J.
Indones.
Math.
Soc.
Vol.
No.
April 2017, pp.
1Ae12.
ON INFINITESIMAL PROJECTIVE
TRANSFORMATIONS OF THE TANGENT BUNDLE
WITH THE COMPLETE LIFT OF A FINSLER METRIC
Bidabad1 .
Rezaii2 and M.
Zohrehvand3 Faculty of Mathematics and Computer Science.
Amirkabir University of Technology.
Tehran.
Iran bidabad@aut.
Faculty of Mathematics and Computer Science.
Amirkabir University of Technology.
Tehran.
Iran mmreza@aut.
Faculty of Mathematical Sciences and Statistics.
Malayer University.
Malayer.
Iran zohrehvand@malayeru.
Abstract.
Let (M, .
be a Finsler manifold and T M0 its slit tangent bundle with the complete lift metric gE.
In this paper, we prove that every infinitesimal complete lift projective transformation on (T M0 , gE), is an infinitesimal affine transformation.
Moreover, if (M, .
is a Landsberg manifold, then there is a one-to-one correspondence between infinitesimal complete lift projective transformations on (T M0 , gE) and infinitesimal affine transformations on (M, .
Key words and Phrases: Finsler manifold.
Complete lift metric.
Tangent bundle.
Infinitesimal projective transformation.
Abstrak.
Misalkan (M, .
adalah manifold Finsler dan T M0 adalah bundel tangen slit-nya dengan metrik lift lengkap gE.
Pada makalah ini, kami membuktikan bahwa setiap transformasi projektif lift lengkap infinitesimal pada (T M0 , gE), adalah transformasi afin infinitesimal.
Lebih jauh, jika (M, .
adalah manifold Landsberg, maka terdapat korespondensi satu-satu antara transformasi projektif lift lengkap infinitesimal pada (T M0 , gE) dengan transformasi afin infinitesimal pada (M, .
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.
Bidabad et.
Introduction Let (M, .
be a Riemannian manifold and I a transformation on M .
Then.
I 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.
Let V be a vector field on M and {It } the local one-parameter group generated by V .
Then.
V is called an infinitesimal projective .
transformation on M if every It is a projective .
Let IE be a transformation of T M , the tangent bundle of M .
IE is called a fiber-preserving transformation if it preserves the fibers.
A vector field XE on T M with the local one-parameter group {IEt } is called an infinitesimal fiberpreserving transformation on T M if each IEt 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.
efer to subsection 2.
2 or .
It is well-known that there are some lift metrics on T M as follows: complete lift metric or g2 , diagonal lift metric or Sasaki metric or g1 g3 , lift metric I II or g1 g2 and lift metic II i or g2 g3 , where g1 := gij dxi dxj , g2 := 2gij dxi y j and g3 := gij y i y j are all bilinear differential forms defined globally on T M .
For more details one can refer to .
The problems of exsisting special infinitesimal transformations on the tangent bundle of a Riemannian manifold with some lift metrics are considered by several authors, e.
, 6, 7, 8, .
, 13, 14, .
The study shows that the special infinitesiaml transformations on T M might lead to some global results.
For example, in .
, it is proved that if T 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 .
, that if T M with the complete lift metric admits a non-affine infinitesimal projective transformation, then M is locally flat.
Therefore, it is meaningful to the study of the infinitesimal transformations on the tangent bundle T M .
Yamauchi in .
, proved the following theorem:
Theorem A: Let M be a non-Euclidean complete n-dimensional Riemannian manifold, and let T 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 manifold are considered by many authors, e.
, 2, 4, .
1A vector field XE on (T M, gE) is called an essential infinitesimal conformal transformation if there exsits a scalar function E on T M such that E depends only y i with respect to the induced coordinate .
i , y i ) on T M and AXE gE = 2EgE,.
Infinitesimal Projective Transformations In this paper, the infinitesimal fiber-preserving projective transformations on the tangent bundle of a Finsler manifold with the complete lift metric are considered.
Then, as a special case, the infinitesimal complete lift projective transformations are studied and the following theorem is proved.
Theorem 1.
Let (M, .
be a C O connected Finsler manifold and T M0 its slit tangent bundle with the complete lift metric gE.
Then, every infinitesimal complete lift projective transformation on (T M0 , gE) is affine and it naturally induced an infinitesimal affine transformation on (M, .
Thus.
Theorem A is true for the Finsler manifold and the infinitesimal complete lift transformations.
From Theorem 1.
1, the following corollary can be immediately found.
Corollary 1.
The Lie algebra of complete lift projective vector field on (T M0 , gE) 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.
Let (M, .
be an n-dimensional Landsberg manifold and T M0 its slit tangent bundle with a complete lift metric gE.
Then, every infinitesimal complete lift transformation V c on (T M0 , gE) is projective if and only if V is an infinitesimal affine transformation on (M, .
Preliminaries Let M be a real n-dimensional C O manifold and T M its tangent bundle.
The elements of T M are denoted by .
, .
with y OO Tx M .
Also T M0 = T M \ .
be the slit tangent bundle of M .
The natural projection A : T M0 Ie M is given by A.
, .
:= x.
A Finsler structure on M is a function of F : T M Ie .
O) with the following properties.
F is C O on T M0 , .
F is positively 1-homogeneous on the fibers of tangent bundle T M and .
the Hessian g of F 2 with elements gij .
, .
:= 21 [F 2 .
, .
]yi yj is positive-definite.
In the sequel of this paper, a Finsler manifold with a Finsler structure F will be denoted by (M, .
instead of (M.
F ).
Let Vv T M := kerAOv be the set of the vectors tangent to the fiber S through v OO T M0 .
Then, the vertical vector bundle on M is defined by V T M := vOOT M0 Vv T M .
A non-linear connection or a horizontal distribution on T M0 is a complementary distribution HT M for V T M on T T M0 .
Therefore, by using a non-linear connection, the following decomposition is resulted:
T T M0 = V T M Oi HT M, .
where HT M is a vector bundle completely determined by the non-linear differentiable functions N ij .
, .
on T M , which is called coefficients of the non-linear B.
Bidabad et.
connection HT M .
The pair (HT M.
ON) is called a Finsler connection on the manifold M , where ON a linear connection to V T M .
Indeed, a Finsler connection is a triple (N ij .
F ijk .
C ijk ) where N ij are the coefficients of a nonlinear connection.
F ijk and C ijk are the horizontal part and the vertical part of this connection, respectively.
Using the local coordinates .
i , y i ) on T M we have the local field of frames field {Xi .
XiE } on T T M .
It is well known that a local field of frames {Xi .
XiE } can be chosen so that it is adapted to the decomposition .
Xi OO e(HT M ) and XiE OO e(V T M ) set of vector fields on HT M and V T M , where Xi := OC Oe N ji j .
XiE := OCx OCy OCy i where the indices i, j, .
and iE, jE, .
run over the range 1, .
, n.
Analogous to Riemannian geometry, the following lemma in Finsler geometry was obtained by straightforward calculations.
Lemma 2.
The Lie brackets of the adapted frame of T M satisfy the following .
[Xi .
Xj ] = Rhij XhE , .
[Xi .
XjE ] = XjE (N hi )XhE , .
[XiE .
XjE ] = 0, where Rhij = Xj (N hi ) Oe Xi (N hj ).
For a Finsler connection (N ij .
F ijk .
C ijk ), the curvature tensor has three component R.
P and S, that are called hh-curvature, hv-curvature and vv-curvature.
They are defined as follows:
Rkhij := Xi (F hkj ) Oe Xj (F hki ) F m kj F mi Oe F ki F mj R ij C jm .
Pk hij := XkE (F hij ) Oe Xj (C hik ) F m ki C mj Oe C kj F mi XjE (N i )C jm .
Skhij := XjE (C hki ) Oe XiE (C hkj ) C mki C hmj Oe C mkj C hmi .
Let (M, .
be a Finsler manifold, the geodesic of g satisfy the following system of differential equations dx d2 xi 2Gi x, = 0, where Gi = Gi .
, .
are called the geodesic coefficients, which are given by Gi = g il [F 2 ]xm yl y m Oe [F 2 ]xl .
The differentiable functions Gi j := XjE (Gi ) determine a non-linear connection which is called the canonical nonlinear connection of Finsler manifold (M, .
what follows, the canonical nonlinear connection Gi j will be used.
There are several Finsler connections on a Finsler manifold, which we present some of them.
The Berwald connection of a Finsler manifold (M, .
is defined by Infinitesimal Projective Transformations triple (Gi j .
Gi jk , .
, where Gi jk := XkE (Gi j ).
The hh-curvature Hkhij and hvcurvature Gkhij of the Berwald connection are obtained as follows:
Hkhij = Xi (Ghkj ) Oe Xj (Ghki ) Gmkj Ghmi Oe Gmki Ghmj .
Gkhij = XkE (Ghij ).
It is obvious that Rhij = y k Hkhij .
Also, the Cartan connection of a Finsler manifold (M, .
is defined by triple (Gij .
F ijk .
C ijk ), where F ijk := 21 g ih {Xj .
kh ) Xk .
jh ) Oe Xh .
jk )} and C ijk := 1 ih 2 g XkE .
jh ).
A Finsler manifold (M, .
is called a Berwald manifold, if the geodesic spray coefficient Gi Aos are quadratic functions of y-coordinates in each tangent space, i.
XlE XkE XjE (Gi ) = 0.
OAj, k, l.
In the other words, the Finsler manifold (M, .
is a Berwald metric if the hvcurvature tensor field of the Berwald connection vanishes.
A Finsler manifold (M, .
is called Landsberg manifold, if the geodesic spary coefficient Gi Aos satisfy the following equations .
y m gim XlE XkE XjE (Gi ) = 0.
OAj, k, l.
It is obvious that every Brwald manifold is a Landsberg manifold.
By defining the tensor field P ijk := Gi jk Oe F ijk , one can see that, a Finsler manifold (M, .
is a Landsberg manifold if and only if P ijk = 0.
For more details, one can refer to .
In the following, all manifolds are supposed to be connected.
Infinitesimal fiber-preserving transformations.
Let XE be a vector field on T M and {IEt } the local one-parameter group of local transformations of T M generated by XE.
Then.
XE is called a fiber-preserving vector field on T M if each IEt is a fiber-preserving transformation of T M .
From .
, we have the following Lemma 2.
Let XE be a vector field on T M with components .
h , v hE ) with respect to the adapted frame {Xh .
XhE }.
Then XE is a fiber-preserving vector field on T M if and only if v h are functions on M Therefore, every fiber-preserving vector field XE on T M induces a vector field V = v h OCh on M , where OCi := OCx Let .
x , y } be the dual basis of {Xh .
XhE }, where y i = dy i Oe Gi h dxh .
Using a straight forward calculation similar to .
, one can obtain the following Lemma 2.
Let XE be a fiber-preserving vector field of T M with the components .
h , v hE ).
Then, the Lie derivatives of the adapted frame and the dual basis are given as follows:
AXE Xh = OeOCh v m Xm .
b Rmbh Oe v bE Gmbh Oe Xh .
hE )}XmE , .
AXE XhE = .
b Gmbh Oe XhE .
mE )}XmE .
Bidabad et.
AXE dxh = OCm v h dxm , .
AXE y h = Oe.
b Rhbm Oe v bE Ghbm Oe Xm .
hE )}dxm Oe .
b Ghbm Oe XmE .
hE )}y m .
For more details in infinitesimal transformations, one can refer to .
Complete Lift Vector Fields and Lie Derivative.
Let V = v i OCi 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 .
t } is the local 1-parameter group of M generated by V and iEt the extended point transformation of it , then .
Et } induces a vector field V c on T M which is called the complete lift of V , .
The complete lift vector field V c of V can be written as V c = v i Xi y (F ja v a OCj v i )XiE .
From the Lemma 2.
2, it concluded that, the class of complete lift vector fields is a subclass of fiber-preserving vector fields.
Let V be a vector field on M and .
t } the local one parameter group of local transformations of M generated by V .
Take any tensor field S on M .
Then, the Lie derivative AV S of S with respect to V is a tensor field on M , defined by iO (S) Oe S OC O it (S).
=0 = lim t tIe0 OCt on the domain of it , where iOt (S) denotes the pull back of S by it .
In local coordinates the Lie derivative of an arbitrary tensor.
T ki , is given locally by.
AV S =
AV T ki = v a OCa T ki y a OCa v b XbE (T ki ) Oe T ia OCa v k T ka OCi v a .
Therefore AV y i = 0.
For the Lie derivatives of the adapted frame and the dual basis with respect to complete lift vector field V c , the following lemma can be presented.
Lemma 2.
Let (M, .
be a Finsler manifold.
V a vector field on M and V c its complete lift, then .
AV c Xi = OeOCi v h Xh Oe AV Ghi XhE , .
AV c XiE = OeOCi v h XhE , .
AV c dxh = OCm v h dxm , .
AV c y h = AV Ghm dxm OCm v h y m .
Infinitesimal projective transformations.
Let M be a Riemannian manifold.
A vector field X on M is said to be an infinitesimal projective transformation, if there exists a 1-form on M such that (AX ON)(Y.
Z) = (Y )Z (Z)Y, or equivalently AX (ONY Z) Oe ONY (AX Z) Oe ON[X,Y ] Z = (Y )Z (Z)Y.
Infinitesimal Projective Transformations where ON is the Riemannian connection of M and Y .
Z OO e(M ) the set of vector fields on M .
In Finsler geometry, a vector field V on (M, .
is called an infinitesimal projective transformation if there exists a function .
, .
on T M such that AV Gi = y i .
Accordingly, the following relations can be resulted.
AV Gij = ji j y i .
AV Gi jk = k ji j ki jk y i .
AV Gi jkl = kl ji jl ki jk li jkl y i , where j := XjE (), jk := XkE .
) and jkl := XlE .
k ).
If = 0, then it can be said that V is an infinitesimal affine transformation.
Riemannian Connection of T M0 With the Complete Lift Metric Let g = .
, .
) 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 from g.
They are called the lift metric of g.
A one of such metrics is gE = 2gij dxi y j , which is called the complete lift metric, .
Thus (T M0 , gE) is a Riemannian u be the Riemannian connection of T M0 with respect to the complete Let ON lift metric gE and eEA BC the coefficients of ON, that is, u X Xj = eEm ji Xm eEji XmE .
ONXi XjE = eEjEi Xm eEjEi XmE , u X Xj = eEm Xm eEmE XmE .
ON u X XjE = eEm Xm eEmE XmE .
ON j iE j iE jE iE jE iE where the coefficients A.
C, .
run over the range 1, .
, n, 1E, .
, nE.
The following lemma is trivial.
Lemma 3.
We have the following equations u X dxh = Oeh dxm Oe eEh y m , .
ON mEi u X y h = OehE dxm Oe eEhE y m , .
ON mEi u X dxh = Oeh dxm Oe eEh y m , .
ON mEiE u X y h = OehE dxm Oe eEhE y m .
mEiE u defined by T (X.
Y ) = ON u XY Oe ON uY X Oe Since the torsion tensor T (X.
Y ) of ON [X.
Y ] vanishes, the following relations can be given by means of Lemma 2.
1 and the relation .
eEm ji = eEij , eEji = eEij R ij , eEm jEi = eEijE , eEjEi = eEijE Gji , eEm jE iE = eEiEjE , eEjE iE = eEiEjE .
Bidabad et.
Lemma 3.
The connection coefficients eEA BC of ON satisfy the following relations.
eEji = F ji Oe P ji , .
eEji = g Rimj , .
eEhjEi = 0, .
eEhjiE = 0, .
eEhEjEi = Ghji , f ) eEhEj iE = 0, .
eEhjEiE = 0, .
eEhEjE iE = 2C hji .
u i.
ONgE u = 0, we Proof.
By means of Lemma 3.
1 and metric compatibility of ON, u X gE = ON u X .
gij dxi y j ) 0=ON = Oe2gia eEaEjm dxi dxj 2.
aj (F aim Oe eEaim ) gia (F ajm Oe eEaEjEm )}dxi y j Oe 2gaj eEaiEm y i y j .
u X gE = ON u X .
gij dxi y j ) 0=ON = Oe2gia eEaEj mE dxi dxj 2.
Cijm Oe gaj eEaimE Oe gia eEaEjE mE }dxi y j Oe 2gaj eEaiEmE y i y j .
It follows that gia eEaEjm gja eEaEim = 0, .
gaj (F aim Oe eEaim ) gia (F ajm Oe eEaEjEm ) = 0, .
gaj eEaiEm gai eEajEm = 0, gia eEaEj mE gja eEaEimE = 0, 2Cijm Oe gaj eEaimE Oe gia eEaEjE mE = 0, gaj eEaiEmE gai eEajEmE = 0.
From .
, we have gia eEaEjm = Oegja eEaEim = Oegja .
EaEmi Rami ) = OeRjmi Oe gja eEaEmi = OeRjmi gma .
EaEij Raij ) = OeRjmi Rmij gma eEaEij = OeRjmi Rmij Oe gia eEaEmj = OeRjmi Rmij Oe gia .
EaEjm Rajm ) = OeRjmi Rmij Oe Rijm Oe gia eEaEjm = Oegia eEaEjm 2Rmij , which show the relation .
in the lemma.
According to .
gaj eEaiEmE = Oegai eEajEmE = Oegai eEamEjE = gam eEaiEjE = gam eEajEiE = Oegaj eEamEiE = Oegaj eEaiEmE , thus, we get the relation .
in the lemma.
From .
, we have gaj (F aim Oe eEaim ) = Oegia (F ajm Oe eEaEjEm ) = Oegia (Gajm Oe P ajm Oe eEaEjEm ) = Oegia (OeP ajm Oe eEaEmjE ).
From above and .
, it can be said that gaj (F aim Oe eEaim ) gaj (F ami Oe eEami ) = gia P ajm gma P aji = 2Pjim , this shows the relations .
, .
and f ).
Infinitesimal Projective Transformations From .
, it can be said that gai eEajEm = Oegaj eEaiEm = Oegaj eEamiE = gma eEaEjE iE Oe 2Cmji .
From .
, .
, the relations .
, .
can be obtained.
This completes the proof.
Remark 3.
From Lemma 3.
2 and the relation .
, the following equations can be obtained.
u X Xj =(F h Oe P h )Xh g hm Rimj XhE .
ON u X XjE = Gh XhE .
ON .
ONXiE Xj = 0.
ONXiE XjE = 2C ji XhE .
It would be mentioned that, when (M, .
is a Riemannian manifold, then the relations .
are reduced to u X Xj =ehji Xh g hm Rimj XhE .
ON u X XjE = ehji XhE .
ON u X Xj = 0.
ON u X XjE = 0.
ON .
where ehji are the coefficients of the Riemannian connection of M .
Main Results Let (M, .
be a Finsler manifold and T M0 its slit tangent bundle with complete lift metric gE.
Here, infinitesimal fiber-preserving projective transformations on (T M0 , gE) are considered and the following proposition is proved.
Proposition 4.
Let (M, .
be a n-dimensional Finsler manifold and T M0 its slit tangent bundle with the complete lift metric gE.
Then, every infinitesimal fiberpreserving projective transformation on (T M0 , gE) induces an infinitesimal projective transformation on (M, .
Proof.
Let XE be an infinitesimal fiber-preserving projective transformation on T M0 .
From .
, it can be concluded that there exists a 1-form o on T M0 such u A ZE Oe ON u ZE Oe ON .
AXE ON [XE,YE ] ZE = o(YE )ZE o(ZE)YE .
YE XE
where YE and ZE OO e(T M ).
Let XE = v h Xh v hE XhE and o = i dxi iE y i .
obtain the following.
u X Xj Oe ON u X A Xj Oe ON Xj = o(XiE )Xj o(Xj )XiE , .
A ON XE
[XE,XiE ] u X XjE Oe ON u X A XjE Oe ON AXE ON XE [XE,XiE ] XjE = o(XiE )XjE o(XjE )XiE , .
u X Xj Oe ON u X A Xj Oe ON AXE ON XE [XE,Xi ] Xj = o(Xi )Xj o(Xj )Xi .
By means of Lemma 2.
Lemma 2.
3, .
, the following relation is obtaind:
Oe v b Hih bj Oe XiE .
bE )Ghbj Oe v bE Ghi bj Oe XiE Xj .
hE ) v b Rmbj Oe v bE Gmbj Oe Xj .
mE ) C hmi XhE = jh iE Xh ih j XhE .
Bidabad et.
Thus, we obtain iE = 0, .
v b Hih bj Oe XiE .
bE )Ghbj Oe v bE Ghi bj Oe XiE Xj .
hE ) v b Rmbj Oe v bE Gmbj Oe Xj .
mE ) C hmi = Oeih j .
By means of Lemma 2.
Lemma 2.
3, .
, .
, the following can be XE(C hij ) C aij .
b Ghba Oe XaE .
hE )) Oe .
b Ghi bj Oe XiE XjE .
hE )) OeC hia .
b Gabj Oe XjE .
aE )) Oe C haj .
b Gabi Oe XiE .
aE )) = 0.
By means of Lemma2.
Lemma2.
3, .
we have XE(F hji Oe P hji ) Oe OCa v h (F aji Oe P aji ) OCi OCj v h OCj v a (F hai Oe P hai ) OCi v a (F haj Oe P haj ) = ih j jh i , .
(F aji Oe Pji ) v b Rhba Oe v bE Ghba Oe Xa .
hE ) X.
ht Ritj ) g at Ritj v b Ghba Oe XaE .
hE ) OCj v a g ht Rita Oe Xi v b Rhbj Oe v bE Ghbj Oe Xj .
hE ) Oe v b Rabj Oe v bE Gabj Oe Xj .
aE ) Ghai OCi v b g ht Rbtj = 0.
From .
, it can be concluded that AV Gh = i y i y h , .
V = v h OCh is an infinitesimal projective transformation on (M, .
This completes the proof.
Proof of Theorem 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.
Lemma 2.
4, relations .
, .
the following can be given AV Ghji 2(AV Gtj )C hit = ih j .
Contracting by y i AV Ghj = y h j , (AV Gtj )C hit = 0, and, we obtain AV Ghji = ih j .
By means of Lemma 2.
1 and Lemma 2.
4 with the relations .
, .
AV C hji = 0.
By taking in to account Lemma 2.
1 and Lemma 2.
4 with the relations .
, we can obtain AV (F hji Oe P hji ) = ih j jh i , .
Infinitesimal Projective Transformations AV Ght (Pji Oe F tji ) OCj v a g ht Rita OCi v b g ht Rbtj Xi (AV Ghj ) Ghti (AV Ghj ) = 0.
Contracting .
by y i y j and using .
leads to obtain AV Gh = 0, thus.
V = v i OCi is an infinitesimal affine transformation on (M, .
and AV Ghj = AV Ghji = 0.
Substituting .
ih j = 0.
Thus, i = 0, i.
V c is an infinitesimal affine transformation on (T M, gE).
This completes the proof.
Proof of Theorem 1.
If V c is an infinitesimal projective transformation then from Theorem 1.
1, one can see that V is an affine.
Let (M, .
be a Landsberg manifold then F hji = Ghji .
If V is an affine, then by use of .
, .
, it can be seen that V c is an infinitesimal affine transformation on (T M0 , gE).
This completes the proof.
From Theorem 1.
3, we have immediately the following remark.
Remark 4.
Let (M, .
be a Landsberg space, then, there is a one-to-one correspondence between complete lift projective vector fields on (T M0 , gE) and affine vector fields on (M, .
References