Harmonic vector fields and the Hodge Laplacian operator on Finsler geometry

We first present the natural definitions of the horizontal differential, the divergence (as an adjoint operator), and a $p$-harmonic form on a Finsler manifold. Next, we prove a Hodge-type theorem for a Finsler manifold in the sense that a horizontal $p$-form is harmonic if and only if the horizontal Laplacian vanishes. This viewpoint provides a new appropriate natural definition of harmonic vector fields in Finsler geometry. This approach leads to a Bochner-Yano type classification theorem based on the harmonic Ricci scalar. Finally, we show that a closed orientable Finsler manifold with a positive harmonic Ricci scalar has a zero Betti number.


Introduction
The existence of harmonic vector fields on the Riemannian manifolds is directly related to the sign of the Ricci tensor. Bochner and Yano have studied the non-existence of harmonic vector fields on the compact Riemannian manifolds with positive Ricci curvature based on the Laplace-Beltrami operator. Next, Bochner proved that if the Ricci curvature on a Riemannian manifold is positive-definite, then all harmonic vector fields vanish [6]. Yano proved that a vector field X is harmonic, if and only if the Laplacian of its corresponding 1-form vanishes [12,13].
In Finsler geometry, Akbar-Zadeh introduced the divergence of horizontal and vertical 1-forms on SM without defining the harmonic forms on a Finsler manifold, where SM := x∈M S x M and S x M := {y ∈ T x M | F (y) = 1}, [1].
Harmonic forms in Finsler geometry are studied in [3,4,8,14]. Recently, the second author introduced a definition of harmonic vector fields on a Finsler manifold, which is slightly modified here in the present work, see [9,10], and Remark 10 in this article. Moreover some natural extensions of Riemannian results, more or less linked to this question are studied in [5].
In the present work, the horizontal differential operator d H and the horizontal co-differential operator δ H , are defined as adjoint operators. The above operators provide a Finslerian version of a well-known Hodge theorem on the Riemannian manifolds in the following sense. (1) We can thus define harmonic p-forms naturally on a Finsler manifold in the sense that, a horizontal p-form is harmonic if and only if the horizontal Laplacian vanishes.
The definition of harmonic p-forms on SM will provide a new definition of a harmonic vector field on a Finsler manifold in the sense that, a vector field on (M , F ) is harmonic if and only if the horizontal Laplacian vanishes.
Finally, we obtain a classification of harmonic vector fields based on the harmonic Ricci scalar Ric defined by the equation (32). • If Ric = 0, then X is parallel.
This theorem is an extension of a well-known result obtained by Bochner and Yano, see Theorem 16. Finally, this brings us to the following fundamental results.  In Section 2, the necessary tools, concepts and definitions of Finsler geometry using the Cartan connection are stated. In Section 3, the definition of Λ H p (SM ) the space of horizontal p-forms and the definition of d H the horizontal divergence operator on the unit fiber bundle SM with an inner product ( · , · ) on Λ H p (SM ) are expressed. In Section 4, the definition of the horizontal (co-differential) divergence, a horizontal Laplacian and a new type of harmonic pform are introduced using the horizontal Laplacian. Section 5 deals with harmonic vector fields on Finsler manifolds where the proof of Theorem 2 is presented. In Section 6, we prove that a closed orientable Finsler manifold with a positive harmonic Ricci scalar has zero Betti number.

Preliminaries and notations
We first recall some Riemannian definitions of harmonic analysis. Let (M , g ) be a compact and orientable Riemannian manifold of dimension n. A p-form on (M , g ) for 1 ≤ p ≤ n is given by where the indices i 1 , . . . , i p run over the range 1, . . . , n and the coefficients are components of the skew-symmetric tensor fields of type (0, p). The differential dϕ is a (p + 1)−form given by where the coefficients are components of the skew-symmetric tensor fields of type (0, p + 1) and ∇ j are the components of Levi-Civita covariant derivative. The co-differential δϕ is a (p −1)−form given by where the coefficients are components of the skew-symmetric tensor fields of type (0, p − 1). The co-differential of a scalar function is defined to be zero. It is easy to verify that d(dϕ) = 0 and δ(δϕ) = 0, see for instance [13]. In Riemannian geometry a differential form ϕ is called harmonic if it satisfies dϕ = 0 and δϕ = 0. A vector field X is said to be harmonic if its associated 1-form is harmonic. It is well known that a necessary and sufficient condition for a p-form ϕ to be harmonic is where ∆ is called Laplacian, see [13] for more details.
The pair δ δx i , ∂ ∂y i forms a horizontal and vertical frame for and we have the following Whitney sum cf. [11, p. 29].
The Cartan connection is a natural extension of the Riemannian connection, which is metric compatible and semi-torsion free. For a global approach to the Cartan connection one can refer to [1]. According to the definition, the 1-forms of Cartan connection with respect to the dual basis {dx i , δy i } are given by where, Γ i j k and C i j k are the horizontal and vertical coefficients of Cartan connection respectively defined by and Let us consider the components of an arbitrary (2,2)-tensor field T j k i s on T M . The horizontal and vertical components of the Cartan connection of T j k i s in a local coordinates are given respectively by The curvature tensor in Cartan connection is given by the hh-curvature, hv-curvature and vvcurvature with the following components, cf. [1]; Trace of the hh-curvature of Cartan connection is denoted by R i j := R l i l j , which is not symmetric in general.
Let (M , F ) be a Finsler manifold, π : T M 0 → M the bundle of non-zero tangent vectors and π * T M the pullback bundle. The tangent space T x M , x ∈ M can be considered as a fiber of the pullback bundle π * T M . Therefore a section X on π * T M is denoted by X = X i (x, y) ∂ ∂x i . The Ricci identity for Cartan connection is given by the following equation cf. [1]. Now we are in a position to define some basic notions on harmonic forms on Finsler manifolds.

The p-forms and horizontal operators
Here and everywhere in this paper, we assume the differential manifold M is compact and without boundary or simply closed. Let (M , F ) be a closed Finsler manifold, u : M → SM a unitary vector field and ω = u i dx i the corresponding 1-form on M . A volume element on SM is given by Let π = a i (z)dx i be a horizontal 1-form on SM . The co-differential or divergence of π concerning the Cartan connection is defined by where, T ki j = C ki j = 1 2 ∂g i j ∂y k , are the components of Cartan tensors and where a i = g i j a j , cf. [1, p. 67]. Let us denote the horizontal part of the differential dπ by cf. [1, p. 224]. According to the above discussion, we are in a position to define a horizontal differential operator in the following sense.
a horizontal p-form on SM . A horizontal differential operator is a differential operator on SM given by where, for 1 ≤ i , i k ≤ n and 1 ≤ k ≤ p, we have Let ϕ and π be the two arbitraries horizontal p-forms on SM with the components ϕ i 1 ...i p and π i 1 ...i p , respectively. We consider an inner product ( · , · ) on Λ H p as follows where, ϕ i 1 ...i p = g i 1 j 1 . . . g i p j p ϕ j 1 ... j p .

The horizontal Laplacian and harmonic p-forms
Using the above concepts, we define the horizontal Laplacian. This definition of Laplacian is different from those given in [1,4] and [11]. Let (M , F ) be a Finsler manifold and ψ a horizontal (p+1)-form on SM , given by We define the horizontal divergence (co-differential) of ψ by Remark 6. If ϕ is a horizontal 1-form on SM , then δ H reduces to δ, and we have

Definition 7. Let (M , F ) be a Finsler manifold. A horizontal Laplacian on SM is defined by
where d H and δ H are horizontal differential and horizontal co-differential operators on SM , respectively.
Now we are able to show the basic equivalence relation in the following theorem.

Horizontal Laplacian of p-forms
Let ϕ be a horizontal p-form on SM , by definitions of horizontal differential and co-differential we can easily see that and On the other hand, by definition we have The equations (24) and (25) yield In particular for an arbitrary horizontal 1-form ϕ = ϕ i (z)dx i on SM , the above equation reduces to This fact gives rise to a new definition of horizontal harmonic vector fields on Finsler manifolds.

Definition 8. A horizontal p-form ϕ on SM is called horizontally harmonic if we have
The horizontal harmonic p-forms will be referred to in the following as h-harmonic p-forms or simply h-harmonic.
Remark 9. C. Bertrand and A. Rauzy, using a horizontal lift of a p-form on M to SM have defined the Laplacian on a Finsler manifold which is different from our point of view. More intuitively, they construct a sub-elliptic operator on the associated unitary bundle and give a lower bound for the first eigenvalue of this operator by using the horizontal Ricci tensor of the Berwald connection, see [4].

The harmonic vector fields on Finsler manifolds
Recently, one of the present authors has introduced in a joint work a definition for harmonic vector fields on Finsler manifolds using the Cartan and Berwald connections in the following sense.

Remark 10. Let (M , F ) be a closed Finsler manifold. A vector field
and ∇ and D are the covariant derivatives of Cartan and Berwald connections, respectively, cf. [9,10].
The above definition of harmonic vector fields and the corresponding harmonic 1-forms have some inconveniences. First, it could not be easily extended to the harmonic p-forms on Finsler manifolds. In particular, the occurrence of the mixed terms of differential and co-differential could not be readily established in the Finsler setting. Second, the both Berwald's and Cartan's covariant derivatives must be considered in this calculations which needs more preliminaries for this definition. Finally, contrary to the definition of harmonic vector fields on the Riemannian manifolds, we do not have the following proper bilateral relation in general; The remedy lies in a slight modification of definition in the following sense. Let X = X i (x) ∂ ∂x i be a vector field on M . One can associate to X a 1-form X on SM defined by , and z ∈ SM [1]. The horizontal part of the associated 1form X on SM is called associate horizontal 1-form and denoted by X = X i (z)dx i . Definition 11. Let (M , F ) be a Finsler manifold. A vector field X = X i (x) ∂ ∂x i on M is called harmonic related to the Finsler structure F if the associate horizontal 1-form X = X i (z)dx i is hharmonic on SM .

Remark 12.
According to this definition of the Finslerian harmonic vector field, if X is a harmonic vector field concerning the Finsler structure F , then the associate horizontal 1-form X = X i (z)dx i , is h-harmonic on SM , where X i (z) is a real function on SM and z = (x, y) ∈ SM .

Theorem 13. Let (M , F ) be a closed Finsler manifold. A vector field ϕ = ϕ i ∂ ∂x i on M is harmonic if and only if
Proof. The Ricci identity (7) yields Substituting the last equation in (27) we get the result.   F ) is Riemannian, then the above equation reduces to the following well known form.
∂x i be a vector field on (M , F ). Inspired by [9] and [10] and based on the Ricci tensor, we define the harmonic Ricci scalar Ric as follows Furthermore, we obtain a classification result given in Theorem 2.
Proof of Theorem 2. Let X = X i (x) ∂ ∂x i be a vector field on (M , F ) and Y and Z two 1-forms on SM defined at z ∈ SM by Y = (X k ∇ k X i )(z)dx i and Z = (X i ∇ j X j )(z)dx i , respectively. Using (9) and similarly The difference of δZ and δY yields On the other hand we have Therefore Replacing (36) and (7) in (35) we obtain If X is a harmonic vector field, then by definition of Ric given by (32) the last equation becomes δZ − δY = ∥∇X ∥ 2 + Ric.
By integration over SM and using (10), we obtain If Ric = 0, or X k X t R t k = X k∇ r X j R r j k + X k ∇ k X j ∇ 0 T j , then (38) yields the first assertion. If Ric > 0, that is, if we have X k X t R t k > X k∇ r X j R r j k + X k ∇ k X j ∇ 0 T j , then using the equation (38) we get the second assertion.  (1) If Ric(X , X ) = X k X t R t k = 0, then X is parallel.
(2) If Ric(X , X ) = X k X t R t k > 0, then X vanishes.

Cohomology class and Betti number
On a smooth manifold M the de Rham cohomology H 1 dR (M ) := Z 1 (M )/B 1 (M ), is an equivalence class of the closed forms on M . The fact that a closed form is not exact indicates that the manifold has a certain global topological structure that prevents the existence of any hole or twist. The de Rham cohomology class is therefore, a way to understand, via the tangent bundle, the global topology of a manifold.
On a compact Riemannian manifold, every equivalence class in H k dR (M ) contains exactly one harmonic form. That is, every member ω of a given equivalence class of closed forms can be written as ω = α + γ where α is exact and γ is harmonic, i.e. ∆γ = 0.
The dimension of the space of all harmonic forms of degree p on a manifold M is called the pth Betti number of the manifold.
Due to Hodge theory, the first Betti number is equal to the dimension of the space of harmonic 1-forms on M , and this space is isomorphic to H 1 dR (M ). As mentioned earlier, on a Finsler manifold (M , F ), a vector field is harmonic if X = X i (x, y)dx i , the associate horizontal 1-form on SM , is h-harmonic. Hence the definition of a harmonic form on (M , F ) is closely related to the Finsler structure F .
The following theorem will be used in the sequel.

Theorem 17 ([7]
). If A is a closed, nonempty, convex subset of a Hilbert space B , then for every y in B there is a unique x in A that minimizes the distance from y to A.
We are now able to prove Theorem 3.
Let θ = θ i (x)dx i ∈ Z 1 (M ) such that θ = θ i (x, y)dx i ∈ Z 1 (SM ) is the associate 1-form on SM . Using Theorem 17, three is a unique minimizer, say f 0 ∈ C ∞ (SM ) such that ∥θ i (x, y)dx i − d H f 0 ∥ 2 is minimized. For all f ∈ C ∞ (SM ) and t ∈ R we have Since ∥θ i (x, y)dx i − d H f 0 − t d H f ∥ 2 has a unique minimum at t = 0, we deduce for all f ∈ C ∞ (SM ). On the other hand The equations (40) and (41) yield δ H (θ i (x, y)dx i − d H f 0 ) = 0 and the proof is complete.

□
We then prove the corollary.
Proof of Corollary 4. Let (M , F ) be a closed orientable Finsler manifold and X a harmonic vector field related to F . Assuming Ric > 0, the second part of Theorem 2 asserts that the harmonic vector field X related to F vanishes identically. Theorem 3 yields that the dimension of the space of all harmonic forms of degree one is the first Betti number of the manifold. Hence the first Betti number is b 1 = 0.