f˜)=ρ£([XV,C]£)(ρ£(YV)f˜)=[ρ£(XV),ρ£(C)](ρ£(YV)f˜)=ρ£(XV)(ρ£(C)(ρ£(YV)f˜))−ρ£(C)(ρ£(XV)(ρ£(YV)f˜))=ρ£(XV)([ρ£(C),ρ£(YV)]f˜+ρ£(YV)(ρ£(C)f˜))−ρ£(C)(ρ£(XV)(ρ£(YV)f˜))=ρ£(XV)(ρ£[C,YV]£f˜+ρ£(YV)(ρL(C)f˜))−ρ£(C)(ρ£(XV)(ρ£(YV)f˜)).
Since f˜ is homogeneous of degree 1, then we have ρ£(C)f˜=f˜. Setting this in the above equation and using (ii) of (2.9) we get
A¯(X^,Y^)=−ρ£(C)(ρ£(XV)(ρ£(YV)f˜)).(4.4)
From (4.3) and (4.4) we have the assertion.
Definition 4.3.
Let h be a horizontal endomorphism and ℋ¯ be a horizontal map of π associated to h. Operator ∇h
with properties
- (i)
- (ii)
- (iii)
(∇X¯hα¯)(Y¯):=ρ£(ℋ¯X¯)(α¯(Y¯))−α¯(∇X¯hY¯),
is called the canonical
h-covariant differential, where
f˜∈C∞(E),
X¯, Y¯∈Γ(π*π),
α¯∈Ω1(π).
Lemma 4.1.
Let H be the tension of h and X˜ be a section of £πE. Then
(∇hδ)(j¯X˜)=𝒱¯H(X˜).(4.5)
Proof.
Using (ii) of the above definition we get
(∇hδ)(j¯X˜)=∇j¯X˜hδ=𝒱¯[ℋ¯j¯X˜,i¯δ]£=𝒱¯[hX˜,C]£=𝒱¯[hX,C]£F−N(X˜)=𝒱¯H(X˜).
Since i𝒱¯=v, (4.5) gives us
i¯(∇hδ)(j¯X˜)=vH(X˜)=H(X˜).
By reason of the above relation, the (1, 1) tensor field
H¯=∇hδ is called the tension of the horizontal map
ℋ¯. Indeed, we have
H¯(X¯)=𝒱¯[ℋ¯X¯,C]£, ∀X¯∈Γ(π*π).(4.6)
Let A¯∈𝒯lk(π). Then we define
(∇X¯hA¯)(α1¯,α2¯,…,αk¯,X1¯,X2¯,…,Xl¯):=ρ£(ℋ¯X¯)(α1¯,α2¯,…,αk¯,X1¯,X2¯,…,Xl¯)−∑i=1kA¯(α1¯,…,∇X¯hαi¯,…,αk¯,X1¯,X2¯,…,Xl¯)−∑i=1lA¯(α1¯,α2¯,…,αk¯X1¯,…,∇X¯hXi¯,…,Xl¯).
Moreover, for
A¯∈𝒯lk(π), the tensor field
∇hA¯∈𝒯l+1k(π) is defined by the following rule
(∇hA¯)(X¯,α1¯,α2¯,…,αk¯,X1¯,X2¯,…,Xl¯):=(∇X¯hA¯)(α1¯,α2¯,…,αk¯,X1¯,X2¯,…,Xl¯).
Definition 4.4.
A map
{D:Γ(£πE)×Γ(π*π)→Γ(π*π),(X˜,Y¯)↦DX˜Y¯,
which satisfies
- (i)
- (ii)
DX˜f˜Z¯=f˜DX˜Z¯+ρ£(X˜)(f˜)Z¯,
- (iii)
is called a
ρ£-covariant derivative in Γ(π
*π).
Theorem 4.1.
Let h be a horizontal endomorphism and ℋ¯ be a horizontal map of π associated to h. Then
∇:Γ(£πE)×Γ(π*π)→Γ(π*π),
given by
∇X˜Y¯:=∇𝒱¯X˜vY¯+∇j¯X˜hY¯,(4.7)
is a ρ£-covariant derivative in Γ(π
*π),
whereX˜∈Γ(£πE) and Y¯∈Γ(π*π).
Proof.
Let f˜∈C∞(E). Then we have
∇X˜f˜Y¯=∇𝒱¯X˜vf˜Y¯+∇j¯X˜hf˜Y¯=ρ£(i¯𝒱¯X˜)f˜+ρ£(ℋ¯j¯X˜)f˜+f˜∇𝒱¯X˜vfY¯+∇j¯X˜hfY¯=ρ£(i¯𝒱¯X˜)f˜+ρ£(ℋ¯j¯X˜)f˜+f˜∇X˜Y¯.
It is easy to show that i¯𝒱¯X˜=vX˜ and ℋ¯j¯X˜=hX˜. Therefore the above equation gives us
∇X˜f˜Y¯=ρ£(vX˜)f˜+ρ£(hX˜)f˜+f˜∇X˜Y¯=ρ£(X˜)f˜+f˜∇X˜Y¯.
Similarly we can show ∇X˜(Y¯+Z¯)=∇X˜Y¯+∇X˜Z¯ and ∇f˜X˜+Y˜Y¯=f˜∇X˜Z¯+∇Y˜Z¯. Therefore ∇ is a ρ£-covariant derivative in Γ(π*π).
The ρ£-covariant derivative ∇ introduced by the above theorem is called Berwald derivative generated by h. Indeed the Berwald derivative is as follows:
∇X˜Y¯=j¯[vX˜,ℋ¯Y¯]£+𝒱¯[hX˜,iY¯]£, ∀X˜∈Γ(£πE), ∀Y¯∈Γ(π*,π).
Using the above equation we can obtain
∇XVY^=0, ∇XhY^=𝒱¯[Xh,YV]£,(4.8)
∇iX¯Y¯=j¯[i¯X¯,ℋY¯]£, ∇ℋX¯Y¯=𝒱¯[ℋX¯,iY¯]£,(4.9)
where
X and
Y are sections of
E and
X¯, Y¯∈Γ(π*π).
Now we consider the local basis {eα} of Γ(E). Then {eα^} is a basis of Γ(π*π), where eα^(u)=(u,eα(π(u))), for all u ∈ E. Using (2.4), Proposition 4.1, and the definition of j, it is easy to check that
ℋ¯eα^=δα, i¯eα^=𝒱α, j¯(δα)=eα^, 𝒱¯(𝒱α)=eα^.(4.10)
Also we deduce 𝒱¯(δα)=0. Therefore using the above equation, (2.4) and (4.8) we obtain
∇δαeβ^=𝒱¯[δα,eβV]£=𝒱¯[δα,𝒱β]£=−∂ℬαγ∂yβeγ^,∇𝒱αeβ^=j¯[𝒱α,eβh]£=j¯[𝒱α,δβ]£=0,
and consequently
∇X˜Y¯=(X˜α((ραi∘π)∂Y¯β∂xi+ℬαγ∂Y¯β∂yγ)−X˜αY¯γ∂ℬαβ∂yγ+X˜α¯∂Y¯β∂yα)eβ^,(4.11)
where
X˜=X˜αδα+X˜α¯𝒱α∈Γ(£πE) and
Y¯=Y¯βeβ^∈Γ(π*π).
Definition 4.5.
A ρ£-covariant derivative operator D in Γ(π*π) is said to be associated to the horizontal map ℋ¯ if Dδ=𝒱¯.
Lemma 4.2.
Let ∇ be the Berwald derivative induced by h. Then
∇δ=H¯∘j¯+𝒱¯.(4.12)
Proof.
Using (ii) of Definition 4.1, (ii) of Definition 4.3 and (4.6) we get
(∇δ)(X˜)=∇𝒱¯X˜vδ+∇j¯X˜hδ=j¯[i¯𝒱¯X˜,ℋ¯δ]£+H¯(j¯X˜)=j[vX˜,ℋ¯δ]£+H¯(j¯X˜).(4.13)
Now let X˜=X˜αδα+X˜α¯𝒱α. It is easy to see that δ=yαeα^. Then using (4.10) we obtain
j¯[vX˜,ℋ¯δ]£=j¯[X˜α¯𝒱α,yβδβ]£=X˜α¯j¯(δα)=X˜α¯eα^=𝒱¯(X˜).
Setting the above equation in (4.13) implies (4.12).
Proposition 4.5.
Let S be a spray on £pE and hSB be the Berwald endomorphism generated by it. If ℋ¯ is the horizontal map generated by hSB and ∇ is the Berwald derivative induced by hSB, then ∇Sδ = 0.
Proof.
From the above lemma we have
∇Sδ=H¯j¯(S)+𝒱¯(S).
Using
(4.10) it is easy to see that
j¯S=yαeα^=δ. Thus we have
∇Sδ=H¯δ+𝒱¯(S).(4.14)
But (4.6) gives us H¯δ=𝒱¯[ℋ¯δ,C]£. On the other hand, from Lemma 2.1 we have hSBS=S. Therefore we get
S=hSBS=ℋ¯j¯S=ℋ¯δ,
and consequently
H¯δ=𝒱¯[S,C]£. Since
S is a spray, [
S,
C]
£ = −
S. Therefore
H¯δ=−𝒱¯(S). Setting this equation in
(4.14) we obtain ∇
Sδ = 0.
4.1. Torsions and partial curvatures
Let D be a ρ£-covariant derivative in Γ(π*π). The (π*π)-valued two-forms
Th(D)(X˜,Y˜):=DX˜j¯Y˜−DY˜j¯X˜−j¯[X˜,Y˜]£,Tv(D)(X˜,Y˜):=DX˜𝒱¯Y˜−DY˜𝒱¯X˜−𝒱¯[X˜,Y˜]£,
are said to be the horizontal and the vertical torsions of
D, respectively, where
X˜ and
Ỹ belong to Γ(
£πE). The maps
A and
B given by
A(X¯,Y¯):=Th(D)(ℋ¯X¯,ℋ¯Y¯), B(X¯,Y¯):=Th(D)(ℋ¯X¯,i¯Y¯),(4.15)
where
X¯,
Y¯∈Γ(π*π), are called the
h-horizontal and the
h-mixed torsions of
D (with respect to
ℋ¯), respectively.
A will also be mentioned as the torsion of
D, while for
B we use the term Finsler torsion as well.
D is said to be symmetric if
A = 0 and
B is symmetric. The maps
R1,
P1 and
Q1 given by
R1(X¯,Y¯):=Tv(D)(ℋ¯X¯,ℋ¯Y¯), P1(X¯,Y¯):=Tv(D)(ℋ¯X¯,i¯Y¯),(4.16)
Q1(X¯,Y¯):=Tv(D)(i¯X¯,i¯Y¯), ∀X¯,Y¯∈Γ(π*π),(4.17)
are called the
v-horizontal, the
v-mixed and the
v-vertical torsions of
D, respectively. Using
(4.9),
(4.15),
(4.16) and
(4.17) we can obtain
Lemma 4.3.
Let D be a ρ£-covariant derivative in Γ(π*π). Then all of the partial torsions of the ρ£-covariant derivative operator D are tensor fields of type (1, 2) on Γ(π*π). Moreover, for any vector fields X¯, Y¯ belong Γ(π*π) we have
A(X¯,Y¯)=Dℋ¯X¯Y¯−Dℋ¯Y¯X¯−j[ℋ¯X¯,ℋ¯Y¯]£,B(X¯,Y¯)=−Di¯Y¯X¯−j¯[ℋ¯X¯,i¯Y¯]£=−DiY¯X¯+∇i¯Y¯X¯,R1(X¯,Y¯)=−𝒱¯[ℋ¯X¯,ℋ¯Y¯]£,P1(X¯,Y¯)=Dℋ¯X¯Y¯−𝒱¯[ℋ¯X¯,i¯Y¯]£=Dℋ¯X¯Y¯−∇ℋ¯X¯Y¯,Q1(X¯,Y¯)=Di¯X¯Y¯−Di¯Y¯X¯−𝒱¯[i¯X¯,i¯Y¯]£,
where ∇
is the Berwald derivative given by (4.7).
Corollary 4.1.
A r£-covariant derivative in Γ(π*E) is the Berwald derivative induced by a given horizontal endomorphism if and only if, its Finsler torsion and v-mixed torsion vanish.
Using the above lemma we get
A(j¯X˜,j¯Y˜)=DhX˜j¯Y˜−DhY˜j¯X˜−j[hX˜,hY˜]£,B(j¯X˜,𝒱¯Y˜)=−DvY˜j¯X˜−j[hX˜,vY˜]£,−B(j¯Y˜,𝒱¯X˜)=DvX˜j¯Y˜+j[hY˜,vX˜]£.
Since [vX˜,vY˜]∈Γ(V£πE), then j¯[vX˜,vY˜]=0. Therefore summing the above equations give us
A(j¯X˜,j¯Y˜)+B(j¯X˜,𝒱¯Y˜)−B(j¯Y˜,𝒱¯X˜)=DX˜j¯Y˜−DY˜j¯X˜−j[X˜,Y˜]£=Th(D)(X˜,Y˜).
Thus we have
Lemma 4.4.
The horizontal torsion Th(D) is completely determined by the torsion A and the Finsler torsion B. Indeed, we have
Th(D)(X˜,Y˜)=A(j¯X˜,j¯Y˜)+B(j¯X˜,𝒱¯Y˜)−B(j¯Y˜,𝒱¯X˜), ∀X˜,Y˜∈Γ(£πE).
Lemma 4.5.
Let D be a r£-covariant derivative in Γ(π*π). If D is associated to the horizontal map ℋ¯, then for every section X¯ of π*π we have
B(δ,X¯)=0, P1(X¯,δ)=−H¯(X¯).
Proof.
Since D is associated to the horizontal map ℋ¯, then Dδ=𝒱¯. Therefore using Lemma 4.3 we get
B(δ,X¯)=−Di¯X¯δ−j¯[ℋ¯δ,i¯X¯]£=−𝒱¯(i¯X¯)−j¯[ℋ¯δ,i¯X¯]£=−X¯−j¯[ℋ¯δ,i¯X¯]£.(4.18)
Now let X¯=X¯αeα^. Then we deduce i¯X¯=X¯α𝒱α and consequently
j¯[ℋ¯δ,i¯X¯]£=j¯[yαδα,X¯β𝒱β]£=−j¯(X¯αδα)=−X¯αeα^=−X¯.
Setting the above equation in (4.18) we derive that B(δ,X¯)=0. Using (4.12) and Lemma 4.3 we get
P1(X¯,δ)=Dℋ¯X¯δ−∇ℋ¯X¯δ=𝒱ℋ¯X¯−(H¯∘j¯+𝒱¯)(ℋ¯X¯)=−H¯∘j¯(ℋ¯X¯).
But we have jℋ¯=1Γ(π*π). Therefore the above equation gives us the second part of the assertion.
Definition 4.6.
Let D be a ρ£-covariant derivative in Γ(π*π). Then the maps R, P and Q given by
R(X¯,Y¯)Z¯:=KD(ℋ¯X¯,ℋ¯Y¯)Z¯,P(X¯,Y¯)Z¯:=KD(ℋ¯X¯,i¯Y¯)Z¯,Q(X¯,Y¯)Z¯:=KD(i¯X¯,i¯Y¯)Z¯,
are said to be the horizontal or Riemann curvature, the mixed or Berwald curvature and the vertical or Berwald-Cartan curvature of
D (with respect to
ℋ¯), respectively.
Lemma 4.6.
Let D be a ρ£-covariant derivative in Γ(π*π). If D is associated to the horizontal map ℋ¯, then we have
R(X¯,Y¯)δ=R1(X¯,Y¯), P(X¯,Y¯)δ=P1(X¯,Y¯), Q(X¯,Y¯)δ=Q1(X¯,Y¯),
where X¯,
Y¯∈Γ(π*π).
Moreover, if the Finsler torsion is symmetric, then Q(·,·)
δ =
Q1 = 0.
Proof.
Since D is associated to the horizontal map ℋ¯, then Dδ=𝒱¯ and therefore
Dℋ¯X¯δ=0, Di¯X¯δ=X¯, ∀X¯∈Γ(π*π).
Using the above equations, the proof of the first part of the assertion is obvious. Now we prove the second part. From the first part we have
Q(X¯,Y¯)δ=Q1(X¯,Y¯)=Di¯X¯Y¯−Di¯Y¯X¯−𝒱¯[i¯X¯,i¯Y¯]£.
Since the Finsler torsion B is symmetric, then
0=B(X¯,Y¯)−B(Y¯,X¯)=Di¯X¯Y¯−Di¯Y¯X¯−j[ℋ¯X¯,i¯Y¯]£+j[ℋ¯Y¯,i¯X¯]£.
Two above equations give us
Q(X¯,Y¯)δ=j¯[ℋ¯X¯,i¯Y¯]£−j¯[ℋ¯Y¯,i¯X¯]£−𝒱¯[i¯X¯,i¯Y¯]£.
Since j¯ is surjective, there exist X˜, Ỹ ∈ Γ(£πE) such that X¯=j¯X˜ and Y¯=j¯Y˜. Setting these equations in the above equation imply
Q(j¯X˜,j¯Y˜)δ=j¯[hX˜,JY˜]£−j¯[hY˜,JX˜]£−𝒱¯[JX˜,JY˜]£,
and consequently
i¯(Q(j¯X˜,j¯Y˜)δ)=J[hX˜,JY˜]£−J[hY˜,JX˜]£−v[JX˜,JY˜]£=J[X˜,JY˜]£+J[JX˜,Y˜]£−[JX˜,JY˜]£=−NJ(X˜,Y˜)=0.
Since ī is injective, the above equation gives us Q(j¯X˜,j¯Y˜)δ=0 and therefore Q(X¯,Y¯)δ=0.
Now we denote the torsions and the curvatures of the Berwald derivative ∇, by A∘, B∘, R1∘, P1∘, Q1∘ and R∘, P∘, Q∘, respectively. Using (4.11) and Lemma 4.3 it is easy to prove the following
Lemma 4.7.
Let ∇ be the Berwald derivative induced by h and {eα} be a basis of E. Then
A∘=12tαβγeα^∧eβ^⊗eγ^,(4.19)
R∘1=−12Rαβγeα^∧eβ^⊗eγ^,(4.20)
B∘=0, P1∘=0, Q1∘=0,(4.21)
where {eα^} is a dual basis of {eα^} and tαβγ and Rαβγ are given by (2.13) and (2.15).
Using A∘ and R1∘ we introduce the following tensor fields:
{A∘∘:Γ(£πE)×Γ(£πE)→Γ(£πE)A∘∘(X˜,Y˜)=i¯A∘(j¯X˜,j¯Y˜),(4.22)
{R1∘∘:Γ(£πE)×Γ(£πE)→Γ(£πE)R1∘∘(X˜,Y˜)=i¯R1∘(j¯X˜,j¯Y˜).(4.23)
Using (4.19) and (4.22) we can obtain
A∘∘(δα,δβ)=tαβγ𝒱γ, A∘∘(𝒱α,δβ)=A∘∘(𝒱α,𝒱β)=0.
Therefore from (2.12) we deduce
A∘∘=12tαβγ𝒳α∧𝒳β⊗𝒱γ=t,
where
t is the weak torsion of
h. Similarly using
(4.20) and
(4.23) we obtain
R1∘∘=−12Rαβγ𝒳α∧𝒳β⊗𝒱γ=Ω,
where Ω is the curvature of
h given in
(2.14).
Proposition 4.6.
Let ∇ be the Berwald derivative induced by h in Γ(π*π). Then A∘∘=t and R1∘=Ω∘, where t and Ω are weak torsion and curvature of h, respectively.
Using (4.10), (4.11) and Definition 4.6 we can deduce
Theorem 4.2.
Let ∇ be the Berwald derivative induced by h in Γ(π*π) and {eα} be a basis of E. Then
R∘=R∘αβγλeλ^⊗eα^⊗eβ^⊗eγ^,P∘=P∘αβγλeλ^⊗eα^⊗eβ^⊗eγ^,Q∘=S∘αβγλeλ^⊗eα^⊗eβ^⊗eγ^,
where
R∘αβγλ=−(ραi∘π)∂2ℬβλ∂xi∂yγ−ℬβμ∂2ℬβλ∂yμ∂yγ+(ρβi∘π)∂2ℬαλ∂xi∂yγ+ℬβμ∂2ℬαλ∂yμ∂yγ+∂ℬβμ∂yγ∂ℬαλ∂yμ−∂ℬαμ∂yγ∂ℬβλ∂yμ+(Lαβμ∘π)∂ℬμλ∂yγ,(4.24)
P∘αβγλ=∂2ℬαλ∂yβ∂yγ,(4.25)
S∘αβγλ=0.(4.26)
Using R∘, P∘ and Q∘ we introduce the following tensor fields:
{R∘∘:Γ(£πE)×Γ(£πE)→Γ(£πE),R∘∘(X˜,Y˜)=i¯R∘(j¯X˜,j¯Y˜),(4.27)
{P∘∘:Γ(£πE)×Γ(£πE)→Γ(£πE),P∘∘(X˜,Y˜)=i¯P∘(j¯X˜,j¯Y˜),(4.28)
{Q∘∘:Γ(£πE)×Γ(£πE)→Γ(£πE),Q∘∘(X˜,Y˜)=i¯Q∘(j¯X˜,j¯Y˜).(4.29)
Using the above theorem, (3.1)–(3.3) and (4.27)–(4.29) we derive that
Proposition 4.7.
Let ∇ be the Berwald derivative induced by h. Then
R∘∘=RB, P∘∘=PB, Q∘∘=QB,
where RB,
PB and QB are the horizontal, mixed and vertical curvatures of the Berwald-type connection (DB,h), respectively.
Proposition 4.8.
Let ∇ be the Berwald derivative induced by h. Then for sections X, Y and Z of E we have
P∘(X^,Y^)Z^=𝒱¯[[Xh,YV]£,ZV]£.
Proof.
Let X = Xαeα, Y = Yβ eβ and Z = Zγeγ be sections of E. Then we have X^=(Xα∘π)e^α, Ŷ = (Yβ π)êβ and Ẑ = (Zγ π)êγ. Therefore, (4.25) implies
P∘(X^,Y^)Z^=((XαYβZγ)∘π)∂2ℬαλ∂yβ∂yγe^λ.
Similarly we can obtain
𝒱¯[[Xh,YV]£,ZV]£=𝒱¯[[(Xα∘π)δα,(Yβ∘π)𝒱β]£,(Zγ∘π)𝒱γ]£((XαYβZγ)∘π)∂2ℬαλ∂yβ∂yγe^λ.
Two above equations give us the assertion.
Definition 4.7.
The covariant derivative operator D given by
DX˜Y¯:=∇X˜Y¯+1n+1(trP∘(j¯X˜,Y¯))δ,(4.30)
is called the Yano derivative induced by
ℋ¯, where
P∘ is the mixed curvature of the Berwald derivative ∇.
Using (4.11) and (4.30) we get
DX˜Y¯=(X˜α((ραi∘π)∂Y¯β∂xi+ℬαγ∂Y¯β∂yγ)−X˜αY¯γ∂ℬαβ∂yγ+X˜α¯∂Y¯β∂yα+1n+1X˜αY¯γyβ∂2ℬαλ∂yλ∂yγ)e^β,
where
X˜=X˜αδα+X˜α¯𝒱α∈Γ(£πE) and
Y¯=Y¯βe^β∈Γ(π*π). In particular case we have
Dδαe^β=(1n+1yγ∂2ℬαλ∂yλ∂yβ−∂ℬαγ∂yβ)e^γ, D𝒱αe^β=0.
5. Application to optimal control
Let us consider the following driftless control affine system with quadratic cost in the space ℝ3:
{x˙1=u2x˙2=u1+u2x2,x˙3=u1+u2x3(5.1)
min∫0Tℒ(u(t))dt, ℒ(u)=12((u1)2+(u2)2),
where
x˙i=dxidt and
u1,
u2 are control variables. We are looking for the optimal trajectories starting from the point (0, 1, 0)
t and parameterized by arclength (minimum time problem) and free endpoint. From the system of differential equations we get
u1 =
x˙2 −
x˙1x2,
u2 =
x˙1 and it results the Lagrangian
ℒ(x,x˙)=12((x˙2−x˙1x2)2+(x˙1)2),
with holonomic constraint
x˙2−x˙3=x˙1(x2−x3),
which leads to the equation ln|
x2 −
x3| =
x1 +
c. Next, using the Lagrange multiplier
λ =
λ(
t) we obtain the total Lagrangian (including the constraints) given by
L(x,x˙)=ℒ(x,x˙)+λ(x˙2−x˙3−x˙1x2+x˙1x3)=12((x˙2−x˙1x2)2+(x˙1)2)+λ(x˙2−x˙3−x˙1x2+x˙1x3).
We notice that the Hessian matrix of L is singular on tangent bundle Tℝ3, and L is a degenerate Lagrangian (not regular). The corresponding Euler-Lagrange equations on tangent bundle lead to a complicated system of second-order differential equations. Moreover, because the Lagrangian is not regular, we cannot obtain the explicit coefficients of the semispray S from the symplectic equation iSωL = −dEL and it is difficult to find the coefficients of the nonlinear connection induced by Lagrangian function in this case. We will use a different approach, considering the framework of Lie algebroids.
The system can be written in the form
x˙=u1X1+u2X2, x=(x1x2x3)∈ℝ3, X1=(011), X2=(1x2x3),min∫0Tℒ(u(t))dt, ℒ(u)=12((u1)2+(u2)2).(5.2)
The vector fields are given by
X1=∂∂x2+∂∂x3, X2=∂∂x1+x2∂∂x2+x3∂∂x3.
The Lie bracket is
[X1,X2]=[∂∂x2+∂∂x3,∂∂x1+x2∂∂x2+x3∂∂x3]=X1,
and it results that the associated distribution Δ =
span{
X1,
X2} is holonomic and has the constant rank 2. Moreover, from the system
(5.1) we obtain
x˙2 −
x˙3 =
x˙1(
x2 −
x3) which yields
ln|x2−x3|=x1+c.(5.3)
(
c is a constant) and it results that Δ determines a foliation on ℝ
3 given by the surfaces
(5.3) and two points can be joined by a optimal trajectory if and only if they are situated on the same leaf. In order to use the framework of Lie algebroids, we consider
E = Δ (holonomic distribution with constant rank), the anchor
ρ :
E →
TM is the inclusion and [,]
E the induced Lie bracket. In the case of our application, the anchor
ρ has the components
ραi=(011x11x2),
Using the structure equation on Lie algebroid
[Xα,Xβ]=LαβγXγ, α,β,γ=1,2,
we obtain the non-zero structure functions
L121=1, L211=−1.
The components of the semispray [10]
Sɛ=gɛβ(ρβi∂ℒ∂xi−ραi∂2ℒ∂xi∂uβyα−Lβαγuα∂ℒ∂uγ),
induced by the Lagrangian
ℒ(u)=12((u1)2+(u2)2) on
E are given by
S1=−u1u2, S2=(u1)2.
The functions Sα are homogeneous of degree 2 in u and it results that S is a spray. The coefficients of the canonical nonlinear connection ℬ = −[S, J]E given by (2.17)
ℬαβ=12(∂Sβ∂uα−uɛLαɛβ),
have the form
ℬ11=−u2, ℬ21=0, ℬ12=u1, ℬ22=0.
Also, the non-zero coefficients of the curvature from (2.15) of nonlinear connection ℬ are given by
R121=u2, R122=u1, R211=−u2, R212=−u1.
The Berwald connection is given by
Dδ1Bδ1=−δ2, Dδ1Bδ2=−δ1,
and the Douglas tensor has the components equal to zero.
The Euler-Lagrange equations on Lie algebroids given by (see [24])
dxidt=ραiuα, ddt(∂ℒ∂uα)=ραi∂ℒ∂xi−Lαβɛuβ∂ℒ∂uɛ,
lead to the following differential equations
u˙1=−u1u2, u˙2=(u1)2,
which can be written in the form
dxidt=ραiuα, duαdt=Sα(x,u),
and give the integral curves of the spray
S.
In order to solve this optimal control problem we can use the Pontryagin Maximum Principle on the cotangent bundle. The Hamiltonian function has the form
H(u,x,p)=pix˙i−ℒ=p1u2+p2(u1+u2x2)+p3(u1+u2x3)−12((u1)2+(u2)2),
and with the equations
∂H∂ui=0 we obtain that
p2 +
p3 =
u1 and
p1 +
p2x2 +
p3x3 =
u2, which replaced into the expression of Hamiltonian, yields
H(x,p)=12((p2+p3)2+(p1+p2x2+p3x3))2.(5.4)
The Hamilton equations on the cotangent bundle
dxidt=∂H∂pi,dpidt=−∂H∂xi,
lead to a very complicated system of implicit differential equations. We can use a different approach, involving the framework of Lie algebroids. First, we need by the following result [
14]:
Proposition 5.1.
The relation between the Hamiltonian H on the cotangent bundle T*M and the Hamiltonian ℋ on dual bundle E* is given by
H(x,p)=ℋ(ρ*(p)), μ=ρ*(p), p∈Tx*M, μ∈Ex*.
Proof.
The Fenchel-Legendre dual of Lagrangian L is the Hamiltonian H given by
H(x,p)=supv{〈p,v〉−L(v)}=supv{〈p,v〉−ℒ(u);ρ(u)=v}=supv{〈p,ρ(u)〉−ℒ(u)}=supu{〈ρ*(p),u〉−ℒ(u)}=ℋ(ρ*(p)),
and we get
H(x,p)=ℋ(μ), μ=ρ*(p),
or locally
μα=ρα*ipi,(5.5)
where the Hamiltonian
H(
x,
p) is degenerate on Ker
ρ★ ⊂
T*M.
Using the Legendre transformation associated to the regular Lagrangian ℒ=12((u1)2+(u2)2) on Lie algebroid E, we can obtain the nondegenerate (regular) Hamiltonian ℋ on E* in the form
ℋ=12(μ12+μ22).
Using (5.5) we find the Hamiltonian H from (5.4) on the cotangent bundle given by H(x, p) = ℋ(μ) with μα=ρα*ipi, α = 1, 2
(μ1μ2)=(0111x1x2)(p1p2p3).
Next, from Hamilton equations on Lie algebroid [4]
dxidt=ραi∂ℋ∂μα, dμαdt=−ραi∂ℋ∂xi−μγLαβγ∂ℋ∂μβ,
we deduce that
x˙1=μ2, x˙2=μ1+x2μ2, x˙3=μ1+x3μ2,μ˙1=−μ1μ2, μ˙2=μ12.
The form of the last relations leads to the following change of variables
μ1(t)=r(t)sechθ(t), μ2(t)=r(t)tanhθ(t),(5.6)
where
sinhθ=eθ−e−θ2, coshθ=eθ+e−θ2, tanhθ=sinhθcoshθ, sechθ=1coshθ.
The differential equations
μ˙1=−μ1μ2,
with the relations
(5.6) yields
r˙r−θ˙tanhθ=−rtanhθ.(5.7)
Also, from the equation
μ˙2=μ12,
and
(5.6) we get
r˙rtanhθ+θ˙sech2θ=rsech2θ).(5.8)
Now, reducing θ˙ from the equations (5.7) and (5.8), we obtain
r˙=0⇒r=c, c∈ℝ,
and
θ˙=r.
Since the optimal trajectories are parameterized by arclength (minimum time problem) the conclusion corresponds exactly to the 1/2 level of the Hamiltonian and we have
ℋ=r22=12,
which yields
r=1, θ=t.
The equation
μ˙1=−μ1x˙1,
implies that
x1(t)=lnc1secht, c1∈ℝ.
Since we are looking for the trajectories starting from the point (0, 1, 0)t, we have x1(0) = 0 and
lnc1=0⇒c1=1,
which leads to
x1(t)=ln1secht=lncosht.
We obtain also that
μ˙2=μ1(x˙2−x2μ2)=μ1x˙2+x2μ˙1,
and, consequently,
μ2 =
μ1x2 +
c2. Further,
x2(t)=sinht+c2secht.
From x2(0) = 1 we obtain that c2 = 1 and this yields
x2(t)=sinht+cosht.
In the same way we get
x3(t)=sinht+c3secht.
From x3(0) = 0 we obtain that c3 = 0 and it results
x3(t)=sinht.
Using (5.1) we have u2 = x˙1, u1 = x˙3 − u2x3 = x˙2 − u2x2 and by direct computation, we obtain the control variables
u2(t)=sinhtcosht, u1(t)=1cosht.
Finally, we obtain the solution of driftless control affine systems given by
x1(t)=lncosht, x2(t)=sinht+cosht, x3(t)=sinht.
The solution is optimal because the Hamiltonian function is convex.
References
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