Energy β-Conformal Change and Special Finsler Spaces

The main aim of the present paper is to establish an intrinsic investigation of the energy β-conformal change of the most important special Finsler spaces, namely, C-recurrent, C-recurrent, C-recurrent, S-recurrent, quasi-C-reducible, semi-C-reducible, C-reducible, P-reducible, C2-like, S3-like, P2-like and h-isotropic, ···, etc. Necessary and sufficient conditions for such special Finsler manifolds to be invariant under an energy β-conformal change are obtained. It should be pointed out that the present work is formulated in a prospective modern coordinate-free form.


Introduction
An important aim of Finsler geometry is the construction of a natural geometric framework of variational calculus and the creation of geometric models that are appropriate for dealing with different physical theories, such as general relativity, relativistic optics, particle physics and others.As opposed to Riemannian geometry, the extra degrees of freedom offered by Finsler geometry, due to the dependence of its geometric objects on the directional arguments, make this geometry potentially more suitable for dealing with such physical theories at a deeper level.
Studying Finsler geometry, however, one encounters substantial difficulties trying to seek analogues of classical global, or sometimes even local, results of Riemannian geometry.These difficulties arise mainly from the fact that in Finsler geometry all geometric objects depend not only on positional coordinates, as in Riemannian geometry, but also on directional arguments.
In Riemannian geometry, there is a canonical linear connection on the manifold M, whereas in Finsler geometry there is a corresponding canonical linear connection due to E. Cartan.However, this is not a connection on M but is a connection on T (TM), the tangent bundle of TM, or on π −1 (TM), the pullback of the tangent bundle TM by :TM M   .The infinitesimal transformations (changes) in Riemannian and Finsler geometry are important, not only in differential geometry, but also in application to other branches of science, especially in the process of geometrization of physical theories [1].
In [2], we investigated intrinsically energy β-conformal change of the fundamental linear connections on the pullback bundle of a Finsler manifold, namely, the Cartan connection, the Berwald connection, the Chern connection and the Hashiguchi connection.Moreover, the change of their curvature tensors is obtained.
The present paper is a continuation of [2] where we present an intrinsic investigation of energy β-conformal change of the most important special Finsler spaces, namely, C h -recurrent, C v -recurrent, C 0 -recurrent, S v -recurrent, quasi-C-reducible, semi-C-reducible.
C-reducible, P-reducible, C 2 -like, S 3 -like, P 2 -like and h-isotropic, ••• , etc.Moreover, we obtain necessary and sufficient conditions for such special Finsler manifolds to be invariant under an energy β-conformal change.
Finally, it should be pointed out that all results obtained are formulated in a prospective modern coordinate-free form.

Notation and Preliminaries
In this section, we give a brief account of the basic concepts of the pullback approach to intrinsic Finsler geometry necessary for this work.For more details, we refer to [3][4][5][6].
We assume, unless otherwise stated, that all geometric objects treated are of class C ∞ .The following notation will be used throughout this paper: M: a real paracompact differentiable manifold of finite , i L being the interior derivative with respect to a vector form L.
Elements of will be called π-vector fields and will be denoted by barred letters.Tensor fields on will be called π-tensor fields.The fundamental π-vector field is the π-vector field for all u M. We have the following short exact sequence of vector bundles, relating the tangent bundle T (TM) and the pullback bundle : , where the bundle morphisms ρ and γ are defined respectively by The vector 1-form J on TM defined by : J     is called the natural almost tangent structure of TM.The vertical vector field C on TM defined by : Let D be a linear connection (or simply a connection) on the pullback bundle .We associate with D the map called the connection (or the deflection) map of D. A tangent vector is called the horizontal space to M at u.The connection D is said to be regular if If M is endowed with a regular connection, then the vector bundle maps .
, this map will be called the horizontal map of the connection D. According to the direct sum decomposition (1), a regular connection D gives rise to a horizontal projector h D and a vertical projector v D , given by , where I is the identity endomorphism on T (TM).The (classical) torsion tensor T of the connection D is defined by

T
The horizontal ((h)h-) and mixed ((h)hv-) torsion tensors, denoted by Q and T respectively, are defined by The (classical) curvature tensor K of the connection D is defined by , , .
The horizontal (h-), mixed (hv-) and vertical (v-) curvature tensors, denoted by R, P and S respectively, are defined by The contracted curvature tensors, denoted by , R ̂, P ̂ and respectively, are also known as the (v)h-, (v)hvand (v)v-torsion tensors and are defined by On a Finsler manifold   , M L , there are canonically associated four linear connections on [7]: the Cartan connection , the Chern (Rund) connection D c , the Hashiguchi connection . The following theorem guarantees the existence and uniqueness of the Cartan connection on the pullback bundle.
Theorem 2.1.[8] Let  , M L  be a Finsler manifold and g the Finsler metric defined by L. There exists a unique regular connection on π −1 (TM) such that (a) is metric : M L be a Finsler manifold and g the Finsler metric dened by L. We define: : the vertical scalar curvature.
Deicke theorem [9] can be formulated globally as follows: M L be a Finsler manifold.The following assertions are equivalent: Concerning the Berwald connection on the pullback bundle, we have Theorem 2.4.[8] Let  ,  M L be a Finsler manifold.There exists a unique regular connection D • on such that Such a connection is called the Berwald connection associated with the Finsler manifold   , M L .We terminate this section by some concepts and results concerning the Klein-Grifone approach to i c Finsler geometry.For more details, we refer to [10 3 A semispray X which is homogeneous of degree 2 in , is called a spray.Proposition 2.5.[12] Let M L be a Finsler manifold.The vector field G on TM defined by d . Such a spray is called the canonical spray.
A nonlinear connection on M is a vector 1-form Γ on TM, C ∞ on M, C 0 on TM, such that , .
The horizontal and vertical projectors h Γ and v Γ associated with  are defined by To each nonlinear connection Γ there is associated a semispray S defined by , where S' is an arbitrary semispray.A nonlinear connection Γis homogeneous if The torsion of a nonlinear connection Γ is the vector 2-form t on TM defined by Theorem 2.6.[11] On a Finsler manifold   , M L , there exists a unique conservative homogenous nonlinear connection with zero torsion.It is given by: where G is the canonical spray.Such a nonlinear connection is called the canonical connection, the Barthel connection or the Cartan nonlinear connection associated with (M,L).
It should be noted that the semispray associated with the Barthel connection is a spray, which is the canonical spray.

Energy β-Conformal Change and Special Finsler Spaces
In [2], we investigated intrinsically a particular β-change, called an energy β-conformal change: where   , M L is a Finsler manifold admitting a con-  is called a concurrent π-vector field if it satisfies the following conditions , 0 In other words,  is a concurrent π-vector field if : , M L be two Finsler manifolds related by the energy β-conformal change (4).Then the associated Cartan connections  and are related by:  where T  is a 2-form on TM, with values in π −1 (T by Moreover, v-torsion T and the π-form C are invariant.(a) The (h) h V is the vector π-form defined by here H is the vector π-form defined by here , where    quently, the hv-curvature P va Hence, .Conse nishes [15].
(b) The proof is similar to that of (a). , in Equation (10), making use of

  
On the other hand, Definition 3.8(a) for X   , yields   . where where and the Cartan tensor T has the form Under the energy β-conformal change (4), we have: si-C-reducible, then the Cartan tenso fies Relation (11).Setting r T satis X Y    into Equation (11) and using the fact that From which together with the given assumption,
The more relaxed condition , , , will be called the T o -condition.Theorem 3.13.Under the energy β-conformal change (4), we have: , together with 3.5(a) and Lemma 3.  of [13]) and Taking the trace of the above equation with respect to , noting that where ω is a (1) π-form (positively homogeneous of degree 0).(b) P-reducible if the π-tensor field , where δ is the π-form defined by

Concluding Remark
It should be pointed out that a global formulatio of different aspects of Finsler geometry may give more insight into the infrastructure of physical theories and make a understanding on the essence of such theories without being trapped into the com This is one of the motivations all results obtained are formulated in a prospective modern coordinate-free form.Moreover, it should be noted that the outcome of this work is twofold.Firstly, the local expressions of the obtained results, when calculated, coincide with the existing local results.Secondly, new global proofs have been established.
dimension n and of class C ∞ , of the tangent bundle TM by π,   1 current π-vector field  , the fundamental π-vector field and σ(x) is a function on M.Moreover, the relation between the two Barthel connections Γ and   , corresponding to this change, is obtained.The energy β-conformal change of the fundamental linear connections on the pullback bundle of a Finsler manifold is studied.In this section, we introduce the effect of energy β-conformal change on some important special Finsler spaces.The intrinsic definitions of the special Finsler spaces treated here are quoted from[14].The following definition and three lemmas are useful for subsequence use.Definition 3.1.[15]Let   , M L be a Finsler manifold.A π-vector field

MLemma 3 . 4 .
L be a Finsler manifold which admits a concurrent π-vector  .Then, we have: (a) The concurrent π-vector eld  is everywhere non -zero.(b) The scalar function  zero and is orthogonal to  .(d) The π-vector fields m and  satisfy [2] Under the energy β-conformal change (4), we have

Now
The above two equations and Theorem 3.5(a) imply th ma 2.3, the result follows.4], together w Definition 3.10.A Finsler manifold (M,L) is said to r T has the from at 0. T T    Hence, by Lem (b) and (c) follow from Theorem 4.7 of [1 ith Theorem 3.5(a).

A
follows that the π-orm C vanishes.Hence, by Lemma 2.3 and Theorem 3.5(a) imply that f Follows from the defining property of C-reducitting X Y    , taking into account Lemma 3.3(e a 2.3, Theorem 3.5(a) and ), Lemm Follows from Equation (17) by setting X   , taking into account that

4 M
 and the v-curvature tensor S has the form: dim

18) Theorem 3 . 15 .
Under the energy β-conforma ( )), the vertical scalar curvature Sc v vanishes.Now, again from Equation (18) togeth with Theorem 3is said to be:(a) P 2 -like if the hv-curvature tensor P has the form:

3 M
and   , M L  are Riemannian.(b) If  , M L is P-reducible, then, by Definition 3.16(b), the (v)hv-to sion P satisfy Relation (20).Setting r  , is said to be h-otropic if there exists a scalar k o such that the horizontal curvature tensor R has the fo is rm Under the energy β-conformal change (4), we have:(a) if   , M L is h-isotropic, provided that   , 0 X Y Z  (defined by Theorem 3.5(d)), then the h-curvature tensors R and R  of the Cartan connection va   (Proposition 3.4(g) of[15]), we have From which and Theorem 3.5(c) (un sumption), the h-curvature tensors R a R 