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  .
In this paper, we use the pulled-back bundle approach  to introduce a 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 be a Finsler manifold of flag scalar curvature. Then we consider a Finslerian Ricci tensor defined by . In this definition, denotes again a Ricci tensor associated with F and a Finsler-Ehresmann connection , is a section of the vector bundle , X is a section of the tangent bundle of , is the hh-curvature of Chern connection, is the special g-orthonormal basis section for and is the orthonormal basis section for . Let be an n-dimensional Finsler manifold and ( being a finite parameter) a family of fundamental tensors of . We consider the following Finslerian Ricci deformation:
The existence and uniqueness for solution of the Equation (1) are known in special cases, particulary in Riemannian spaces and Berwald spaces  . This Finslerian Ricci deformation generalizes the classical Riemannian one.
A Finsler metric on M is said to be conformally equivalent to F, if there exists a function u on M such that . In this paper, denotes the conformal class of F. We prove the following results.
Theorem 1. Let be an n-dimensional compact Finsler manifold, a Ricci deformation of F for and the fundamental tensor of .
If for all then there exists a family of functions on M satisfying the following equation.
with , where is the horizontal scalar curvature, is the
Chern connection, denotes a horizontal lift of a section of , is the horizontal Laplacian, is the gradient and is the -tensor on M measuring the variation of during the conformal change of F.
Corollary 1. Assume . Then the Equation (2) in Theorem 1 has a unique solution on a parameterized-interval for some .
Finally, using the trace-free horizontal Ricci tensor we prove the following
Theorem 2. Let be an n-dimensional compact Finsler-Einstein manifold, g its fundamental tensor and the unique solution of Equation (2) on for some , where is the maximal parameterized-interval on which (1) has a solution. Assume that the tensor , defined for a and in local coordinate by
is conformally equivalent to g. Then there exists a unique Ricci deformation of F such that .
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.
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  and  .
Let be the tangent bundle of a connected manifold of dimension . We denote by the slit tangent bundle of M. We introduce a coordinate system on TM as follows. Let be an open
set with a local coordinate on U. By setting for every , we introduce a local coordinate on .
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 is called a Finsler metric on M if:
1) F is on the entire slit tangent bundle ,
2) F is positively 1-homogeneous on the fibers of TM, that is ,
3) the Hessian matrix with elements
is positive definite at every point of .
Given a manifold M and a Finsler metric F on TM, the pair is called a Finsler manifold.
Remark 2.1. for all and for every .
The pulled-back bundle is a vector bundle over the slit tangent bundle . The fiber at a point is defined by
By the objects (4), the vector bundle admits a Riemannian metric.
called fundamental tensor. Likewise, there is the Finslerian Cartan tensor
where . is a symmetric section of . Note that,
and are respectively regarded as basis sections of and (see  ).
2.1. Finsler-Ehresmann Connection and Chern Connection
For the differential mapping of the submersion , the vertical subbundle of is defined by and it is locally spanned
by the set , on each . Then it induces the short exact sequence
A horizontal subspace of is by definition any complementary to . The bundles and give a smooth splitting 
The vertical bundle is uniquely determined but the horizontal bundle is not canonically determined. An Ehresmann connection is a selection of horizontal subspace of .
In this paper, we consider the choice of Ehresmann connection which arises from the Finsler metric F and it is call Finsler-Ehresmann connection  , constructed as follows. As explained in  , all Finsler structure F on M induces a vector field on in the form
where and the elements
are y-homogeneous of degree two. The vector field G is called spray on M and the are called spray coefficients of G.
Consider the functions
one defines a -valued form on by
This -valued form is globally well defined on  .
From the form , called Finsler-Ehresmann form, defined in (9), one defines a Finsler-Ehresmann connection as follows.
Definition 2.2. A Finsler-Ehresmann connection associated with the submersion.
is the subbundle of given by , where is the bundle morphism from to defined in (9), and which is complementary to the vertical subbundle .
Now we define horizontal lift and vertical lift of a section of as follows.
Definition 2.3. Let be the restricted projection.
1) The form induces a linear map
, for each point ; (10)
The vertical lift of a section of is a unique section of such that for every
and . (11)
2) The differential projection induces a linear map
, for each point ; (12)
The horizontal lift of a section of is a unique section of such that for every
and . (13)
Remark 2.2. The vector bundle can be naturally identified with the horizontal subbundle of or with the vertical bundle . Thus any section of is considered as a section of or a section of . In fact
are respectively horizontal and vertical lifts of the natural basis for .
The following theorem defines the Chern connection on the bundle .
Theorem 3.  Let be a Finsler manifold, g the fundamental tensor associated with F and the differential mapping of the submersion . There exist a unique linear connection on the pulled-back tangent bundle such that, for all
one has the following properties:
1) Symmetry: ,
2) Almost g-compatibility:
where is the Cartan tensor defined in (6) and is the Finsler-Ehresmann form defined in (9).
2.2. On Finslerian Curvatures of Chern Connection
Let be a vector bundle over . Then, one denotes by the -module of differentiable sections of , where
denotes the fibered product of . By convention . The tensors that will be considered are defined as follows:
Definition 2.4. Let be a Finsler manifold. A tensor field T of type on is a mapping
which is -linear in each arguments.
The full curvature , of Chern connection , is the -tensor defined by
where and . Using the decomposition (8), we have
where with and .
The full curvature can be written as
1) As in the Riemannian case, one can define a version of as follows:
where and are respectively the hh- and hv-curvature tensor of the Chern connection. One has
2) The hh-curvature tensor is a generalization of the usual Riemannian curvature.
3) The hv-curvature tensor is a Finsler non Riemannian curvature.
Definition 2.5. Let 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 of by
for every . In g-orthonormal basis sections of , we have
2) the horizontal scalar curvature of is the trace of the horizontal Ricci tensor. is a function on . In g-orthonormal basis
sections of ,
3) Let be a Finsler manifold and g its fundamental tensor. Consider a flag , that is a 2-dimensional subspace of , a flag-pole and a transverse edge . A flag curvature is defined by
where is a noncolinear to the vector y, with X and Y such that and .
Remark 2.4. If is independent of the transverse edge , then the Finsler manifold is called of scalar flag curvature. Denote this scalar by . When has no dependence on either x nor y, 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 is a -tensor on given by
where that is the pullback of g by the submersion ; and
for every and for any , ,
is the horizontal Ricci tensor and is the horizontal scalar curvature of .
2) An n-dimensional Finsler manifold is horizontally Einstein if the trace-free horizontal Ricci tensor associated with F vanishes, that is
In this case, is a function on M for .
2.3. Horizontal Differential Operators on a Finsler Manifold
In this paragraph, we give fundamental horizontal differential operators on .
Remark 2.5. A differential operator O of order 2m defined on a differentiable manifold M is written as
where and f is assumed to be a differentiable function of its arguments.
1) Let be the canonical mapping defined by . For a smooth function u on M, the gradient of u, noted by , is the section of , given by
for any and for every . Locally, one has
2) For a section , we define the horizontal divergence by
where g is the fundamental tensor associated with F and is the Chern connection.
Remark 2.6. In the local basis sections of the bundle , we have:
1) Let be a Finsler manifold and the Chern connection on the pulled-back bundle . The horizontal Hessian of a function u on M is the map
2) Let be a Finsler manifold and u a function on M. The horizontal Laplacian of u are respectively defined by the following relation.
Lemma 4. Let . The horizontal Laplacian of u can be given in term of the horizontal Hessian of u by
Furthermore, in g-orthonormale basis sections , one has
Proof. By definition of horizontal Laplacian.
3. Finslerian Ricci Deformation and Conformal Metrics
In this section we prove the main results.
Lemma 5. Let be an n-dimensional Finsler manifold. If is Finsler metric conformal to F then the horizontal Ricci tensors and associated with and F respectively are conformally related by the equation:
where is the special g-orthonormal basis sections for and is the -tensor on given by
with the dual section of to the Cartan tensor and the -tensor
Proof. The proof is straightforward from the definitions (0.5) and using the conformal change of the Chern connection given by Theorem 3.
Lemma 6. Let be an n-dimensional Finsler manifold. If is Finsler metric conformal to F then the horizontal scalar curvatures and associated with and F respectively are conformally related by the equation:
Proof. The proof is straightforward from the definitions (0.5) and using the conformal change of the Chern connection given by Theorem 3.
3.1. 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 be an n-dimensional Finsler manifold of scalar flag curvature
and the fundamental tensors family of . We consider the following Finslerian Ricci deformation
where is the horizontal Ricci tensor defined by (19). The existence of solutions is known in special cases, particulary in Riemannian and Berwald spaces,  .
3.2. Main Results
We first prove the Theorem 1 on the necessary condition for to be
conformal equivalent to F. We find the existence of a family of
functions on M satisfying a parabolic type Equation (2) with initial function equal to zero.
Proof of Theorem 1.1. We denote by and g the fundamental tensors of and F respectively. If then there exists a function on M such that or equivalently . Then by definition of Finslerian Ricci flow given by relation (1), we have
By Lemma 31 and using the Equation (36), we obtain
for every and for every where is the special g-orthonormal basis sections for . We have
where . We then have
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. (  ) Let be a bundle of tensors over a smooth compact Riemannian manifold . We seek a smooth family of smooth tensor fields on M which satisfies the equation
is given and belongs to , the components of a double contravariant symmetric tensor field on M are, in local chart, smooth functions in its arguments, and , with values in , is smooth in its components. If the tensor field is everywhere positive definite, then there exists a unique smooth solution f on for some .
Fom the Proposition 7, we get
Corollary 2. Let be an n-dimensional compact Finsler manifold. Assume . Then the Equation (2) in Theorem 1 has a unique solution on a parameterized-interval for some .
Proof. We put: ,
Then the relations (41) and (2) are equivalent with .
which is positive definite since is positive and is nonnegative. Applying the Proposition 7 we obtain the result.
When a Finsler metric F is of Einstein type, we obtain the existence on a compact Finsler-Einstein manifold of a Ricci flow which preserves the conformal class.
Proof of Theorem 1.2. For , . Since , we have
It follows that, since is solution of Equation (2),
If F is a Finsler-Einstein metric, the trace-free horizontal Ricci tensor associated to F vanishes and then have
If the tensor is conformally equivalent to g, then it holds
Putting (43) and (44) in (42), we obtain
Hence, is a Ricci deformation of F, by construction of , . The uniqueness derives from Theorem 1 and Corollary 2.
Remark 3.1. If F is a Riemannian metric then, by the relation (32), the -tensor vanishes. The Theorem 2 becomes the result in  for the Riemannian case.
In this present work, we use the pulled-back bundle approach  to introduce a Finslerian horizontal Ricci flow, called Finslerian Ricci deformation. This approach is natural and is very important because it facilitates the analogy with the Riemannian geometry.
Using the results of the previous sections we plan to study, in the future, the evolution of the Chern hh-curvature and its various traces under the Finslerian Ricci deformation.
Conflicts of Interest
The authors declare no conflicts of interest regarding the publication of this paper.
 Azami, S. and Razavi, A. (2013) Existence and Uniqueness for Solution of Ricci Flow on Finsler Manifolds. International Journal of Geometric Methods in Modern Physics, 10, 1-21.