Vol. 23, No. 1, April 2017, pp.1–12.
ON INFINITESIMAL PROJECTIVE
TRANSFORMATIONS OF THE TANGENT BUNDLE
WITH THE COMPLETE LIFT OF A FINSLER METRIC
B. Bidabad
1, M. M. Rezaii
2and M. Zohrehvand
31
Faculty of Mathematics and Computer Science, Amirkabir University of Technology, Tehran, Iran
Faculty of Mathematics and Computer Science, Amirkabir University of Technology, Tehran, Iran
Faculty of Mathematical Sciences and Statistics, Malayer University, Malayer, Iran
Abstract. Let (M, g) be a Finsler manifold andT M0 its slit tangent bundle with the complete lift metric ˜g. In this paper, we prove that every infinitesimal complete lift projective transformation on (T M0,˜g), is an infinitesimal affine transformation. Moreover, if (M, g) is a Landsberg manifold, then there is a one-to-one correspon-dence between infinitesimal complete lift projective transformations on (T M0,˜g) and infinitesimal affine transformations on (M, g).
Key words and Phrases: Finsler manifold, Complete lift metric, Tangent bundle, Infinitesimal projective transformation.
Abstrak. Misalkan (M, g) adalah manifold Finsler dan T M0 adalah bundel tan-gen slit-nya dengan metrik lift lengkap ˜g. Pada makalah ini, kami membuktikan bahwa setiap transformasi projektif lift lengkap infinitesimal pada (T M0,˜g), adalah transformasi afin infinitesimal. Lebih jauh, jika (M, g) adalah manifold Landsberg, maka terdapat korespondensi satu-satu antara transformasi projektif lift lengkap infinitesimal pada (T M0,˜g) dengan transformasi afin infinitesimal pada (M, g).
Kata kunci: Manifold Finsler, Metric lift lengkap, Tangent bundle, Infinitesimal projective transformation.
2000 Mathematics Subject Classification: 53B40, 53B21, 53C60. Received: 5-Jan-2017, revised: 1-Feb-2017, accepted: 1-Feb-2017.
B. Bidabad
1. Introduction
Let (M, g) be a Riemannian manifold and φa transformation onM. Then,
φ is called a projective transformation if it preserves the geodesics as point sets. Also, an affine transformation may be characterized as a projective transformation which preserves geodesics with the affine parameter.
LetV be a vector field onM and{φt}the local one-parameter group gener-ated byV. Then,V is called an infinitesimal projective (affine) transformation on
M if everyφt is a projective (affine) transformation.
Let ˜φ be a transformation of T M, the tangent bundle of M; then, ˜φ is called a fiber-preserving transformation if it preserves the fibers. A vector field ˜X
on T M with the local one-parameter group {φ˜t} is called an infinitesimal fiber-preserving transformation on T M if each ˜φt is a fiber-preserving transformation. Infinitesimal fiber-preserving transformations is an important class of infinitesimal transformations on T M which include infinitesimal complete lift transformations as a special subclass(refer to subsection 2.2 or [4]).
It is well-known that there are some lift metrics onT M as follows: complete lift metric or g2, diagonal lift metric or Sasaki metric or g1+g3, lift metric I+II org1+g2 and lift metic II+III org2+g3, whereg1:=gijdxidxj,g2:= 2gijdxiδyj andg3:=gijδyiδyj are all bilinear differential forms defined globally onT M. For more details one can refer to [17].
The problems of exsisting special infinitesimal transformations on the tan-gent bundle of a Riemannian manifold with some lift metrics are considered by several authors, e.g. [5, 6, 7, 8, 9] and [12, 13, 14, 15]. The study shows that the special infinitesiaml transformations onT M might lead to some global results. For example, in [7] and [15], it is proved that ifT M with the complete lift metric or the lift metric I+II admits an essential infinitesimal conformal1 transformation, then
M is isomorphic to the standard sphere. Also, it is proved in [12], that ifT M with the complete lift metric admits a non-affine infinitesimal projective transformation, thenM is locally flat. Therefore, it is meaningful to the study of the infinitesimal transformations on the tangent bundleT M.
Yamauchi in [14], proved the following theorem:
Theorem A:LetMbe a non-Euclidean complete n-dimensional Riemannian manifold, and letT M be its tangent bundle with the complete lift metric. Then, every infinitesimal fiber-preserving projective transformation on T M is an affine one and it naturally induced an infinitesimal affine transformation on M.
Special infinitesimal transformations on the tangent bundle of a Finsler man-ifold are considered by many authors, e.g. [1, 2, 4, 10].
1A vector field ˜Xon (T M,˜g) is called an essential infinitesimal conformal transformation if there
exsits a scalar function Ω on T M such that Ω depends only yi
with respect to the induced coordinate (xi
, yi
In this paper, the infinitesimal fiber-preserving projective transformations on the tangent bundle of a Finsler manifold with the complete lift metric are consid-ered. Then, as a special case, the infinitesimal complete lift projective transforma-tions are studied and the following theorem is proved.
Theorem 1.1. Let (M, g)be a C∞ connected Finsler manifold and
T M0 its slit
tangent bundle with the complete lift metric ˜g. Then, every infinitesimal complete lift projective transformation on (T M0,g˜)is affine and it naturally induced an
in-finitesimal affine transformation on (M, g).
Thus, Theorem A is true for the Finsler manifold and the infinitesimal com-plete lift transformations. From Theorem 1.1, the following corollary can be imme-diately found.
Corollary 1.2. The Lie algebra of complete lift projective vector field on(T M0,˜g)
is reduced to an affine one.
The Landsberg manifolds form an important class of the Finsler manifolds which include the Berwald manifolds. In the next theorem, it is shown that the inverse of Theorem 1.1 is true for the Landsberg manifolds.
Theorem 1.3. Let (M, g) be an n-dimensional Landsberg manifold and T M0 its
slit tangent bundle with a complete lift metricg˜. Then, every infinitesimal complete lift transformationVc on(T M
0,g˜)is projective if and only ifV is an infinitesimal
affine transformation on (M, g).
2. Preliminaries
Let M be a real n-dimensional C∞
manifold and T M its tangent bundle. The elements ofT M are denoted by (x, y) withy∈TxM. AlsoT M0=T M\ {0} be the slit tangent bundle of M. The natural projection π :T M0 → M is given byπ(x, y) :=x. AFinsler structureonM is a function ofF :T M →[0,∞) with the following properties; (i) F isC∞ on
T M0, (ii)F is positively 1-homogeneous on the fibers of tangent bundle T M and (iii) the Hessian g of F2 with elements
gij(x, y) :=12[F2(x, y)]yiyj is positive-definite. In the sequel of this paper, a Finsler manifold with a Finsler structureF will be denoted by (M, g) instead of (M, F).
LetVvT M :=kerπ∗vbe the set of the vectors tangent to the fiber throughv∈
T M0. Then, the vertical vector bundle onMis defined byV T M :=S
v∈T M0VvT M.
A non-linear connection or a horizontal distribution on T M0 is a complementary distribution HT M forV T M onT T M0. Therefore, by using a non-linear connec-tion, the following decomposition is resulted:
T T M0=V T M⊕HT M, (1) where HT M is a vector bundle completely determined by the non-linear differ-entiable functions Ni
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connectionHT M. The pair (HT M,∇) is called a Finsler connection on the man-ifold M, where ∇ a linear connection to V T M. Indeed, a Finsler connection is a triple (Ni
j, Fijk, Cijk) whereNij are the coefficients of a nonlinear connection,
Fi
jk and Cijk are the horizontal part and the vertical part of this connection, respectively[3].
Using the local coordinates (xi, yi) onT M we have the local field of frames field {Xi, X¯i}onT T M. It is well known that a local field of frames{Xi, X¯i} can be chosen so that it is adapted to the decomposition (1) i.e. Xi ∈Γ(HT M) and
X¯i∈Γ(V T M) set of vector fields onHT M andV T M, where
Xi:=
∂ ∂xi −N
j i
∂
∂yj, X¯i:=
∂ ∂yi,
where the indicesi, j, . . .and ¯i,¯j, . . .run over the range 1, . . . , n.
Analogous to Riemannian geometry, the following lemma in Finsler geometry was obtained by straightforward calculations.
Lemma 2.1. [14]The Lie brackets of the adapted frame ofT M satisfy the following
(1) [Xi, Xj] =RhijXh¯, (2) [Xi, X¯j] =Xj¯(Nhi)X¯h, (3) [X¯i, X¯j] = 0,
whereRh
ij =Xj(Nhi)−Xi(Nhj).
For a Finsler connection (Ni
j, Fijk, Cijk), the curvature tensor has three com-ponentR, P andS, that are called hh-curvature, hv-curvature and vv-curvature, respectively. They are defined as follows:
Rk ijh :=Xi(Fhkj)−Xj(Fhki) +FmkjFhmi−FmkiFhmj+RmijChjm,
Pk ijh :=X¯k(Fhij)−Xj(Chik) +FmkiChmj−CmkjFhmi+X¯j(Nmi)Chjm,
Sk ijh :=X¯j(Chki)−X¯i(Chkj) +CmkiChmj−CmkjChmi.
Let (M, g) be a Finsler manifold, the geodesic ofg satisfy the following system of differential equations
d2xi
dt2 + 2G ix,dx
dt
= 0,
whereGi=Gi(x, y) are called the geodesic coefficients, which are given by
Gi= 1 4g
iln[F2]
xmylym−[F2]xl
o .
The differentiable functionsGi
j:=X¯j(Gi) determine a non-linear connection which is called the canonical nonlinear connection of Finsler manifold (M, g). In what follows, the canonical nonlinear connectionGi
j will be used.
triple (Gi
j, Gijk,0), where Gijk := X¯k(Gij). The hh-curvature Hk ijh and hv -curvatureGh
k ij of the Berwald connection are obtained as follows:
Hk ijh =Xi(Ghkj)−Xj(Ghki) +GmkjGhmi−GmkiGhmj,
Gk ijh =X¯k(Ghij), respectively. It is obvious thatRh
ij =ykHk ijh .
Also, the Cartan connection of a Finsler manifold (M, g) is defined by triple (Gi
j, Fijk, Cijk), where Fijk := 12gih{Xj(gkh) +Xk(gjh)−Xh(gjk)} and Cijk := 1
2gihX¯k(gjh).
A Finsler manifold (M, g) is called a Berwald manifold, if the geodesic spray coefficientGi’s are quadratic functions ofy-coordinates in each tangent space, i.e.
X¯lX¯kX¯j(Gi) = 0, ∀j, k, l.
In the other words, the Finsler manifold (M, g) is a Berwald metric if the hv-curvature tensor field of the Berwald connection vanishes.
A Finsler manifold (M, g) is called Landsberg manifold, if the geodesic spary coefficientGi’s satisfy the following equations [11]
ymgimX¯lX¯kX¯j(Gi) = 0, ∀j, k, l.
It is obvious that every Brwald manifold is a Landsberg manifold. By defining the tensor fieldPi
jk :=Gijk−Fijk, one can see that, a Finsler manifold (M, g) is a Landsberg manifold if and only ifPi
jk= 0. For more details, one can refer to [3]. In the following, all manifolds are supposed to be connected.
2.1. Infinitesimal fiber-preserving transformations. Let ˜X be a vector field on T M and {φ˜t} the local one-parameter group of local transformations of T M generated by ˜X. Then, ˜X is called a fiber-preserving vector field on T M if each
˜
φt is a fiber-preserving transformation of T M. From [13], we have the following lemma.
Lemma 2.2. LetX˜ be a vector field onT M with components(vh, v¯h)with respect
to the adapted frame {Xh, X¯h}. Then X˜ is a fiber-preserving vector field onT M
if and only if vh are functions onM
Therefore, every fiber-preserving vector field ˜X onT M induces a vector field
V =vh∂
honM, where ∂i :=∂x∂i.
Let {dxi, δyi} be the dual basis of {Xh, X¯h}, where δyi = dyi −Gihdxh. Using a straight forward calculation similar to [14], one can obtain the following lemma.
Lemma 2.3. Let X˜ be a fiber-preserving vector field of T M with the components
(vh, v¯h). Then, the Lie derivatives of the adapted frame and the dual basis are given
as follows:
(1)£X˜Xh=−∂hvmXm+{vbRmbh−v ¯ bGm
bh−Xh(v ¯ h)}X
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(3)£X˜dxh=∂mvhdxm,
(4)£X˜δyh=−{vbRhbm−v¯bGhbm−Xm(v¯h)}dxm− {vbGhbm−Xm(¯ v¯h)}δym.
For more details in infinitesimal transformations, one can refer to [8].
2.2. Complete Lift Vector Fields and Lie Derivative. Let V = vi∂ i be a vector field on M. Then, V induces an infinitesimal point transformation on M. This is naturally extended to a point transformation of the tangent bundle T M
which is called extended point transformation. If {ϕt} is the local 1-parameter group ofM generated byV and ˜ϕtthe extended point transformation ofϕt, then
{ϕ˜t} induces a vector fieldVc onT M which is called the complete lift of V, (c.f. [4]).
The complete lift vector field Vc of V can be written as Vc = viX i +
yj(Fi
java +∂jvi)X¯i. From the Lemma 2.2, it concluded that, the class of com-plete lift vector fields is a subclass of fiber-preserving vector fields.
LetV be a vector field onM and{ϕt}the local one parameter group of local transformations of M generated byV. Take any tensor field S onM. Then, the Lie derivative£VS ofS with respect toV is a tensor field onM, defined by
£VS=
∂ ∂tϕ
∗
t(S)|t=0= lim t→0
ϕ∗
t(S)−S
t ,
on the domain ofϕt, whereϕ∗
t(S) denotes the pull back ofS byϕt.
In local coordinates the Lie derivative of an arbitrary tensor, Tki, is given locally by[16];
£VTki=va∂aTki+ya∂avbX¯b(Tki)−Tai∂avk+Tka∂iva.
Therefore
£Vyi = 0.
For the Lie derivatives of the adapted frame and the dual basis with respect to complete lift vector fieldVc, the following lemma can be presented.
Lemma 2.4. [4]Let (M, g) be a Finsler manifold, V a vector field onM andVc
its complete lift, then
(1)£VcXi=−∂ivhXh−£VGh iX¯h, (2)£VcX¯i=−∂ivhX¯h,
(3)£Vcdxh=∂mvhdxm,
(4)£Vcδyh=£VGmhdxm+∂mvhδym.
2.3. Infinitesimal projective transformations. LetM be a Riemannian mani-fold. A vector fieldX onM is said to be aninfinitesimal projective transformation, if there exists a 1-form θonM such that
(£X∇)(Y, Z) =θ(Y)Z+θ(Z)Y,
or equivalently
where ∇ is the Riemannian connection of M and Y, Z ∈ Γ(M) the set of vector fields onM.
In Finsler geometry, a vector fieldV on (M, g) is called an infinitesimal projec-tive transformation if there exists a function Ψ(x, y) onT M such that£VGi= Ψyi [2]. Accordingly, the following relations can be resulted.
£VGij= Ψδji+ Ψjyi,
£VGijk= Ψkδji+ Ψjδki + Ψjkyi,
£VGijkl= Ψklδij+ Ψjlδik+ Ψjkδli+ Ψjklyi,
where Ψj:=X¯j(Ψ), Ψjk:=X¯k(Ψj) and Ψjkl:=X¯l(Ψjk). If Ψ = 0, then it can be said thatV is an infinitesimal affine transformation.
3. Riemannian Connection ofT M0 With the Complete Lift Metric
Let g = (gij(x, y)) be a Finsler metric on M. As we said that, there are several Riemannian or pseudo-Riemannian metrics on T M0 which can be defined fromg. They are called the lift metric ofg. A one of such metrics is ˜g= 2gijdxiδyj, which is called the complete lift metric, (c.f. [17]). Thus (T M0,g˜) is a Riemannian manifold.
Let ˜∇ be the Riemannian connection of T M0 with respect to the complete lift metric ˜g and ˜ΓA
Lemma 3.1. We have the following equations
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Lemma 3.2. The connection coefficientsΓ˜A
BC of∇˜ satisfy the following relations.
a) ˜Γh which show the relation b) in the lemma.
According to (4) and (10):
gajΓ˜¯aim¯ =−gaiΓ˜a¯jm¯ =−gaiΓ˜am¯¯j =gamΓ˜¯ai¯j=gamΓ˜j¯a¯i=−gajΓ˜am¯¯i=−gajΓ˜¯aim¯,
thus, we get the relationg) in the lemma. From (4) and (6), we have
gaj(Faim−Γ˜aim) =−gia(Fajm−Γ˜¯jma¯ ) =−gia(Gajm−Pajm−Γ˜¯a¯jm)
=−gia(−Pajm−˜Γ¯am¯j). From above and (8), it can be said that
gaj(Faim−Γ˜a
From (7) and (9), it can be said that
gaiΓ˜a¯jm=−gajΓ˜a¯im=−gajΓ˜ma¯i =gmaΓ˜¯ja¯¯i−2Cmji. (11)
From (4), (9) and (11), the relationsc),d) andh) can be obtained. This completes the proof.
Remark 3.1. From Lemma 3.2 and the relation (3), the following equations can be obtained.
˜
∇XiXj =(F h
ji−Phji)Xh+ghmRimjX¯h, ∇˜XiX¯j =G h
jiXh¯, ˜
∇X¯iXj= 0, ∇˜X¯iX¯j = 2C
h jiX¯h.
(12)
It would be mentioned that, when (M, g) is a Riemannian manifold, then the rela-tions (12) are reduced to
˜
∇XiXj =Γ h
jiXh+ghmRimjXh¯, ∇˜XiX¯j= Γ h
jiXh¯, ˜
∇X¯iXj= 0, ∇˜X¯iX¯j = 0,
(13)
whereΓh
ji are the coefficients of the Riemannian connection ofM.
4. Main Results
Let (M, g) be a Finsler manifold andT M0its slit tangent bundle with com-plete lift metric ˜g. Here, infinitesimal fiber-preserving projective transformations on (T M0,g˜) are considered and the following proposition is proved.
Proposition 4.1. Let (M, g) be a n-dimensional Finsler manifold and T M0 its
slit tangent bundle with the complete lift metric ˜g. Then, every infinitesimal fiber-preserving projective transformation on(T M0,g˜)induces an infinitesimal projective
transformation on (M, g).
Proof. Let ˜X be an infinitesimal fiber-preserving projective transformation on
T M0. From (2), it can be concluded that there exists a 1-form Θ on T M0 such that
£X˜∇˜Y˜Z˜−∇˜Y˜£X˜Z˜−∇˜[ ˜X,Y˜]Z˜= Θ( ˜Y) ˜Z+ Θ( ˜Z) ˜Y , (14)
where ˜Y and ˜Z ∈ Γ(T M). Let ˜X = vhX h+v
¯ hX
¯
h and Θ = θidxi +θ¯iδyi. We obtain the following.
£X˜∇˜X¯iXj−∇˜X¯i£X˜Xj−∇˜[ ˜X,X¯i]Xj = Θ(X¯i)Xj+ Θ(Xj)X¯i, (15)
£X˜∇˜X¯iX¯j−∇˜X¯i£X˜X¯j−∇˜[ ˜X,X¯i]X¯j = Θ(X¯i)X¯j+ Θ(X¯j)X¯i, (16)
£X˜∇˜XiXj−∇˜Xi£X˜Xj−∇˜[ ˜X,Xi]Xj = Θ(Xi)Xj+ Θ(Xj)Xi. (17)
By means of Lemma 2.1, Lemma 2.3, (12) and (15), the following relation is obtaind:
−nvbHi bjh −X¯i(v¯b)Ghbj−v¯bGhi bj−X¯iXj(v¯h)
+ vbRmbj−v¯bGmbj−Xj(vm¯)
B. Bidabad By means of Lemma 2.1, Lemma 2.3, (12), (16) and (18), the following can be presented: By means of Lemma2.1, Lemma2.3, (12) and (17) we have
˜ From (21), it can be concluded that
£VGh=θiyiyh, (23)
i.e. V =vh∂
h is an infinitesimal projective transformation on (M, g). This com-pletes the proof.
Proof of Theorem 1.1. Infinitesimal complete lift transformations is a subclass of infinitesimal fiber-preserving transformations. So, the similar method as in the proof of Proposition 4.1 is used. By means of Lemma 2.1, Lemma 2.4, relations (12), (15) and (18) the following can be given
£VGhji+ 2(£VGtj)Chit=δihθj.
By means of Lemma 2.1 and Lemma 2.4 with the relations (12), (16) and (18):
£VChji= 0. (25)
By taking in to account Lemma 2.1 and Lemma 2.4 with the relations (12) and (17), we can obtain
and
£VGht(Pjit −Ftji) +∂jvaghtRita+∂ivbghtRbtj
+Xi(£VGhj) +Ghti(£VGhj) = 0. Contracting (26) byyiyj and using (24) leads to obtain
£VGh= 0,
thus,V =vi∂
i is an infinitesimal affine transformation on (M, g) and
£VGhj =£VGhji= 0. (27)
Substituting (27) in (24)
δhiθj= 0.
Thus, θi = 0, i.e. Vc is an infinitesimal affine transformation on (T M,g˜). This completes the proof.
Proof of Theorem 1.3. IfVc is an infinitesimal projective transformation then from Theorem 1.1, one can see that V is an affine. Let (M, g) be a Landsberg manifold then
Fh
ji=Ghji. (28)
If V is an affine, then by use of (24), (26) and (28), it can be seen thatVc is an infinitesimal affine transformation on (T M0,˜g). This completes the proof.
From Theorem 1.3, we have immediately the following remark.
Remark 4.1. Let (M, g) be a Landsberg space, then, there is a one-to-one cor-respondence between complete lift projective vector fields on (T M0,g˜) and affine
vector fields on (M, g).
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