The concept of a concurrent vector field in Riemannian geometry had been introduced and investigated by K. Yano [1]. Concurrent vector fields in Finsler geometry had been studied locally by S. Tachibana [2], M. Matsumoto and K. Eguchi [3]. In [4], 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 [5]. Concircular vector fields in Finsler geometry have been studied locally by Prasad et al. [6].

In this paper, we introduce and 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 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-creducible, quasi-c-reducible, c₂-like, s₃-like, p-reducible, p₂-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.

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 [7] and [8]. We shall use the same notations of [7].

In what follows, we denote by the tangent bundle to, the algebra of functions on, the -module of differentiable sections of the pullback bundle. The elements of will be called -vector fields and will be denoted by barred letters. The tensor fields on will be called -tensor fields. The fundamental -vector field is the -vector field defined by for all.

We have the following short exact sequence of vector bundles

with the well known definitions of the bundle morphisms and. The vector space is the vertical space to at.

Let be a linear connection on the pullback bundle. We associate with the map called the connection map of. The vector space is the horizontal space to at. The connection is said to be regular if

If is endowed with a regular connection, then the vector bundle maps and are vector bundle isomorphisms. The map will be called the horizontal map of the connection. We have.

The horizontal ((h)h-) and mixed ((h)hv-) torsion tensors of, denoted by and respectively, are defined by

where is the (classical) torsion tensor field associated with.

The horizontal (h-), mixed (hv-) and vertical (v-) curvature tensors of, denoted by and respectively, are defined by

where is the (classical) curvature tensor field associated with.

The contracted curvature tensors of, denoted by and respectively, known also as the (v)h-, (v)hvand (v)v-torsion tensors, are defined by

If is endowed with a metric on, we write

The following theorem guarantees the existence and uniqueness of the Cartan connection on the pullback bundle.

Theorem 2.1. [9] Let be a Finsler manifold and the Finsler metric defined by. There exists a unique regular connection on such that 

a) is metric:;

b) The (h)h-torsion of vanishes:;

c) The (h)hv-torsion of satisfies:

.

Such a connection is called the Cartan connection associated with the Finsler manifold.

One can show that the (h)hv-torsion of the Cartan connection is symmetric and has the property that for all [9].

Concerning the Berwald connection on the pullback bundle, we have Theorem 2.2. [9] Let be a Finsler manifold. There exists a unique regular connection on such that a);

b) is torsion-free:;

c) The (v)hv-torsion tensor of vanishes:

.

Such a connection is called the Berwald connection associated with the Finsler manifold.

Theorem 2.3. [9] Let be a Finsler manifold. The Berwald connection is expressed in terms of the Cartan connection as

In particular, we have: 

a)

b).

Finally, for a Finsler manifold, we use the following definitions and notations:

,

the angular metric tensor,

the Cartan tensor,

the contracted torsion,

is the -vector field associated with the -form,

the v-curvature (hv-crvature, h-curvature) tensor of Cartan connection.

the vertical Ricci tensor,

the vertical Ricci map

: the vertical Scalar curvature,

: the h-covariant derivative associated with the Cartan connection,

: the v-covariant derivative associated with the Cartan connection.

The notion of a concircular vector field has been studied in Riemannian geometry by Adati and Miyazawa [5]. The notion of a concurrent vector field has been investigated locally (resp. intrinsically) in Finsler geometry by Matsumoto and Eguchi [3], Tachibana [2] (resp. Youssef et al. [4]). In this section, we investigate intrinsically the notion of a concircular -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.

Definition 3.1. Let be a Finsler manifold. A π-vector field is called a concircular π-vector field (with respect to the Cartan connection) if it satisfies the following conditions:

where; and are two non-zero scalar functions on.

In particular, if is constant and, then is a concurrent -vector field.

The following two Lemmas are useful for subsequent use.

Lemma 3.2. Let be a Finsler manifold. If is a concircular -vector field and is the -form defined by, then has the properties:

a)b).

Proof.

a) Using the fact that, we have

b) The proof is similar to that of (a). □

Lemma 3.3. Let be a Finsler manifold and the Berwald connection on. Then, we have 

a) A -vector field is independent of the directional argument if, and only if, for all;

b) A scalar (vector) -form is independent of the directional argument if, and only if, for all.

Proof. We prove (a) only; the proof of (b) is similar. Let. Then,

where and are respectively the bases of the horizontal space and the pullback fibre. As, we have , and so

is in dependent of.□

Remark 3.4. From Definition 2.1, Lemma 2.3 and Theorem 1.3, we conclude that a);

b);

c)where.

Now, we have the following 

Theorem 3.5. Let be a concircular -vector field on.

For the v-curvature tensor, the following relations hold¹:

a);

b);

c);

d)

For the hv-curvature tensor, the following relations hold:

e);

f);

g);

h);

For the h-curvature tensor, the following relations hold1:

i);

j);

k);

l)

m)

Proof. The proof follows from the properties of the curvature tensors and, investigated in [10], 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 [4] concerning concurrent -vector fields.

Corollary 3.6. Let be a concurrent -vector field on.

For the v-curvature tensor, the following relations hold:

a);

b)

;

c).

For the hv-curvature tensor, the following relations hold:

d)

;

e)

;

f).

For the h-curvature tensor, the following relations hold:

g);

h)

;

i).

Proof. The proof follows from Theorem 2.5 by letting be a constant function on and. □

Proposition 3.7. Let be a concircular -vector field. For every, we have:  

a);

b);

c);

d);

e);

f).

Proof.

a) From Theorem 2.5(e), by setting and making use of the symmetry of and the identity [10], we obtain

From which, since, the result follows.

b) We have [10]

From which, setting, it follows that

Hence, making use of (a), the symmetry of and the fact that, the result follows.

c) Clear.

d) We have from [10],

From which, by setting in (3), using (b) and the symmetry of, we conclude that. Similarly, setting in (3), using (a) and the symmetry of, we get.

e) The proof follows from Theorem 2.5 (j) by setting, taking into account the fact that [10].

f) We have

Hence, there exists a scalar function such that

Consequently, using (a) and the symmetry of, we get

This completes the proof. □

Theorem 3.8. A concircular -vector field and its associated -form are independent of the directional argument.

Proof. By Theorem 1.3(a), we have

From which, by setting and taking into account (2), Proposition 2.7(a) and Lemma 2.3, we conclude that and is thus independent of the directional argument.

 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, imply that is also independent of the directional argument. □

In view of Theorem 1.3 and Proposition 2.7, we have Theorem 3.9. A -vector field on is concircular with respect to Cartan connection if, and only if, it is concircular with respect to Berwald connection.

Remark 3.10. As a consequence of the above results, we retrieve a result of [4] concerning concurrent - vector fields: A concurrent -vector field and its associated -form are independent of the directional argument. Moreover, a -vector field on is concurrent with respect to Cartan connection if, and only if, it is concurrent with respect to Berwald connection.

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 [7].

In this section, we investigate the effect of the existence of a concircular -vector field on some important special Finsler spaces. The intrinsic definitions of the special Finsler spaces treated here are quoted from [7].

For later use, we need the following lemma.

Lemma 4.1. Let be a Finsler manifold admitting a concircular -vector field. Then, we have:

a) The concircular -vector field is everywhere non-zero.

b) The scalar function is everywhere nonzero.

c) The -vector field is everywhere non-zero and is orthogonal to.

d) The -vector fields and satisfy.

e) The scalar function is everywhere non-zero.

Proof.

a) Follows by Definition 2.1.

b) Suppose that, then

Hence, as is nondegenerate, vanishes, which contradicts (a). Consequently,.

c) If, then. Differentiating covariantly with respect to, we get

From which,

By (1), using (2), we obtain

From which, since, we are led to a contradiction:. Consequently,.

On the other hand, the orthogonality of the two - vector fields and follows from the identities and.

d) Follows from (c).

e) Follows from (d), (c) and the fact that . □

Definition 4.2. A Finsler manifold is said to be:

a) Riemannian if the metric tensor is independent of or, equivalently, if;

b) Berwald if the torsion tensor is horizontally parallel:;

c) Landsberg if the -torsion tensor or, equivalently, if.

Theorem 4.3. A Landsberg manifold admitting a concircular -vector field is Riemannian.

Proof. Suppose that is Landsberg, then. Consequently, the hv-curvature vanishes [10]. Hence, by Theorem 2.5(e),

From which, taking into account the fact that is a non-zero function, it follows that. Hence the result follows. □

As a consequence of the above result, we get Corollary 4.4. The existence of a concircular - vector field implies that the three notions of being Landsberg, Berwald and Riemannian coincide.

Definition 4.5. A Finsler manifold is said to be:

a) -like if and the Cartan tensor has the form

b) -reducible if and the Cartan tensor has the form2

c) semi--reducible if and the Cartan tensor has the form

where, and are scalar functions satisfying.

d) quasi--reducible if and the Cartan tensor has the from 

where is a symmetric -tensor field satisfying.

Theorem 4.6. Let be a Finsler manifold admitting a concircular -vector field.

a) If is quasi-C-reducible, then it is Riemannian, provided that.

b) If is -reducible, then it is Riemannian.

c) If is semi--reducible, then it is -like.

Proof. 

a) Follows from the defining property of quasi-Creducibility by setting and using the fact that and the given assumption;

b) Setting in (3.3), taking into account Proposition 2.7(a), Lemma 3.1(e) and, it follows that, which is equivalent to (Deicke theorem [11]);

c) Let be semi--reducible. Setting and in (3.4), taking into account Proposition 2.7(a) and, we get

From which, since (Lemma 3.1(e)) and, it follows that.

Consequently, is -like. □

Definition 4.7. A Finsler manifold is said to be -like if and the v-curvature tensor has the form:

Theorem 4.8. If an -like manifold admits a concircular -vector field, then the v-curvature tensor vanishes.

Proof. Setting in (3.5), taking Theorem 2.5 into account, we immediately get

Taking the trace of the above equation, we have

Consequently,

From which, since (Lemma 3.1(e)), the vertical scalar curvature vanishes. Now, again, from (3.5), the result follows. □

Definition 4.9. A Finsler manifold, where, is said to be:

a) -like if the hv-curvature tensor has the form:

where is -form, positively homogeneous of degree.

b) p-reducible if the π-tensor field

has the form

where is the -form defined by

Theorem 4.10. Let be a Finsler manifold admitting a concircular -vector field.

a) If is -like, then it is Riemannian, provided that.

b) If is -reducible, then it is Landsbergian.

Proof.

a) Setting in (3.6), taking into account Theorem 2.5 and Proposition 2.7, we immediately get

Hence, the result follows.

b) Setting in (3.7) and using the identity, we conclude that

, with (Lemma 3.1

(e)). Consequently,. Hence, again, from Definition 3.9(b), the (v)hv-torsion tensor. □

Definition 4.11. A Finsler manifold of is said to be -isotropic if there exists a scalar function such that the horizontal curvature tensor has the form

where is called the scalar curvature.

Theorem 4.12. For an -isotropic Finsler manifold admitting a concircular -vector field, the scalar curvature is given by

where.

Proof. From Definition 3.11, by setting and, we have

On the other hand, using Theorem 2.5(i), we have

From (8) and (9), it follows that

Taking the trace of the above equation, we get

Hence, the scalar is given by

This completes the proof. □

Corollary 4.13. For an h-isotropic Finsler manifold admitting a concurrent π-vector field, the hcurvature vanishes.

Proof. If is concurrent, then the -form vanishes. Hence, using (10), the scalar vanishes. Consequently, from Definition 3.11, the -curvature vanishes. □

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.

Definition 5.1. A Finsler manifold is said to be -recurrent if the (h)hv-torsion tensor has the property that

where is a scalar (1)π-form, positively homogenous of degree zero in, called the -recurrence form.

Similarly, is called -recurrent if the (h)hv-torsion tensor has the property that

where is a scalar (1) π-form, positively homogenous of degree in, called the -recurrence form.

Theorem 5.2. If a -recurrent Finsler manifold admits a concircular π-vector field, then it is Riemannian, provided that.

Proof. We have [10]

Setting, making use of Theorem 2.5, Proposition 2.7 and the identity [10]

we get

On the other hand, Definition 4.1 yields

Under the given assumption, the above two equations imply that. Hence, is Riemannian. □

In view of the above theorem, we have.

Corollary 5.3. In the presence of a concircular - vector field, the three notions of being -recurrent, -recurrent and Riemannian coincide, provided that.

Proof. By Theorem 4.7 of [7], regardless of the existence of concircular -vector fields, a -recurrent Finsler space is necessarily Riemannian. On the other hand, a Riemannian space is trivially both -recurrent and -recurrent. □

Remark 5.4. Corollary 4.3 remains true if in particular a concircular -vector field replaced by a concurrent -vector field [4].

The following definition gives the second type of recurrence related to the -curvature tensor.

Definition 5.5. If we replace by in Definition 4.1, then is said to be -recurrent (-recurrent).

Theorem 5.6. If an -recurrent Finsler manifold admits a concircular -vector field, then its - curvature tensor vanishes.

Proof. Suppose that is an -recurrent manifold which admits a concircular -vector field. Then, by Definition 4.5 and Theorem 2.5(a), we have

On the other hand, by Theorem 2.5(c), we get

From the above two equations, since, the -curvature tensor vanishes. □

Corollary 5.7. Let be a Finsler manifold which admits a concircular -vector field. The following assertions are equivalent: 

a) is -recurrent, b) is -recurrent, c) the -curvature tensor vanishes.

In fact, for an -recurrent Finsler manifold the - curvature tensor vanishes [7] regardless of the existence of concircular -vector fields.

Remark 5.8. We retrieve here a result of [4] concerning concurrent -vector fields: Corollary 4.7 remains true if in particular a concircular -vector field replaced by a concurrent -vector field.

In the following we give the third type of recurrence related to the -curvature tensor.

Definition 5.9. If we replace by in Definition 4.1, then is said to be -recurrent (- recurrent).

In view of the above definition, we have Theorem 5.10. Let be a -recurrent Finsler manifold admitting a concircular -vector field. Theneither (a) is Riemannian, or

(b) has the property that

.

Proof. By Theorem 2.5(g), we have

On the other hand, by Definition 4.9 and Theorem 2.5(e), we get

From which together with (1), it follows that

By setting and noting that

[10], the above equation gives

Now, we have two cases: either and consequently is Riemannian, or

. This completes the proof. □

Lemma 5.11. For a -recurrent Finsler manifold, the -curvature tensor vanishes.

Proof. Suppose that is -recurrent, then, by Definition 4.9, we get

From which, together with the fact that [10] and, the result follows. □

In view of Theorem 4.10 and Lemma 4.11, we have Theorem 5.12. Let be a Finsler manifold admitting a concircular -vector field. Then, the following assertions are equivalent:

a) is -recurrent;

b) is -recurrent;

c) is Riemannianprovided that in the -recurrence case.

Remark 5.13. In view of Theorem 4.12, we conclude that under the presence of a concurrent -vector field, the three notions of being -recurrent, -recurrent and Riemannian coincide, provided that.

Finally, we focus our attention to the fourth type of recurrent Finsler manifolds related to the -curvature tensor.

Definition 5.14. If we replace by in Definition 4.1, then is said to be -recurrent (- recurrent).

Theorem 5.15. An -recurrent Finsler manifold admitting a concircular -vector field is -isotropic with scalar curvature

where

Moreover, if is -recurrent with

, then the -curvature tensor vanishes.

Proof. Firstly, suppose that is an - recurrent manifold which admits a concircular -vector field. Then, by Theorem 2.5 (l), we have

On the other hand, by Definition 4.14 and Theorem 2.5(i), we get

The above two equations imply that

Consequently,

Hence,

From the above two relations, noting that

[10], we get

Taking the trace of the above relation with respect to the two arguments and, we obtain

From which, together with (4.2), we obtain

This means that is -isotropic (Definition 3.11) with scalar curvature

Finally, the second part of the theorem follows from Definition 4.14 and the identity [10]. □

As a consequence of the above theorem, we have Corollary 5.16. For an -recurrent Finsler manifold admitting a concurrent -vector field, the - curvature tensor vanishes.

• The concept of a concircular -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 -vector fields have been obteined.

• The effect of the existence of concircular -vector fields on some of the most important special Finsle spaces has been investigated.

• Different types of recurrent Finsler manifolds admitting concircular -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.