1. Introduction
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, Th-recurrent, Tv-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.
2. 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 [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
(1)
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.
3. 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 [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:
(1)
(2)
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],
(3)
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.
4. 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 [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
(1)
From which,
(2)
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
(3)
c) semi-
-reducible if 
and the Cartan tensor 
has the form
(4)
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:
(5)
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:
(6)
where 
is 
-form, positively homogeneous of degree
.
b) p-reducible if the π-tensor field
has the form
(7)
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
(8)
On the other hand, using Theorem 2.5(i), we have
(9)
From (8) and (9), it follows that
Taking the trace of the above equation, we get
Hence, the scalar 
is given by
(10)
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. □
5. 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
.
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
(1)
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,
(2)
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.
• 6. Concluding Remarks
• 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.
NOTES
1
denotes the alternate sum
.
2
denotes the cyclic sum over the arguments 
and
.