Concircular $\pi$-Vector Fields and Special Finsler Spaces

The aim of the present paper is to investigate intrinsically the notion of a concircular $\pi$-vector field in Finsler geometry. This generalizes the concept of a concircular vector field in Riemannian geometry and the concept of a concurrent vector field in Finsler geometry. Some properties of concircular $\pi$-vector fields are obtained. Different types of recurrence are discussed. The effect of the existence of a concircular $\pi$-vector field on some important special Finsler spaces is investigated. The whole work is formulated in a coordinate-free form.


Introduction
The concept of a concurrent vector field in Riemannian geometry had been introduced and investigated by K. Yano [6]. Concurrent vector fields in Finsler geometry had been studied locally by S. Tachibana [5], M. Matsumoto and K. Eguchi [3]. In [9], we investigated intrinsically concurrent vector fields in Finsler geometry. On the other hand, the notion of a concircular vector field in Riemannian geometry has been studied by Adat and Miyazawa [1]. Concircular vector fields in Finsler geometry have been studied locally by Prasad et. al. [4].
In this paper, we introduce and investigate intrinsically the notion of a concircular πvector field in Finsler geometry, which generalizes the concept of a concircular vector field in Riemannian geometry and the concept of a concurrent vector field in Finsler geometry. Some properties of concircular π-vector fields are obtained. These properties, in turn, play a key role in obtaining other interesting results. Different types of recurrence are discussed. The effect of the existence of a concircular π-vector field on some important special Finsler spaces is investigated : Berwald, Landesberg, C-reducible, semi-C-reducible, quasi-Creducible, C 2 -like, S 3 -like, P -reducible, P 2 -like, h-isotropic, T h -recurrent, T v -recurrent, etc.
Global formulation of different aspects of Finsler geometry may help better understand these aspects without being trapped into the complications of indices. This is one of the motivations of the present work, where almost all results obtained are formulated in a 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 [8] and [10]. We shall use the same notations of [8].
In what follows, we denote by π : T M −→ M the tangent bundle to M, F(T M) the algebra of C ∞ functions on T M, X(π(M)) the F(T M)-module of differentiable sections of the pullback bundle π −1 (T M). The elements of X(π(M)) will be called π-vector fields and will be denoted by barred letters X. The tensor fields on π −1 (T M) will be called π-tensor fields. The fundamental π-vector field is the π-vector field η defined by η(u) = (u, u) for all u ∈ T M.
We have the following short exact sequence of vector bundles with the well known definitions of the bundle morphisms ρ and γ. The vector space Let D be a linear connection on the pullback bundle π −1 (T M). We associate with D the map K : T (T M) −→ π −1 (T M) : X −→ D X η, called the connection map of D. The vector space H u (T M) = {X ∈ T u (T M) : K(X) = 0} is 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 γ, ρ| H(T M ) and K| V (T M ) are vector bundle isomorphisms. The map β := (ρ| H(T M ) ) −1 will be called the horizontal map of the connection D. We have K • γ = id π −1 (T M ) .
The horizontal ((h)h-) and mixed ((h)hv-) torsion tensors of D, denoted by Q and T respectively, are defined by where T is the (classical) torsion tensor field associated with D.
The horizontal (h-), mixed (hv-) and vertical (v-) curvature tensors of D, denoted by R, P and S respectively, are defined by where K is the (classical) curvature tensor field associated with D.
The contracted curvature tensors of D, denoted by R, P and S respectively, known also as the (v)h-, (v)hv-and (v)v-torsion tensors, are defined by If M is endowed with a metric g on π −1 (T M), we write (1.1) The following theorem guarantees the existence and uniqueness of the Cartan connection on the pullback bundle.
Theorem 1.1. [7] Let (M, L) be a Finsler manifold and g the Finsler metric defined by L. There exists a unique regular connection ∇ on π −1 (T M) such that (a) ∇ is metric : ∇g = 0, Such a connection is called the Cartan connection associated with the Finsler manifold (M, L).
One can show that the (h)hv-torsion of the Cartan connection is symmetric and has the property that T (X, η) = 0 for all X ∈ X(π(M)) [7].
Concerning the Berwald connection on the pullback bundle, we have , Y ∈ X(π(M)). In particular, we have:

Concircular π-vector fields on a Finsler manifold
The notion of a concircular vector field has been studied in Riemannian geometry by Adati and Miyazawa [1]. The notion of a concurrent vector field has been investigated locally (resp. intrinsically) in Finsler geometry by Matsumoto and Eguchi [3], Tachibana [5] (resp. Youssef et al. [9]). In this section, we investigate intrinsically the notion of a concircular π-vector field in Finsler geometry, which generalizes the concept of a concircular vector field in Riemannian geometry and the concept of concurrent vector field in Finsler geometry.
The following two Lemmas are useful for subsequence use.
(a) Using the fact that ∇g = 0, we have The proof is similar to that of (a).  (T M).
Proof. . We prove (a) only; the proof of (b) is similar.
where ∂ i = ∂ ∂x i ,∂ i = ∂ ∂y i and δ i , ∂ i are respectively the bases of the horizontal space and the pullback fibre. As ρ( where µ(X) := dψ(βX).
For the v-curvature tensor S, the following relations hold : For the hv-curvature tensor P , the following relations hold : For the h-curvature tensor R, the following relations hold 1 : Proof. The proof follows from the properties of the curvature tensors S, P and R, investigated in [11], together with Definition 2.1 and Remark 2.4, taking into account the fact that the (h)h-torsion of the Cartan connection vanishes.
In view of the above theorem, we retrieve a result of [9] concerning concurrent π-vector fields.
Corollary 2.6. Let ζ ∈ X(π(M)) be a concurrent π-vector field on (M, L). For the v-curvature tensor S, the following relations hold : For the hv-curvature tensor P , the following relations hold : For the h-curvature tensor R, the following relations hold : Proof. The proof follows from Theorem 2.5 by letting σ(x) be a constant function on M and ψ(x) = −1.
(a) From Theorem 2.5(e), by setting Z = ζ and making use of the symmetry of T and the identity g(P (X, Y )Z, Z) = 0 [11], we obtain From which, since ψ(x) = 0, the result follows.
From which, setting X = ζ, it follows that Hence, making use of (a), the symmetry of P and the fact that T (X, η) = 0, the result follows.
(d) We have from [11], From which, by setting Y = ζ in (2.3), using (b) and the symmetry of T , we conclude that P (X, ζ)Z =. Similarly, setting X = ζ in (2.3), using (a) and the symmetry of T , we get P (ζ, Y )Z = 0.
(e) The proof follows from Theorem 2.5(j) by setting Z = ζ , taking into account the fact that g(R(X, Y )Z, Z) = 0 [11].
Hence, there exists a scalar function λ such that Consequently, using (a) and the symmetry of T , we get This completes the proof.
Theorem 2.8. A concircular π-vector field ζ and its associated π-form ω are independent of the directional argument y.
Proof. By Theorem 1.3(a), we have . From which, by setting Y = ζ and taking into account (2.2), Proposition 2.7(a) and Lemma 2.3, we conclude that D • γX ζ = 0 and ζ is thus independent of the directional argument y.
On the other hand, we have from the above relation . This, together with Lemma 2.2(b), Proposition 2.7(a) and the symmetry of T , imply that ω is also independent of the directional argument y.
In view of Theorem 1.3 and Proposition 2.7, we have Theorem 2.9. A π-vector field ζ on (M, L) is concircular with respect to Cartan connection if, and only if, it is concircular with respect to Berwald connection. Remark 2.10. As a consequence of the above results, we retrieve a result of [9] concerning concurrent π-vector fields: A concurrent π-vector field ζ and its associated π-form ω are independent of the directional argument y. Moreover, a π-vector field ζ on (M, L) is concurrent with respect to Cartan connection if, and only if, it is concurrent with respect to Berwald connection.

Special Finsler spaces admitting concircular π-vector fields
Special Finsler manifolds arise by imposing extra conditions on the curvature and torsion tensors available in the space. Due to the abundance of such geometric objects in the context of Finsler geometry, special Finsler spaces are quite numerous. The study of these spaces constitutes a substantial part of research in Finsler geometry. A complete and systematic study of special Finsler spaces, from a global point of view, has been accomplished in [8].
In this section, we investigate the effect of the existence of a concircular π-vector field on some important special Finsler spaces. The intrinsic definitions of the special Finsler spaces treated here are quoted from [8].
On the other hand, the orthogonality of the two π-vector fields m and η follows from the identities g(η, η) = L 2 and g(η, ζ) = B.
As a consequence of the above result, we get Corollary 3.4. The existence of a concircular π-vector field ζ implies that the three notions of being Landsberg, Berwald and Riemannian coincide.

(d) quasi-C-reducible if dim M ≥ 3 and the Cartan tensor T has the from
where A is a symmetric π-tensor field satisfying A(X, η) = 0. (a) If (M, L) is quasi-C-reducible, then it is Riemannian, provided that A(ζ, ζ) = 0. Proof.
(a) Follows from the defining property of quasi-C-reducibility by setting X = Y = ζ and using the fact that C(ζ) = 0 and the given assumption A(ζ, ζ) = 0.
(a) Setting Z = ζ in (3.6), taking into account Theorem 2.5 and Proposition 2.7, we immediately get Hence, the result follows.
where k o is called the scalar curvature.
Theorem 3.12. For an h-isotropic Finsler manifold admitting a concircular π-vector field ζ, the scalar curvature k o is given by Proof. From Definition 3.11, by setting Z = ζ and X = m, we have On the other hand, using Theorem 2.5(i), we have From (3.8) and (3.9), it follows that Taking the trace of the above equation, we get Hence, the scalar k o is given by This completes the proof.
Corollary 3.13. For an h-isotropic Finsler manifold admitting a concurrent π-vector field ζ, the h-curvature R vanishes.
Proof. If ζ is concurrent, then the π-form A vanishes. Hence, using (3.10), the scalar k o vanishes. Consequently, from Definition 3.11, the h-curvature R vanishes.

Different types of recurrent Finsler manifolds admitting concircular π-vector fields
In this section, we investigate intrinsically the effect of the existence of a concircular π-vector field on recurrent Finsler manifolds. We study different types of recurrence (with respect to Cartan connection).
Let us begin with the first type of recurrence related to the Cartan tensor T .

Definition 4.1. A Finsler manifold (M, L) is said to be T h -recurrent if the (h)hv-torsion tensor T has the property that
where λ 1 is a scalar (1) π-form, positively homogenous of degree zero in y, called the h-recurrence form.
where λ 2 is a scalar (1) π-form, positively homogenous of degree −1 in y, called the v-recurrence form.
Proof. We have [11] Setting W = ζ, making use of Theorem 2.5, Proposition 2.7 and the identity [11] g On the other hand, Definition 4.1 yields Under the given assumption, the above two equations imply that T = 0. Hence, (M, L) is Riemannian.
In view of Theorem 4.10 and Lemma 4.11, we have Theorem 4.12. Let (M, L) be a Finsler manifold admitting a concircular π-vector field. Then, the following assertions are equivalent : Remark 4.13. In view of Theorem 4.12, we conclude that under the presence of a concurrent π-vector field ζ, the three notions of being P h -recurrent, P v -recurrent and Riemannian coincide, provided that λ 1 (ζ) = 0.
Finally, we focus our attention to the fourth type of recurrent Finsler manifolds related to the h-curvature tensor R. Theorem 4.15. An R h -recurrent Finsler manifold admitting a concircular π-vector field ζ is h-isotropic with scalar curvature Moreover, if (M, L) is R v -recurrent with λ 2 (η) = 0, then the h-curvature tensor R vanishes.
This means that (M, L) is h-isotropic (Definition 3.11) with scalar curvature k o = φo n . Finally, the second part of the theorem follows from Definition 4.14 and the identity (∇ γη R)(X, Y , Z) = 0 [11].
As a consequence of the above theorem, we have Corollary 4.16. For an R h -recurrent Finsler manifold admitting a concurrent π-vector field ζ, the h-curvature tensor R vanishes.
• The concept of a concircular π-vector field in Finsler geometry has been introduced and investigated from a global point of view. This generalizes, on one hand, the concept of a concircular vector field in Riemannian geometry and, on the other hand, the concept of a concurrent vector field in Finsler geometry. Various properties of concircular π-vector fields have been obteined.
• The effect of the existence of concircular π-vector fields on some of the most important special Finsle spaces has been investigated.
• Different types of recurrent Finsler manifolds admitting concircular π-vector fields have been studied.
• Almost all results of this work have been obtained in a coordinate-free form, without being trapped into the complications of indices.