Finslerian Ricci Deformation and Conformal Metrics

In this paper, a new Ricci flow is canonically introduced in Finsler Geometry and, under the variance of Finsler-Ehresmann form, conformal changes of Finsler metrics are studied. Some existence conditions of this Finslerian Ricci flow on a compact manifold which preserves the conformal class of the initial metric are obtained as an application.


Introduction
The Ricci flow is a very powerful tool in studying of the geometry of manifolds and has many applications in Mathematics and Physics.In Finsler Geometry, the problems on Ricci flow are very interesting.In 2017, to study deformation of Cartan curvature, Bidabad, Shahi and Ahmadi considered the Akbar-Zadeh's Ricci curvature and introduced certain Ricci flow for Finsler n-manifolds [1].
In this paper, we use the pulled-back bundle approach [2] to introduce a The existence and uniqueness for solution of the Equation (1) are known in special cases, particulary in Riemannian spaces and Berwald spaces [3].This Finslerian Ricci deformation generalizes the classical Riemannian one.
A Finsler metric F  on M is said to be conformally equivalent to F, if there exists a C ∞ function u on M such that u F e F =  . In this paper, [ ] F denotes the conformal class of F. We prove the following results. with , where Scal is the horizontal scalar curvature, ∇ is the Chern connection, h denotes a horizontal lift of a section of * TM π ∆ is the horizontal Laplacian, ∇ is the gradient and Θ is the ( ) 0, 2;1 -tensor on M measuring the variation of ∇ during the conformal change of F. , M F be an n-dimensional compact Finsler-Einstein manifold, g its fundamental tensor and t u the unique solution of Equation ( 2) , where [ ) is conformally equivalent to g.Then there exists a unique Ricci deformation The rest of this paper is organised as follows.In Section 2, we give some basic notions on Finsler manifolds.In Section 3, we prove the main results given above.

Preliminaries
In order to deal with the Finslerian Ricci deformation, it is preferable to use a global definition of Chern connection.We adopt the notations given in [2] and [4].
Let : TM M π → be the tangent bundle of a connected C ∞ manifold of dimension ( ) x y Throughout this paper, we use Einstein summation convention for the expressions with indices when an index appears twice as a subscript as well as a superscript in a term.Definition 2.1.A function 2) F is positively 1-homogeneous on the fibers of TM, that is 3) the Hessian matrix , 1 , : is positive definite at every point ( ) Given a manifold M and a Finsler metric F on TM, the pair ( ) .The fiber at a point ( ) By the objects (4), the vector bundle (see [2]).

Finsler-Ehresmann Connection and Chern Connection
For the differential mapping * and it is locally spanned by the set ,1 A horizontal subspace  of T TM  is by definition any complementary to  .The bundles  and  give a smooth splitting [5] .
The vertical bundle  is uniquely determined but the horizontal bundle  is not canonically determined.An Ehresmann connection is a selection of horizontal subspace  of T TM  .
In this paper, we consider the choice of Ehresmann connection which arises from the Finsler metric F and it is call Finsler-Ehresmann connection [6], constructed as follows.As explained in [7], all Finsler structure F on M induces a vector field on TM  in the form are y-homogeneous of degree two.The vector field G is called spray on M and the , 1, , This * TM π -valued C ∞ form θ is globally well defined on TM  [4].
The vertical lift of a section ξ of * TM π is a unique section ( )

T TM
 such that for every ( ) 2) The differential projection , : ,  x x y π = .
The horizontal lift of a section ξ of * TM π is a unique section ( )

T TM
 such that for every ( ) where 1, , : are respectively horizontal and vertical lifts of the natural basis 1, , 2) Almost g-compatibility: where  is the Cartan tensor defined in (6) and θ is the Finsler-Ehresmann form defined in (9).

On Finslerian Curvatures of Chern Connection
Let  be a vector bundle over TM  .Then, one denotes by . The tensors that will be considered are defined as follows: Definition 2.4.Let ( ) , M F be a Finsler manifold.A tensor field T of type , ; p p q on ( ) The full curvature φ , of Chern connection ∇ , is the ( ) where The full curvature φ can be written as , , 1) As in the Riemannian case, one can define a ( ) 2, 2;0 version of φ as follows: where R and P are respectively the hhand hv-curvature tensor of the Chern connection.One has 2) The hh-curvature tensor R is a generalization of the usual Riemannian curvature.
3) The hv-curvature tensor P is a Finsler non Riemannian curvature.Definition 2.5.Let ( ) , M F be a Finsler manifold, R the horizontal part of the full curvature tensor associated with the Chern connection.We define 1) the horizontal Ricci tensor , , , .

Y x y
x y g y y g g y where ξ ∈  is a noncolinear to the vector y, with X and Y such that  , x y τ has no dependence on either x nor y, ( ) , M F is said to be of constant scalar curvature.Now, we define the trace-free horizontal Ricci tensor and an Finsler-Einstein metric as follows.
1) The trace-free horizontal Ricci tensor of an n-dimensional Finsler manifold ( ) Scal .
In this case,

Horizontal Differential Operators on a Finsler Manifold
In this paragraph, we give fundamental horizontal differential operators on ( ) , M F .Remark 2.5.A differential operator O of order 2m defined on a differentiable manifold M is written as and f is assumed to be a differentiable function of its arguments.
2) For a C ∞ section ( ) , we define the horizontal divergence by where g is the fundamental tensor associated with F and ∇ is the Chern Definition 2.8.

1) Let ( )
, M F be a C ∞ Finsler manifold and ∇ the Chern connection on the pulled-back bundle * TM π .The horizontal Hessian of a C ∞ function u on M is the map 2) Let ( ) , M F be a C ∞ Finsler manifold and u a C ∞ function on M. The horizontal Laplacian of u are respectively defined by the following relation.

Finslerian Ricci Deformation and Conformal Metrics
In this section we prove the main results.
where { } 1, , a a n e =  is the special g-orthonormal basis sections for * TM π and Θ is the ( ) 0, 2;1 -tensor on ( ) with the dual section ( ) to the Cartan tensor and the ( ) .
Proof.The proof is straightforward from the definitions (0.5) and using the conformal change of the Chern connection given by Theorem 3.   Proof.The proof is straightforward from the definitions (0.5) and using the conformal change of the Chern connection given by Theorem 3.

Finslerian Ricci Flow
One of the advantages of the Finslerian Ricci tensor obtained by contraction of the Chern hh-curvature is its relation with the second covariant derivative and hence the horizontal Laplacian operators.

Let ( )
, t M F be an n-dimensional Finsler manifold of scalar flag curvature and { } [ )

Main Results
We first prove the Theorem 1 on the necessary condition for t F to be conformal equivalent to F. We find the existence of a family { } [ ) From the Equation (39) we obtain the result.
The following Proposition refers to the existence and uniqueness of solution of the Equation (2).We have Proposition 7. ( [8]) Let  be a bundle of tensors over a smooth compact Riemannian manifold ( ) , M g .We seek a smooth family [ )   ( ) Then the relations (41) and ( 2) are equivalent with Applying the Proposition 7 we obtain the result.

Finslerian,
horizontal Ricci flow, called Finslerian Ricci deformation.This approach is natural and is very important.The problem of construction of the Finslerian Ricci deformation contains a number of new conceptual.Let ( ) , M F be a Finsler manifold of flag scalar curvature.Then we consider a Finslerian Ricci tensor defined by Ric denotes again a Ricci tensor associated with F and a Finsler-Ehresmann connection  , ξ is a section of the vector bundle * TM π , X is a section of the tangent bundle T TM  of R is the hh-curvature of Chern

Corollary 1. Assume 2 n
≥ .Then the Equation(2) in Theorem 1 has a unique solution on a parameterized-interval [ ) 0, s for some s τ ≤ .Finally, using the trace-free horizontal Ricci tensor we prove the following Theorem 2. Let ( )

3 n
≥ .We denote by{ } \ 0 TM TM = the slit tangent bundle of M. We introduce a coordinate system on TM as follows.Let U M ⊂ back bundle * TM π is a vector bundle over the slit tangent bundle TM 

π
of the submersion : TM M

Definition 2 . 2 .
A Finsler-Ehresmann connection associated with the submersion.horizontal lift and vertical lift of a section of * TM π

.
The following theorem defines the Chern connection on the bundle * TM π .Theorem 3. [4] Let ( ) , M F be a Finsler manifold, g the fundamental tensor associated with F and * π the differential mapping of the submersion : TM M π →  .There exist a unique linear connection ∇ on the pulled-back tangent bundle * TM π such that, for all , edge ξ .A flag curvature is defined by

Remark 2 . 4 .
If K is independent of the transverse edge ξ , then the Finsler Journal of Applied Mathematics and Physics manifold ( ) , M F is called of scalar flag curvature.Denote this scalar by function u on M, the gradient of u, noted by u ∇ , is the section of *

.
The horizontal Laplacian H u ∆ of u can be given in term of the horizontal Hessian of u by definition of horizontal Laplacian.
Ric associated with F  and F respectively are conformally related by the equation:

Lemma 6 .
Let ( ) , M F be an n-dimensional Finsler manifold.If u F e F =  is Finsler metric conformal to F then the horizontal scalar curvatures  H F  Scal and H F Scal associated with F  and F respectively are conformally related by the equation: is the special g-orthonormal basis sections for * TM π.We have

2 n
≥ .Then the Equation (2) in Theorem 1 has a unique solution on a parameterized-interval [ ) 0, s for some s T ≤ .Proof.We put: a Finsler-Einstein metric, the trace-free horizontal Ricci tensor associated to F vanishes and then have and (44) in (42), we obtain Ricci deformation of F, by construction of [ ] [ ] derives from Theorem 1 and Corollary 2. Remark 3.1.If F is a Riemannian metric then, by the relation (32), the ( ) 0, 2,1 -tensor Θ vanishes.The Theorem 2 becomes the result in [9] for the Riemannian case.