1. 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 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:
(1)
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
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.
(2)
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
(3)
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.
2. 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
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
(4)
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
(5)
By the objects (4), the vector bundle
admits a Riemannian metric.
called fundamental tensor. Likewise, there is the Finslerian Cartan tensor
(6)
where
.
is a symmetric section of
. Note that,
and
are respectively regarded as basis sections of
and
(see [2] ).
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
(7)
A horizontal subspace
of
is by definition any complementary to
. The bundles
and
give a smooth splitting [5]
(8)
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 [6] , constructed as follows. As explained in [7] , 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
(9)
This
-valued
form
is globally well defined on
[4] .
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)
where
.
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)
where
.
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
and
where
and
are respectively horizontal and vertical lifts of the natural basis
for
.
The following theorem defines the Chern connection on the bundle
.
Theorem 3. [4] 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
and
,
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
(14)
where
with
and
.
The full curvature
can be written as
where
Remark 2.3.
1) As in the Riemannian case, one can define a
version of
as follows:
(15)
where
and
are respectively the hh- and hv-curvature tensor of the Chern connection. One has
(16)
and
(17)
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
(18)
for every
. In g-orthonormal basis sections
of
, we have
(19)
2) the horizontal scalar curvature
of
is the trace of the horizontal Ricci tensor.
is a function on
. In g-orthonormal basis
sections
of
,
(20)
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
(21)
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.
Definition 2.6.
1) The trace-free horizontal Ricci tensor of an n-dimensional Finsler manifold
is a
-tensor on
given by
(22)
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.
Definition 2.7.
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
(23)
for any
and for every
. Locally, one has
(24)
2) For a
section
, we define the horizontal divergence by
(25)
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:
(26)
Definition 2.8.
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
such that
(27)
2) Let
be a
Finsler manifold and u a
function on M. The horizontal Laplacian of u are respectively defined by the following relation.
(28)
Lemma 4. Let
. The horizontal Laplacian
of u can be given in term of the horizontal Hessian of u by
(29)
Furthermore, in g-orthonormale basis sections
, one has
(30)
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:
(31)
where
is the special g-orthonormal basis sections for
and
is the
-tensor on
given by
(32)
with the dual section
of
to the Cartan tensor and the
-tensor
(33)
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:
(34)
where
.
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
(35)
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] .
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
(36)
By Lemma 31 and using the Equation (36), we obtain
(37)
for every
and for every
where
is the special g-orthonormal basis sections for
. We have
(38)
where
. We then have
(39)
with
(40)
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
. We seek a smooth family
of smooth tensor fields on M
which satisfies the equation
(41)
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:
,
and
Then the relations (41) and (2) are equivalent with
.
Moreover,
,
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),
(42)
If F is a Finsler-Einstein metric, the trace-free horizontal Ricci tensor associated to F vanishes and then have
(43)
If the tensor
is conformally equivalent to g, then it holds
(44)
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 [9] for the Riemannian case.
4. Conclusions
In this present work, we use the pulled-back bundle approach [2] 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.