A Global Approach to the Theory of Special Finsler Manifolds
A GLOBAL APPROACH TO THE
THEORY OF
SPECIAL FINSLER MANIFOLDS
Nabil L. Youssef†, S. H. Abed† and A. Soleiman‡
†Department of Mathematics, Faculty of Science,
Cairo University, Giza, Egypt.
nyoussef@frcu.eun.eg, sabed@frcu.eun.eg
‡Department of Mathematics, Faculty of Science,
Benha University, Benha, Egypt.
soleiman@mailer.eun.eg
Dedicated to the memory of Prof. Dr. A. TAMIM
Abstract. The aim of the present paper is to provide a global presentation of the
theory of special Finsler manifolds. We introduce and investigate globally (or in-
trinsically, free from local coordinates) many of the most important and most com-
monly used special Finsler manifolds : locally Minkowskian, Berwald, Landesberg,
general Landesberg, P -reducible, C-reducible, semi-C-reducible, quasi-C-reducible,
P ∗-Finsler, Ch-recurrent, Cv-recurrent, C0-recurrent, Sv-recurrent, Sv-recurrent of
the second order, C2-like, S3-like, S4-like, P2-like, R3-like, P -symmetric, h-isotropic,
of scalar curvature, of constant curvature, of p-scalar curvature, of s-ps-curvature.
The global definitions of these special Finsler manifolds are introduced. Various
relationships between the different types of the considered special Finsler manifolds
are found. Many local results, known in the literature, are proved globally and
several new results are obtained. As a by-product, interesting identities and properties
concerning the torsion tensor fields and the curvature tensor fields are deduced.
Although our investigation is entirely global, we provide; for comparison rea-
sons, an appendix presenting a local counterpart of our global approach and the local
definitions of the special Finsler spaces considered. 1
Keywords and phrases. Berwald, Landesberg, P -reducible, C-reducible, Semi-C-
reducible, Quasi-C-reducible, P ∗-Finsler, Ch-recurrent, Cv-recurrent, Sv-recurrent,
C2-like, S3-like, S4-like, P2-like, R3-like, P -symmetric, h-isotropic, Of scalar curva-
ture, Of constant curvature, Of p-scalar curvature, Of s-ps-curvature.
2000 AMS Subject Classification. 53C60, 53B40.
ArXiv Number: 0704.0053
http://arxiv.org/abs/0704.0053v3
Introduction
In Finsler geometry all geometric objects depend not only on positional coordi-
nates, as in Riemannian geometry, but also on directional arguments. In Riemannian
geometry there is a canonical linear connection on the manifold M , while in Finsler
geometry there is a corresponding canonical linear connection, due to E. Cartan,
which is not a connection on M but is a connection on π−1(TM), the pullback of
the tangent bundle TM by π : TM −→ M (the pullback approach). Moreover, in
Riemannian geometry there is one curvature tensor and one torsion tensor associated
with a given linear connection on the manifold M , whereas in Finsler geometry there
are three curvature tensors and five torsion tensors associated with a given linear
connection on π−1(TM).
Most of the special spaces in Finsler geometry are derived from the fact that
the π-tensor fields (torsions and curvatures) associated with the Cartan connection
satisfy special forms. Consequently, special spaces of Finsler geometry are more
numerous than those of Riemannian geometry. Special Finsler spaces are investigated
locally (using local coordinates) by many authors: M. Matsumoto [16], [18], [15], [14]
and others [6], [19], [8], [7]. On the other hand, the global (or intrinsic, free from
local coordinates) investigation of such spaces is very rare in the literature. Some
considerable contributions in this direction are due to A. Tamim [24], [25].
In the present paper, we provide a global presentation of the theory of special
Finsler manifolds. We introduce and investigate globally many of the most important
and most commonly used special Finsler manifolds : locally Minkowskian, Berwald,
Landesberg, general Landesberg, P -reducible, C-reducible, semi-C-reducible, quasi-
C-reducible, P ∗-Finsler, Ch-recurrent, Cv-recurrent, C0-recurrent, Sv-recurrent, Sv-
recurrent of the second order, C2-like, S3-like, S4-like, P2-like, R3-like, P -symmetric,
h-isotropic, of scalar curvature, of constant curvature, of p-scalar curvature, of s-ps-
curvature.
The paper consists of two parts, preceded by a preliminary section (§1), which
provides a brief account of the basic concepts of the pullback approach to Finsler
geometry necessary to this work. For more detail, the reader is referred to [1], [3], [5]
and [24].
In the first part (§2), we introduce the global definitions of the aforementioned
special Finsler manifolds in such a way that, when localized, they yield the usual
local definitions current in the literature (see the appendix). The definitions are
arranged according to the type of the defining property of the special Finsler manifold
concerned.
In the second part (§3), various relationships between the different types of the
considered special Finsler manifolds are found. Many local results, known in the
literature, are proved globally and several new results are obtained. As a by-product
of some of the obtained results, interesting identities and properties concerning the
torsion tensor fields and the curvature tensor fields are deduced, which in turn play
a key role in obtaining other results.
Among the obtained results are: a characterization of Riemannian manifolds, a
characterization of Sv-recurrent manifolds, a characterization of P -symmetric
manifolds, a characterization of Berwald manifolds (in certain cases), the equivalence
of Landsberg and general Landsberg manifolds under certain conditions, a classifica-
tion of h-isotropic Ch-recurrent manifolds and a presentation of different conditions
under which an R3-like Finsler manifold becomes a Finsler manifold of s-ps curvature.
The above results are just a non-exhaustive sample of the global results obtained in
this paper.
It should finally be noted that some important results of [8], [9], [11], [13], [19],
[20],...,etc. (obtained in local coordinates) are immediately derived from the obtained
global results (when localized).
Although our investigation is entirely global, we conclude the paper with an ap-
pendix presenting a local counterpart of our global approach and the local definitions
of the special Finsler spaces considered. This is done to facilitate comparison and to
make the paper more self-contained.
1. Notation and Preliminaries
In this section, we give a brief account of the basic concepts of the pullback
formalism of Finsler geometry necessary for this work. For more details refer to [1], [3],
[5] and [24]. We make the general assumption that all geometric objects we consider
are of class C∞. The following notations will be used throughout this paper:
M : a real differentiable manifold of finite dimension n and of class C∞,
F(M): the R-algebra of differentiable functions on M ,
X(M): the F(M)-module of vector fields on M ,
πM : TM −→M : the tangent bundle of M ,
π : TM −→M : the subbundle of nonzero vectors tangent to M ,
V (TM): the vertical subbundle of the bundle TTM ,
P : π−1(TM) −→ TM : the pullback of the tangent bundle TM by π,
P ∗ : π−1(T ∗M) −→ TM : the pullback of the cotangent bundle T ∗M by π,
X(π(M)): the F(TM)-module of differentiable sections of π−1(TM).
Elements of X(π(M)) will be called π-vector fields and will be denoted by barred
letters X. Tensor fields on π−1(TM) 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 ∈ TM . The
lift to π−1(TM) of a vector field X on M is the π-vector field X defined by X(u) =
(u,X(π(u))). The lift to π−1(TM) of a 1-form ω on M is the π-form ω defined by
ω(u) = (u, ω(π(u))).
The tangent bundle T (TM) is related to the pullback bundle π−1(TM) by the
short exact sequence
0 −→ π−1(TM)
−→ T (TM)
−→ π−1(TM) −→ 0,
where the bundle morphisms ρ and γ are defined respectively by ρ = (πT M , dπ) and
γ(u, v) = ju(v), where ju is the natural isomorphism ju : TπM (v)M −→ Tu(TπM (v)M).
Let ∇ be a linear connection (or simply a connection) in the pullback bundle
π−1(TM). We associate to ∇ the map
K : TTM −→ π−1(TM) : X 7−→ ∇Xη,
called the connection (or the deflection) map of ∇. A tangent vector X ∈ Tu(TM)
is said to be horizontal if K(X) = 0 . The vector space Hu(TM) = {X ∈ Tu(TM) :
K(X) = 0} of the horizontal vectors at u ∈ TM is called the horizontal space to M
at u . The connection ∇ is said to be regular if
Tu(TM) = Vu(TM)⊕Hu(TM) ∀u ∈ TM.
If M is endowed with a regular connection, then the vector bundle maps
γ : π−1(TM) −→ V (TM),
ρ|H(T M) : H(TM) −→ π
−1(TM),
K|V (T M) : V (TM) −→ π
−1(TM)
are vector bundle isomorphisms. Let us denote β = (ρ|H(T M))
−1, then
ρoβ = idπ−1(T M), βoρ =
idH(T M) on H(TM)
0 on V(TM)
(1.1)
For a regular connection∇ we define two covariant derivatives
∇ and
∇ as follows:
For every vector (1)π-form A, we have
∇ A)(øX, øY ) := (∇βøXA)(øY ) , (
∇ A)(øX, øY ) := (∇γøXA)(øY ).
The classical torsion tensor T of the connection ∇ is defined by
T(X, Y ) = ∇XρY −∇Y ρX − ρ[X, Y ] ∀X, Y ∈ X(TM).
The horizontal ((h)h-) and mixed ((h)hv-) torsion tensors, denoted respectively by Q
and T , are defined by
Q(X, Y ) = T(βXβY ), T (X, Y ) = T(γX, βY ) ∀X, Y ∈ X(π(M)).
The classical curvature tensor K of the connection ∇ is defined by
K(X, Y )ρZ = −∇X∇Y ρZ +∇Y∇XρZ +∇[X,Y ]ρZ ∀X, Y, Z ∈ X(TM).
The horizontal (h-), mixed (hv-) and vertical (v-) curvature tensors, denoted respec-
tively by R, P and S, are defined by
R(X, Y )øZ = K(βXβY )øZ, P (X, Y )øZ = K(βX, γY )øZ, S(X, Y )øZ = K(γX, γY )øZ.
We also have the (v)h-, (v)hv- and (v)v-torsion tensors, denoted respectively by R̂,
P̂ and Ŝ, defined by
R̂(X, Y ) = R(X, Y )øη, P̂ (X, Y ) = P (X, Y )øη, Ŝ(X, Y ) = S(X, Y )øη.
Theorem 1.1. [25] Let (M,L) be a Finsler manifold. There exists a unique regular
connection ∇ in π−1(TM) such that
(a) ∇ is metric : ∇g = 0,
(b) The horizontal torsion of ∇ vanishes : Q = 0,
(c) The mixed torsion T of ∇ satisfies g(T (X, Y ), Z) = g(T (X,Z), Y ).
Such a connection is called the Cartan connection associated to the Finsler man-
ifold (M,L).
One can show that the torsion T of the Cartan connection has the property that
T (X, η) = 0 for all X ∈ X(π(M)) and associated to T we have:
Definition 1.2. [25] Let ∇ be the Cartan connection associated to (M,L). The
torsion tensor field T of the connection ∇ induces a π-tensor field of type (0, 3),
called the Cartan tensor and denoted again T , defined by :
T (X, Y , Z) = g(T (X, Y ), Z), for all X, Y , Z ∈ X(TM).
It also induces a π-form C, called the contracted torsion, defined by :
C(X) := Tr{Y 7−→ T (X, Y )}, for all X ∈ X(TM).
Definition 1.3. [25] With respect to the Cartan connection ∇ associated to (M,L),
we have
– The horizontal and vertical Ricci tensors Rich and Ricv are defined respectively by:
Rich(X, Y ) := Tr{Z 7−→ R(X,Z)Y }, for all X, Y ∈ X(TM),
Ricv(X, Y ) := Tr{Z 7−→ S(X,Z)Y }, for all X, Y ∈ X(TM).
– The horizontal and vertical Ricci maps Rich0 and Ric
0 are defined respectively by:
g(Rich0(X), Y ) := Ric
h(X, Y ), for all X, Y ∈ X(TM),
g(Ricv0(X), Y ) := Ric
v(X, Y ), for all X, Y ∈ X(TM).
– The horizontal and vertical scalar curvatures Sch , Scv are defined respectively by:
Sch := Tr(Rich0), Sc
v := Tr(Ricv0),
where R and S are respectively the horizontal and vertical curvature tensors of ∇.
Proposition 1.4. [12] Let (M,L) be a Finsler manifold. The vector field G deter-
mined by iGΩ = −dE is a spray, called the canonical spray associated to the energy
E, where E := 1
L2 and Ω := ddJE.
One can show, in this case, that G = βoη, and G is thus horizontal with respect
to the Cartan connection ∇.
Theorem 1.5. [26] Let (M,L) be a Finsler manifold. There exists a unique regular
connection D in π−1(TM) such that
(a) D is torsion free,
(b) The canonical spray G = βoη is horizontal with respect to D,
(c) The (v)hv-torsion tensor P̂ of D vanishes.
Such a connection is called the Berwald connection associated to the Finsler
manifold (M,L).
2. Special Finsler spaces
In this section, we introduce the global definitions of the most important and
commonly used special Finsler spaces in such a way that, when localized, they yield
the usual local definitions existing in the literature (see the Appendix). Here we
simply set the definitions, postponing investigation of the mutual relationships be-
tween these special Finsler spaces to the next section. The definitions are arranged
according to the type of defining property of the special Finsler space concerned.
Throughout the paper, g, ĝ, ∇ and D denote respectively the Finsler metric in
π−1(TM), the induced metric in π−1(T ∗M), the Cartan connection and the Berwald
connection associated to a given Finsler manifold (M,L). Also, T denotes the torsion
tensor of the Cartan connection (or the Cartan tensor) and R, P and S denote
respectively the horizontal curvature, the mixed curvature and the vertical curvature
of the Cartan connection.
Definition 2.1. A Finsler manifold (M,L) is :
(a) Riemannian if the metric tensor g(x, y) is independent of y or, equivalently, if
T (X, Y ) = 0, for all X, Y ∈ X(π(M)).
(b) locally Minkowskian if the metric tensor g(x, y) is independent of x or, equiva-
lently, if
∇βX T = 0 and R = 0.
Definition 2.2. A Finsler manifold (M,L) is said to be :
(a) Berwald [24] if the torsion tensor T is horizontally parallel. That is,
∇βX T = 0.
(b) Ch-recurrent if the torsion tensor T satisfies the condition
∇βX T = λo(X) T,
where λo is a π-form of order one.
(c) P ∗-Finsler manifold if the π-tensor field ∇βηT is expressed in the form
∇βη T = λ(x, y) T,
where λ(x, y) =
bg(∇βη C,C)
g(∇βηøC,øC)
and C2 := ĝ(C,C) = C(C) 6= 0; C
being the π-vector field defined by g(C,X) = C(X).
Definition 2.3. A Finsler manifold (M,L) is said to be:
(a) Cv-recurrent if the torsion tensor T satisfies the condition
(∇γXT )(Y , Z) = λo(X)T (Y , Z).
(b) C0-recurrent if the torsion tensor T satisfies the condition
(DγXT )(Y , Z) = λo(X)T (Y , Z).
Definition 2.4. [25] A Finsler manifold (M,L) is said to be :
(a) semi-C-reducible if dimM ≥ 3 and the Cartan tensor T has the form
T (X, Y , Z) =
{~(X, Y )C(Z) + ~(Y , Z)C(X) + ~(Z,X)C(Y )}+
C(X)C(Y )C(Z),
where µ and τ are scalar functions satisfying µ + τ = 1, ~ = g − ℓ ⊗ ℓ and ℓ(X) :=
L−1g(X, η).
(b) C-reducible if dimM ≥ 3 and the Cartan tensor T has the form
T (X, Y , Z) =
{~(X, Y )C(Z) + ~(Y , Z)C(X) + ~(Z,X)C(Y )}.
(c) C2-like if dimM ≥ 2 and the Cartan tensor T has the form
T (X, Y , Z) =
C(X)C(Y )C(Z).
Definition 2.5. A Finsler manifold (M,L), where dimM ≥ 3, is said to be quasi-C-
reducible if the Cartan tensor T is written as :
T (X, Y , Z) = A(X, Y )C(Z) + A(Y , Z)C(X) + A(Z,X)C(Y ),
where A is a symmetric indicatory (2) π-form (A(X, η) = 0 for all X).
Definition 2.6. [25] A Finsler manifold (M,L) is said to be :
(a) S3-like if dim(M) ≥ 4 and the vertical curvature tensor S(X, Y , Z,W )
:= g(S(X, Y )Z,W ) has the form :
S(X, Y , Z,W ) =
(n− 1)(n− 2)
{~(X,Z)~(Y ,W )− ~(X,W )~(Y , Z)}.
(b) S4-like if dim(M) ≥ 5 and the vertical curvature tensor S(X, Y , Z,W ) has the
form :
S(X, Y , Z,W ) =~(X,Z)F(Y ,W )− ~(Y , Z)F(X,W )+
+ ~(Y ,W )F(X,Z)− ~(X,W )F(Y , Z),
(2.1)
where F is the (2)π-form defined by F =
{Ricv −
Scv ~
2(n− 2)
Definition 2.7. A Finsler manifold (M,L) is said to be :
(a) Sv-recurrent if the v-curvature tensor S satisfies the condition
(∇γXS)(Y , Z,W ) = λ(X)S(Y , Z)W,
where λ is a π-form of order one.
(b) Sv-recurrent of the second order if the v-curvature tensor S satisfies the condition
∇ S)(øY, øX,Z,W,U) = Θ(X, Y )S(Z,W )U,
where Θ is a π-form of order two.
Definition 2.8. [24] A Finsler manifold (M,L) is said to be :
(a) a Landsberg manifold if
P̂ (X, Y ) = P (X, Y )η = 0 ∀X, Y ∈ X(π(M)), or equivalently ∇βη T = 0.
(b) a general Landsberg manifold if
Tr{Y −→ P̂ (X, Y )} = 0 ∀X,∈ X(π(M)), or equivalently ∇βη C = 0.
Definition 2.9. A Finsler manifold (M,L) is said to be P -symmetric if the mixed
curvature tensor P satisfies
P (X, Y )Z = P (Y ,X)Z, ∀ øX, øY, øZ ∈ X(π(M)).
Definition 2.10. A Finsler manifold (M,L), where dimM ≥ 3, is said to be P2-like
if the mixed curvature tensor P has the form :
P (X, Y , Z, øW ) = α(Z)T (X, Y , øW )− α(W ) T (X, øY, Z),
where α is a (1) π-form (positively homogeneous of degree 0).
Definition 2.11. [25] A Finsler manifold (M,L), where dimM ≥ 3, is said to be
P -reducible if the π-tensor field P (X, Y , Z) := g(P (X, Y )η, Z) can be expressed in
the form :
P (X, Y , Z) = δ(X)~(Y , Z) + δ(Y )~(Z,X) + δ(Z)~(X, Y ),
where δ is a (1) π-form satisfying δ(øη) = 0.
Definition 2.12. [2] A Finsler manifold (M,L), where dimM ≥ 3, is said to be
h-isotropic if there exists a scalar ko such that the horizontal curvature tensor R has
the form
R(X, Y )Z = ko{g(Y , Z)X − g(X,Z)Y }.
Definition 2.13. [2] A Finsler manifold (M,L), where dimM ≥ 3, is said to be :
(a) of scalar curvature if there exists a scalar function k : TM −→ R such that
the horizontal curvature tensor R(X, Y , Z,W ) := g(R(X, Y )Z,W ) satisfies the
relation
R(η,X, η, Y ) = kL2~(X, Y ).
(b) of constant curvature if the function k in (a) is constant.
Definition 2.14. A Finsler manifold (M,L) is said to be R3-like if dimM ≥ 4 and
the horizontal curvature tensor R(X, Y , Z,W ) is expressed in the form
R(X, Y , Z,W ) =g(X,Z)F (Y ,W )− g(Y , Z)F (X,W )+
+ g(Y ,W )F (X,Z)− g(X,W )F (Y , Z),
(2.2)
where F is the (2)π-form defined by F = 1
{Rich − Sc
2(n−1)
3. Relationships between different types of special
Finsler spaces
This section is devoted to global investigation of some mutual relationships
between the special Finsler spaces introduced in the preceding section. Some conse-
quences are also drawn from these relationships.
We start with some immediate consequences from the definitions:
(a) A Locally Minkowskian manifold is a Berwald manifold.
(b) A Berwald manifold is a Landsberg manifold.
(c) A Landsberg manifold is a general Landsberg manifold.
(d) A Berwald manifold is Ch-recurrent (resp. P ∗-Finsler).
(e) A P ∗-manifold is a Landsberg manifold.
(f) A C-reducible (resp. C2-like) manifold is semi-C-reducible.
(g) A semi-C-reducible manifold is quasi-C-reducible.
(h) A Finsler manifold of constant curvature is of scalar curvature.
The following two lemmas are useful for subsequent use.
Lemma 3.1. [25] For every øX, øY ∈ X(π(M)), we have:
(a) P (øη, øX)øY = 0, (b) P (øX, øη)øY = 0, (c) P (øX, øY )øη = (∇βøηT )(øX, øY ).
Lemma 3.2. If φ is the vector π-form defined by
φ(øX) := øX − L−1ℓ(øX)øη, or φ := I − L−1ℓ⊗ øη, (3.1)
where ℓ is the π-form given by ℓ(X) = L−1g(X, η), then we have:
(a) ~(øX, øY ) = g(φ(øX), øY ), (b) φ(øη) = 0, (c) φ o φ = φ,
(d) Tr(φ) = n− 1, (e) ∇βøX φ = 0, (f) ∇βøX ~ = 0.
As we have seen, a Landsberg manifold is general Landsberg. The converse is
not true. Nevertheless, we have
Proposition 3.3. A C-reducible general Landsberg manifold (M,L) is a Landsberg
manifold.
Proof. Since (M,L) is a C-reducible manifold, then, by Definition 2.4, Lemma 3.2,
the symmetry of ~ and the non-degeneracy of g, we get
T (øX, øY ) =
{~(øX, øY )øC + C(øX)φ(øY ) + C(øY )φ(øX)},
where øC is the π-vector field defined by g(øC, øX) := C(øX). Taking the h-covariant
derivative ∇βøZ of both sides of the above equation, we obtain
(∇βøZ T )(øX, øY ) =
{(∇βøZ ~)(øX, øY )øC + ~(øX, øY )∇βøZ øC + C(øX)(∇βøZ φ)(øY ) +
+(∇βøZ C)(øX)φ(øY ) + C(øY )(∇βøZ φ)(øX) + (∇βøZ C)(øY )φ(øX)},
from which, by setting øZ = øη and taking into account the fact that ∇βøZ ~ = 0 and
that ∇βøZ φ = 0 ( Lemma 3.2), we get
(∇βøη T )(øX, øY ) =
{~(øX, øY )∇βøη øC+(∇βøη C)(øX)φ(øY )+(∇βøη C)(øY )φ(øX)}.
Now, under the given assumption that the (M,L) is a general Landsberg manifold,
then ∇βøη C = 0 (Definition 2.8) and hence ∇βøη øC = 0. Hence ∇βøη T = 0 and the
result follows. �
Also, a Berwald manifold is Landsberg. The converse is by no means true,
although we have no counter-examples. Finding a Landsberg manifold which is not
Berwald is still an open problem. Nevertheless, we have
Proposition 3.4. [25] A C-reducible Landsberg manifold (M,L) is a Berwald
manifold.
Combining the above two Propositions, we obtain the more powerful result :
Proposition 3.5. A C-reducible general Landsberg manifold (M,L) is a Berwald
manifold.
Summing up, we get:
Theorem 3.6. Let (M,L) be a C-reducible Finsler manifold. The following assertion
are equivalent :
(a) (M,L) is a Berwald manifold.
(b) (M,L) is a Landsberg manifold.
(c) (M,L) is a general Landsberg manifold.
We retrieve here a result of Matsumuoto [15], namely
Corollary 3.7. If the h-curvature tensor R and hv-curvature tensor P of a C-
reducible manifold vanish, then the manifold is Locally Minkowskian.
Remark 3.8. [15] It may be conjectured that a Finsler manifold will be Minkowskian
if the h-curvature tensor R and hv-curvature tensor P vanish. As above seen the
conjecture is verified already under somewhat strong condition “C-reducibility”.
Theorem 3.9. Let (M,L) be a Finsler manifold. Then we have :
(a) A C-reducible manifold is P -reducible.
(b) A P -reducible general Landsberg manifold is Landsberg.
Proof.
(a) Since (M,L) is C-reducible, then by Definition 2.4, we have
T (øX, øY, øZ) =
SøX,øY,øZ{~(øX, øY )C(øZ)}.
Applying the h-covariant derivative ∇βøW on both sides of the above equation, taking
into account the fact that (∇βøW T )(øX, øY, øZ) = g((∇βøW T )(øX, øY ), øZ) and that
∇βøW ~ = 0, we obtain
g((∇βøWT )(øX, øY ), øZ) =
SøX,øY,øZ{~(øX, øY )(∇βøW C)(øZ)}.
From which, by setting øW = øη and noting that P (øX, øY )øη = (∇βøη T )(øX, øY ),
the result follows.
(b) Since (M,L) is a P -reducible manifold, then by Definition 2.11, taking into
account the fact that g is nondegenerate, we obtain
P (øX, øY )øη = δ(øX)φ(øY ) + δ(øY )φ(øX) + ø~(øX, øY ) øζ, (3.2)
where øζ is the π-vector field defined by g(øζ, øX) := δ(øX).
Since δ(øη) = 0, then Tr{øY 7−→ δ(øY )φ(øX) + ~(øX, øY ) øζ} = 2δ(øX). Taking
the trace of both sides of (3.2), using the fact that P (øX, øY )øη = (∇βøη T )(øX, øY )
(Lemma 3.1) and that Tr{øY 7−→ (∇βøη T )(øX, øY )} = (∇βøη C)(øX), we get
δ(øX) =
n + 1
(∇βøη C)(øX). (3.3)
Now, from Equations (3.2) and (3.3), we have
g(P (øX, øY )øη, øZ) =
n + 1
SøX,øY,øZ{~(øX, øY )(∇βøη C)(øZ)}. (3.4)
According to the given assumption that the manifold is general Landsberg, then
∇βøη C = 0. Therefore, from (3.4), we get P (øX, øY )øη = 0 and hence the manifold
is Landsberg. �
Proposition 3.10.
(a) A Ch-recurrent manifold is a P ∗-Finsler manifold.
(b) A general Landsberg P ∗-Finsler manifold is a Landsberg manifold.
Proof. The proof is straightforward and we omit it. �
Proposition 3.11. A C2-like Finsler manifold is a Berwald manifold if, and only if,
the π-tensor field C is horizontally parallel.
Proof. Let (M,L) be C2-like. Then, T (øX, øY, øZ) =
C(øC)
C(øX)C(øY )C(øZ),
from which T (øX, øY ) = 1
C(øC)
C(øX)C(øY )øC. Taking the h-covariant derivative of
both sides, we get
(∇βøZT )(øX, øY ) =
−∇βøZC(øC)
C(øX)C(øY )øC +
C(øC)
(∇βøZC)(øX)C(øY )øC +
C(øC)
(∇βøZC)(øY )C(øX)øC +
C(øC)
C(øX)C(øY )∇βøZøC.
In view of this relation, ∇βøZ T = 0 if, and only if, ∇βøZ C = 0. Hence the result. �
Corollary 3.12. A C2-like general Landsberg manifold is a Landsberg manifold.
In view of the above Theorems, we have:
Corollary 3.13. The two notions of being Landsberg and general Landsberg coincide
in the case of C-reducibility, P -reducibility, C2-likeness or P
∗-Finsler.
As we know, a C-reducible Landsberg manifold is a Berwald manifold (Proposi-
tion 3.4 ). Moreover, A C2-like Finsler manifold is a Berwald manifold if, and only
if, the π-tensor field C is horizontally parallel (Proposition 3.11). We shall try to
generalize these results to the case of semi-C-reduciblity.
Theorem 3.14. A semi-C-reducible Finsler manifold is a Berwald manifold if, and
only if, the characteristic scalar µ and the π-tensor field C are horizontally parallel.
Proof. Firstly, if (M,L) is semi-C-reducible, then
T (øX, øY, øZ) =
SøX,øY,øZ{~(øX, øY )C(øZ)}+
C(øC)
C(øX)C(øY )C(øZ).
Taking the h-covariant derivative of both sides, noting that ∇βøX~ = 0, we get
(∇βøWT )(øX, øY, øZ) =
n + 1
SøX,øY,øZ{~(øX, øY ){µ(∇βøWC)(øZ) + (∇βøWµ)C(øZ)}}+
SøX,øY,øZ{(∇βøWC)(øX)C(øY )C(øZ)} −
∇βøW µ
τ ∇βøWC(øC)
}C(øX)C(øY )C(øZ).
Now, if the characteristic scalar µ and the π-tensor field C are horizontally par-
allel, then ∇βøWT = 0 and (M,L) is a Berwald manifold.
Conversely, if (M,L) is a Berwald manifold, then∇βøXT = 0 and hence ∇βøXC =
0, ∇βøXøC = 0. These, together with the above equation, give
∇βøWµ{
SøX,øY,øZ{~(øX, øY )C(øZ)} −
C(øX)C(øY )C(øZ)} = 0,
which implies immediately that ∇βøWµ = 0. �
The following lemmas are useful for subsequent use
Lemma 3.15. For all X, Y ∈ X(π(M)), we have :
(a) [γX, γY ] = γ(∇γXY −∇γYX)
(b) [γX, βY ] = −γ(P (Y ,X)η +∇βYX) + β(∇γXY − T (X, Y ))
(c) [βX, βY ] = γ(R(X, Y )η) + β(∇βXY −∇βYX)
Lemma 3.16. For all øX, øY, øZ, øW ∈ X(π(M)) and W ∈ X(TM), we have :
(a) g((∇WT )(øX, øY ), øZ) = g((∇WT )(øX, øZ), øY ),
(b) g(S(øX, øY )øZ, øW ) = −g(S(øX, øY )øW, øZ).
Proof.
(a) From the definition of the covariant derivative, we get
g((∇WT )(øX, øY ), øZ) = g(∇WT (øX, øY ), øZ)− g(T (∇WøX, øY ), øZ)−
−g(T (øX,∇WøY ), øZ).
(3.5)
Now, we have
g(∇WT (øX, øY ), øZ) = W · g(T (øX, øY ), øZ)− g(T (øX, øY ),∇WøZ)
= W · g(T (øX, øY ), øZ)− g(T (øX,∇WøZ), øY ),
Similarly,
g(T (øX,∇WøY ), øZ) = W · g(T (øX, øZ), øY )− g(∇WT (øX, øZ), øY ).
Substituting these two equations into (3.5), noting the property that g(T (∇WøX, øY ), øZ)
= g(T (∇WøX, øZ), øY ) (cf. §1), the result follows.
(b) follows directly from the general formula (which can be easily proved)
g(K(X, Y )øZ, øW ) + g(K(X, Y )øW, øZ) = 0
by setting X = γøX and Y = γøY , where K is the classical curvature tensor of the
Cartan connection as a linear connection in the pull-back bundle (cf. §1). �
Proposition 3.17. Let (M,L) be a Ch-recurrent Finsler manifold (∇βøXT = λ0(øX)T ).
Then, we have:
(a) If Ko := λo(øη) = 0, then the hv-curvature tensor P is expressed in the form:
P (øX, øY, øZ, øW ) = λo(øZ)T (øX, øY, øW )− λo(øW )T (øX, øY, øZ)
and the (v)hv-torsion P̂ vanishes.
(b) If Ko 6= 0, then the v(hv)-torsion tensor P̂ is recurrent:
(∇βøZP̂ )(øX, øY ) = (λo(øZ) +
∇βøZKo
)P̂ (øX, øY ).
Proof.
(a) The hv-curvature tensor P can be written in the form [25]:
P (øX, øY, øZ, øW ) = g((∇βøZT )(øX, øY ), øW )− g((∇βøWT )(øX, øY ), øZ)+
+g(T (øX, øZ), P̂(øW, øY ))− g(T (øX, øW ), P̂(øZ, øY )).
Then, by using P̂ (øX, øY ) = (∇βøηT )(øX, øY ) (Lemma 3.1) and the C
h-recurrence
condition, we get
P (øX, øY, øZ, øW ) = λo(øZ)T (øX, øY, øW )− λo(øW )T (øX, øY, øZ)−
−λo(øη){g(T (øX, øW ), T (øY, øZ))− g(T (øX, øZ), T (øY, øW ))}
= λo(øZ)T (øX, øY, øW )− λo(øW )T (øX, øY, øZ)− λo(øη)S(øX, øY, øZ, øW ).
Now, if λo(øη) = 0, then (a) follows from the above relation.
(b) If Ko := λo(øη) 6= 0, then by Lemma 3.1 and the recurrence condition, we have
P̂ (øX, øY ) = KoT (øX, øY ),
from which
(∇βøZP̂ )(øX, øY ) = {∇βøZKo +Koλo(øZ)}T (øX, øY ).
Then, (b) follows from the above two equations. �
Theorem 3.18. Assume that (M,L) is Ch-recurrent. Then, the v-curvature tensor S
is recurrent with respect to the h-covariant differentiation : ∇βøXS = θ(øX)S, where
θ is a π-form of order one.
Proof. One can easily show that : For all X, Y, Z ∈ X(TM),
SX,Y,Z{K(X, Y )ρZ +∇XT(Y, Z) +T(X, [Y, Z])} = 0.
Setting X = γøX, Y = γøY and Z = βøZ in the above equation, we get
S(øX, øY )øZ = ∇γøY T (øX, øZ)−∇γøXT (øY, øZ)−∇βøZT(γøX, γøZ)−
−T(γøX, [γøY, βøZ]) +T(γøY, [γøX, βøZ]) +T([γøX, γøY ], βøZ).
Using Lemma 3.15 and the fact that T(γøX, γøZ) = 0, the above equation reduces
S(øX, øY )øZ = (∇γøY T )(øX, øZ)− (∇γøXT )(øY, øZ)+
+T (øX, T (øY, øZ))− T (øY, T (øX, øZ)).
(3.6)
From which, since g(T (øX, øY ), øZ) = g(T (øX, øZ), øY ), we have
g(S(øX, øY )øZ, øW ) = g((∇γøY T )(øX, øZ), øW )− g((∇γøXT )(øY, øZ), øW )+
+g(T (øX, øW ), T (øY, øZ))− g(T (øY, øW ), T (øX, øZ)).
Similarly,
g(S(øX, øY )øW, øZ) = g((∇γøY T )(øX, øW ), øZ)− g((∇γøXT )(øY, øW ), øZ)+
+g(T (øX, øZ), T (øY, øW ))− g(T (øY, øZ), T (øX, øW )).
The above two equations, together with Lemma 3.16, yield
g((∇γøXT )(øY, øZ), øW ) = g((∇γøY T )(øX, øZ), øW ). (3.7)
By (3.6) and (3.7), we obtain
S(øX, øY, øZ, øW ) = g(T (øX, øW ), T (øY, øZ))− g(T (øY, øW ), T (øX, øZ)). (3.8)
Now, using the given assumption that the manifold is Ch-recurrent, Equation
(3.8) implies that
(∇βøXS)(øY, øZ, øV, øW ) = ∇βøXS(øY, øZ, øV, øW )−
−S(∇βøXøY, øZ, øV, øW )− S(øY,∇βøXøZ, øV, øW )−
−S(øY, øZ,∇βøXøV, øW )− S(øY, øZ, øV,∇βøXøW ).
= +∇βøXg(T (øY, øW ), T (øZ, øV ))−∇βøXg(T (øZ, øW ), T (øY, øV ))−
−g(T (∇βøXøY, øW ), T (øZ, øV )) + g(T (øZ, øW ), T (∇βøXøY, øV ))−
−g(T (øY, øW ), T (∇βøXøZ, øV )) + g(T (∇βøXøZ, øW ), T (øY, øV ))−
−g(T (øY, øW ), T (øZ,∇βøXøV )) + g(T (øZ, øW ), T (øY,∇βøXøV ))−
−g(T (øY,∇βøXøW ), T (øZ, øV )) + g(T (øZ,∇βøXøW ), T (øY, øV )).
= g((∇βøXT )(øY, øW ), T (øZ, øV )) + g(T (øY, øW ), (∇βøXT )(øZ, øV ))−
−g((∇βøXT )(øZ, øW ), T (øY, øV ))− g(T (øZ, øW ), (∇βøXT )(øY, øV )).
= 2λo(øX)S(øY, øZ, øV, øW ) =: θ(øX)S(øY, øZ, øV, øW ).
Hence, the result follows. �
Corollary 3.19. In the course of the proof of Theorem 3.18, we have shown that
(Equations (3.7) and (3.8)) :
(a) (∇γøXT )(øY, øZ) = (∇γøY T )(øX, øZ),
(b) S(øX, øY, øZ, øW ) = g(T (øX, øW ), T (øY, øZ))− g(T (øY, øW ), T (øX, øZ)).
Corollary 3.20. Let (M,L) be a C2-like Finsler manifold. Then the the v-curvature
tensor S vanishes.
Proof. Substituting T (øX, øY ) = 1
C(øC)
C(øX)C(øY )øC in Corollary 3.19(b), we
get the result. �
Corollary 3.21. Let (M,L) be a C-reducible manifold. Then,
(a) the v-curvature tensor S has the form
S(øX, øY, øZ, øW ) =
(n + 1)2
{C2~(øX, øW )~(øY, øZ)− C2~(øY, øW )~(øX, øZ) +
+~(øX, øW )C(øY )C(øZ) + ~(øY, øZ)C(øX)C(øW )−
−~(øY, øW )C(øX)C(øZ)− ~(øX, øZ)C(øY )C(øW )}.
(b) the vertical Ricc tensor Ricv has the form
Ricv(øX, øY ) =
(3− n)
(n + 1)2
C(øX)C(øY )−
(n− 1)
(n+ 1)2
C2~(øX, øY ).
(c) the vertical scalar curvature Scv has the form
Scv =
(2− n)
(n+ 1)
Theorem 3.22. A Finsler manifold (M,L) is P -Symmetric if, and only if, the
v-curvature tensor S satisfies the equation ∇βøηS = 0.
Proof. One can show that: For all X, Y, Z ∈ X(TM),
SX,Y,Z{∇ZK(X, Y )−K(X, Y )∇Z −K([X, Y ], Z)} = 0. (3.9)
Setting X = γøX, Y = γøY and Z = βøZ in the above equation, we get
∇βøZS(øX, øY )øW +∇γøY P (øZ, øX)øW −∇γøXP (øZ, øY )øW−
−S(øX, øY )∇βøZøW + P (øZ, øY )∇γøXøW − P (øZ, øX)∇γøY øW−
−K([γøX, γøY ], βøZ)øW −K([γøY, βøZ], γøX)øW −K([βøZ, γøX ], γøY )øW = 0.
By using Lemma 3.15, the above relation reduces to
(∇βøZS)(øX, øY, øW ) + (∇γøY P )(øZ, øX, øW )− (∇γøXP )(øZ, øY, øW )+
+S(P (øZ, øY )øη, øX)øW − S(P (øZ, øX)øη, øY )øW+
+P (T (øY, øZ), øX)øW − P (T (øX, øZ), øY )øW = 0.
(3.10)
Setting øZ = øη in the above equation, taking into account Lemma 3.1 and the fact
that T (øX, øη) = 0 and that (∇γøXP )(øη, øY, øZ) = −P (øX, øY )øZ, we get
P (øX, øY )øZ = P (øY, øX)øZ − (∇βøηS)(øX, øY, øZ). (3.11)
The result follows immediately from (3.11). �
According to (3.11) and Lemma 3.1, we have :
Corollary 3.23. Let P̂ (øX, øY ) := P (øX, øY )øη and T̂ (øX, øY ) := (∇βøηT )(øX, øY ).
Then the π-tensor fields P̂ and T̂ are symmetric.
Theorem 3.18 and Theorem 3.22 give rise the following result.
Theorem 3.24. Assume that a Finsler manifold (M,L) is Ch-recurrent and P -
symmetric. If θ(øη) 6= 0, then the v-curvature tensor S vanishes identically.
Now, we shall prove the following lemma which provides some important and
useful properties of the torsion tensor T and the v-curvature S :
Lemma 3.25. For every øX, øY, øZ and øW ∈ X(π(M)), we have
(a) T (øX, øY ) = T (øY, øX),
(b) T (øη, øX) = 0,
(c) SøX,øY,øZS(øX, øY )øZ = 0,
(d) g(S(øX, øY )øZ, øW ) = g(S(øZ, øW )øX, øY ),
(e) S(øη, øX)øY = 0 = S(øX, øη)øY ,
(f) (∇γøXS)(øη, øY )øZ = −S(øX, øY )øZ, (∇γøXS)(øη, øX)øη = 0 .
(g) S(øX, øY )øZ = −1
{(DγXT )(Y , Z)− (DγY T )(X,Z)}.
Consequently, S vanishes if and only if (DγXT )(Y , Z) = (DγY T )(X,Z).
Proof.
(a) From Corollary 3.19(a), we have
(∇γøXT )(øY, øZ) = (∇γøY T )(øX, øZ).
Setting øZ = øη and using the fact that T (øX, øη) = 0 and that K oγ = idX(π(M)),
the result follows.
(b) Follows from (a) together with the relation T (øX, øη) = 0.
(c) Setting X = γøX, Y = γøY and Z = γøZ in (3.9) and using Lemma 3.15, we
SøX,øY,øZ(∇γøXS)(øY, øZ, øW ) = 0.
Again, setting øW = øη in the above equation and using the fact that S(øX, øY )øη =
0 and that K oγ = idX(π(M)), the result follows.
(d) Follows from Corollary 3.19(b), noting that T is symmetric.
(e) and (f) are clear.
(g) From the relation DγXøY = ∇γXøY − T (øX, øY ) [27], we get
(DγXT )(øY, øZ) = (∇γXT )(øY, øZ)−T (øX, T (øY, øZ))+T (T (øX, øY ), øZ)+T (øY, T (øX, øZ)),
(DγY T )(øX, øZ) = (∇γY T )(øX, øZ)−T (øY, T (øX, øZ))+T (T (øY, øX), øZ)+T (øX, T (øY, øZ)).
The result follows from the above two equations, using Corollary 3.19 and the sym-
metry of T . �
As a direct consequence of the above lemma, we have the
Corollary 3.26. A P2-like Finsler manifold is P -symmetric.
Proposition 3.27. Assume that (M,L) is Cv-recurrent. Then, the v-curvature ten-
sor S is v-recurrent : ∇γøXS = Ψ(øX)S, Ψ being a (1)π-form. Consequently, S
vanishes identically.
Proof. Taking the v-covariant derivative of both sides of the relation in Corollary
3.19(b) and, then, using the assumption that ∇γXT = λ0(X)T , we get
(∇γøXS)(øY, øZ, øV, øW ) = 2λo(øX)S(øY, øZ, øV, øW ) =: ψ(øX)S(øY, øZ, øV, øW ),
which shows that S is v-recurrent.
Now, setting øV = øη in the last equation, using the properties of S and noting
that K oγ = idX(π(M)), we conclude that S = 0. �
The following result gives a characterization of Riemannian manifolds in terms
of Cv-recurrence and C0-recurrence.
Theorem 3.28.
(a) A Cv-recurrent Finsler manifold is Riemannian,
(b) A C0-recurrent Finsler manifold is Riemannian.
Proof. (a) Since (M,L) is Cv-recurrent, then (∇γXT )(Y , Z) = λo(X)T (Y , Z), from
which, by setting øX = øη and noting that ∇γøηT = −T , we get
T (Y , Z) = −λo(η)T (Y , Z). (3.12)
But since (∇γøXT )(øY, øZ) = (∇γøY T )(øX, øZ) (Corollary 3.19), then λo(øX)T (øY, øZ) =
λo(øY )T (øX, øZ). Hence,
λo(η)T (Y , Z) = 0. (3.13)
Then, the result follows from (3.12) and (3.13).
(b) can be proved similarly. �
Theorem 3.29. For a Finsler manifold (M,L), the following assertions are
equivalent :
(a) (M,L) is Sv-recurrent.
(b) The v-curvature tensor S vanishes identically.
(c) (M,L) is Sv-recurrent of the second order.
Proof.
(a) =⇒ (b) : If (M,L) is Sv-recurrent, then by Definition 2.7(a) we have
(∇γøWS)(øX, øY, øZ) = λ(øW )S(øY, øX)øZ,
from which, by setting øZ = øη, taking into account the fact that S(øX, øY )øη = 0
and that Koγ = idπ−1(TM), the result follows.
(b) =⇒ (a) : Trivial.
(b) =⇒ (c) : Trivial.
(c) =⇒ (b) : If the given manifold (M,L) is Sv-recurrent of the second order, then
by Definition 2.7(b) we get
Θ(øX, øY )S(øZ, øV )øW = (
∇ S)(øY, øX, øZ, øV, øW )
= ∇γøY (∇γøXS)(øZ, øV, øW )− (∇γ∇γøY øXS)(øZ, øV, øW )−
−(∇γøXS)(∇γøY øZ, øV, øW )− (∇γøXS)(øZ,∇γøY øV, øW )−
−(∇γøXS)(øZ, øV,∇γøY øW ).
By substituting øZ = øη = øW in the above equation and using Lemma 3.25 and the
fact that S(øX, øY )øη = 0, we get
S(øX, øY )øZ = −S(øZ, øY )øX and S(øX, øY )øZ = −S(øX, øZ)øY.
From this, together with the identity SøX,øY,øZS(øX, øY )øZ = 0, the v-curvature
tensor S vanishes identically. �
In view of the above theorem we have :
Corollary 3.30.
(a) An Sv-recurrent (resp. a second order Sv-recurrent) manifold (M,L) is S3-like,
provided that dimM ≥ 4.
(b) An Sv-recurrent (resp. a second order Sv-recurrent) manifold (M,L) is S4-like,
provided that dimM ≥ 5.
Theorem 3.31. If (M,L) is a P2-like Finsler manifold, then the v-curvature tensor
S vanishes or the hv-curvature tensor P vanishes. In the later case, the h-covariant
derivative of S vanishes.
Proof. As (M,L) is P2-like, then P (X, Y , η, øW ) = α(η)T (X, Y , øW ) =: αoT (X, Y , øW )
and hence
P̂ (øX, øY ) = αoT (X, Y ). (3.14)
Now, setting øW = øη into (3.10), we get
(∇γøY P̂ )(øZ, øX)− (∇γøX P̂ )(øZ, øY )− P (øZ, øX)øY + P (øZ, øY )øX−
−P̂ (T (øX, øZ), øY ) + P̂ (T (øY, øZ), øX) = 0.
Hence,
g((∇γøY P̂ )(øZ, øX), øW )− g((∇γøXP̂ )(øZ, øY ), øW )− P (øZ, øX, øY, øW )+
+P (øZ, øY, øX, øW )− g(P̂ (T (øX, øZ), øY ), øW ) + g(P̂ (T (øY, øZ), øX), øW ) = 0.
From which, together with (3.14) and Definition 2.10, taking into account the relation
(∇γøY P̂ )(øZ, øX) = (∇γøY αo)T (øZ, øX) + αo(∇γøY T )(øZ, øX), we obtain
g((∇γøY αo)T (øZ, øX) + αo(∇γøY T )(øZ, øX), øW )− g((∇γøXαo)T (øZ, øY )+
+αo(∇γøXT )(øZ, øY ), øW ) + α(X)T (Z, Y , øW )− α(W ) T (Z, øY,X)− α(Y )T (Z,X, øW )
+α(W ) T (X, øY, Z)− g(αoT (T (øX, øZ), øY ), øW ) + g(αoT (T (øY, øZ), øX), øW ) = 0.
Therefore, using Corollary 3.19,
(∇γøY α)(øη)T (øX, øZ, øW )− (∇γøXα)(øη)T (øY, øZ, øW ) = αoS(øX, øY, øW, øZ).
It is to be observed that the left-hand side of the above equation is symmetric in
the arguments øZ and øW while the right-hand side is skew-symmetric in the same
arguments. Hence we have
αoS(øX, øY, øW, øZ) = 0, (3.15)
ε(øY )T (øX, øZ, øW )− ε(øX)T (øY, øZ, øW ) = 0, (3.16)
where ε is the π-form defined by ε(øY ) := (∇γøY α)(øη).
Now, If ε 6= 0, it follows from (3.16) that there exists a scalar function Υ such that
T (øX, øY, øZ) = Υ ε(øX)ε(øY )ε(øZ). Consequently, T (øX, øY ) = Υ ε(øX)ε(øY )øε,
where g(øε, øX) := ε(øX). From which
S(øX, øY, øZ, øW ) = g(T (øX, øW ), T (øY, øZ))− g(T (øY, øW ), T (øX, øZ))
= Υ ε(øX)ε(øY )ε(øZ)ε(øW )g(øε, øε)−Υ ε(øX)ε(øY )ε(øZ)ε(øW )g(øε, øε) = 0.
On the other hand, if the v-curvature tensor S 6= 0, then it follows from (3.15)
that ε = 0 and α(øη) = 0. Hence, α = 0 and the hv-curvature tensor P vanishes. In
this case, it follows from the identity (3.10) that ∇βøXS = 0. �
Proposition 3.32. A P2-like Finsler manifold (M,L) is a P
∗-Finsler manifold.
Proof. As (M,L) is P2-like, then from (3.14), we have P̂ (X, Y ) = αoT (X, Y ). Using
Lemma 3.1, we get (∇βøηT )(øX, øY ) = α0T (øX, øY ), from which, by taking the trace,
∇βøηC = α0T , where α0 =
bg(∇βη C,C)
. Hence the result. �
The next definition will be useful in the sequel.
Definition 3.33. A π-tensor field Θ is positively homogenous of degree r in the
directional argument y (symbolically, h(r)) if it satisfies the condition
∇γη Θ = rΘ, or Dγη Θ = rΘ.
Lemma 3.34. Let (M,L) be a Finsler manifold, then we have
(a) The Finsler metric g (the angular metric tensor ~) is homogenous of degree 0,
(b) The v-curvature tensor S is homogenous of degree −2,
(c) The hv-curvature tensor P is homogenous of degree −1,
(d) The h-curvature tensor R is homogenous of degree 0,
(e) The (h)hv-torsion tensor T is homogenous of degree −1,
(f) The (v)hv-torsion tensor P̂ is homogenous of degree 0,
(g) The (v)h-torsion tensor R̂ is homogenous of degree 1.
Lemma 3.35. For every vector (1)π-form A, we have
∇ A)(øX, øY, øZ)− (
∇ A)(øY, øX, øZ) = A(R(øX, øY )øZ)− R(øX, øY )A(øZ)+
γ bR(øX,øY )
A)(øZ).
Deicke theorem [4] can be formulated globally as follows:
Lemma 3.36. Let (M,L) be a Finsler manifold. The following assertions are
equivalent:
(a) (M,L) is Riemannian,
(b) The (h)hv-torsion tensor T vanishes,
(c) The π-form C vanishes.
Theorem 3.37. Let (M,L) be Finsler manifold which is h-isotropic (of scalar k0)
and Ch-recurrent (of recurrence vector λ0). Then, (M,L) is necessarily one of the
following:
(a) A Riemannian manifold of constant curvature,
(b) A Finsler manifold of dimension 2,
(c) A Finsler manifold of dimensions n ≥ 3 with vanishing scalar k0 and
(∇βøXλo)(øY ) = (∇βøY λo)(øX).
Proof. For a Ch-recurrent manifold, one can easily show that
∇ T )(øX, øY, øZ, øW )− (
∇ T )(øY, øX, øZ, øW ) =
= {(∇βøXλo)(øY )− (∇βøY λo)(øX)}T (øZ, øW ) =: Ψ(øX, øY )T (øZ, øW ).
From which, taking into account Lemma 3.35, we obtain
Ψ(øX, øY )T (øZ, øW ) = T (R(øX, øY )øZ, øW ) + T (øZ,R(øX, øY )øW )−
−R(øX, øY )T (øZ, øW ) + (∇
γ bR(øX,øY )
T )(øZ, øW ).
Now, as (M,L) is h-isotropic of scalar k0, then the h-curvature tensor R has the form
R(øX, øY )øZ = k0{g(øX, øZ)øY − g(øY, øZ)øX} ; (n ≥ 3).
From the above two equations, we get
Ψ(øX, øY )T (øZ, øW ) = k0g(øX, øZ)T (øY, øW )− k0g(øY, øZ)T (øX, øW ) + k0g(øX, øW )T (øZ, øY )−
−k0g(øY, øW )T (øZ, øX)− k0g(øX, T (øZ, øW ))øY + k0g(øY, T (øZ, øW ))øX
+k0g(øX, øη)(∇γøY T )(øZ, øW )− k0g(øY, øη)(∇γøXT )(øZ, øW ).
(3.17)
Setting øY = øη, noting that T is h(−1) and g(øη, øη) = L2, we get
Ψ(øX, øη)T (øZ, øW ) = −k0g(øη, øZ)T (øX, øW )− k0g(øη, øW )T (øZ, øX)− k0T (øX, øZ, øW )øη −
−k0g(øX, øη)T (øZ, øW )− k0L
2(∇γøXT )(øZ, øW ).
From which, we have
g(øY, øη)Ψ(øX, øη)T (øZ, øW ) = −k0g(øY, øη)g(øη, øZ)T (øX, øW )− k0g(øY, øη)g(øη, øW )T (øZ, øX)−
−k0g(øY, øη)T (øX, øZ, øW )øη− k0g(øY, øη)g(øX, øη)T (øZ, øW )−
2g(øY, øη)(∇γøXT )(øZ, øW ),
(3.18)
whereas
g(øX, øη)Ψ(øY, øη)T (øZ, øW ) = −k0g(øX, øη)g(øη, øZ)T (øY, øW )− k0g(øX, øη)g(øη, øW )T (øZ, øY )−
−k0g(øX, øη)T (øY, øZ, øW )øη− k0g(øX, øη)g(øY, øη)T (øZ, øW )−
2g(øX, øη)(∇γøY T )(øZ, øW ).
(3.19)
Now, from (3.17), (3.18) and (3.19), we obtain
T (øZ, øW ){L2Ψ(øX, øY )− g(øY, øη)Ψ(øX, øη) + g(øX, øη)Ψ(øY, øη)} =
= UøX,øY k0L
2{~(øX, øZ)T (øY, øW ) + ~(øX, øW )T (øY, øZ)− φ(øY ) T (øX, øZ, øW )}.
Taking the trace of both sides of the above equation, we get
C(øZ){L2Ψ(øX, øY )− g(øY, øη)Ψ(øX, øη) + g(øX, øη)Ψ(øY, øη)} =
= 2k0L
2{~(øX, øZ)C(øY )− ~(øY, øZ)C(øX)}.
(3.20)
Setting øZ = øC, taking into account the fact that ~(øX, øC) = C(øX), the above
equation reduces to
C(øC){L2Ψ(øX, øY )− g(øY, øη)Ψ(øX, øη) + g(øX, øη)Ψ(øY, øη)} = 0.
Now, if C(øC) = g(øC, øC) = 0, then øC = 0 and so C = 0. Consequently, by
Lemma 3.36, (M,L) is a Riemannian manifold of constant curvature.
On the other hand, if (M,L) is not Riemannian, then we have
L2Ψ(øX, øY )− g(øY, øη)Ψ(øX, øη) + g(øX, øη)Ψ(øY, øη) = 0.
From which, together with (3.20), we get
k0{~(øX, øZ)C(øY )− ~(øY, øZ)C(øX)} = 0. (3.21)
If k0 6= 0, then, by (3.21), ~(øX, øZ)C(øY ) = ~(øY, øZ)C(øX). Setting øY = øC,
we get ~(øX, øZ) = 1
C(øX)C(øZ), which implies that dimM = 2.
If k0 = 0, then R = 0 and (3.17) yields Ψ(øX, øY ) = 0, which means that
(∇βøXλo)(øY ) = (∇βøY λo)(øX). �
Now, we focus our attention to the interesting case (c) of the above theorem. In
this case, the h-curvature tensor R = 0 and hence the (v)h-torsion tensor R̂ = 0.
Therefore, the equation (deduced from (3.9))
(∇γøXR)(øY, øZ, øW ) + (∇βøY P )(øZ, øX, øW )− (∇βøZP )(øY, øX, øW )−
−P (øZ, P (øY, øX)øη)øW +R(T (øX, øY ), øZ)øW − S(R(øY, øZ)øη, øX)øW+
+P (øY, P (øZ, øX)øη)øW − R(T (øX, øZ), øY )øW = 0.
reduces to
(∇βøY P )(øZ, øX, øW )− (∇βøZP )(øY, øX, øW )−
−P (øZ, P̂ (øY, øX))øW + P (øY, P̂ (øZ, øX))øW = 0.
Setting øW = øη, we get
(∇βøY P̂ )(øZ, øX)− (∇βøZP̂ )(øY, øX)− P̂ (øZ, P̂ (øY, øX)) + P̂ (øY, P̂ (øZ, øX)) = 0.
(3.22)
Since (M,L) is Ch-recurrent, then, by Proposition 3.17, the (v)hv-torsion tensor
P̂ satisfies the relations (∇βøZP̂ )(øX, øY ) = (Koλo(øZ) + ∇βøZKo)T (øX, øY ) and
P̂ (øX, øY ) = λo(øη)T (øX, øY ) = KoT (øX, øY ). From these, together with (3.22),
we get
(Koλo(øY ) +∇βøYKo)T (øZ, øX)− (Koλo(øZ) +∇βøZKo)T (øX, øY )−
−K2oT (øZ, T (øX, øY )) +K
oT (øY, T (øX, øZ)) = 0.
Hence, by Corollary 3.19,
K2oS(øY, øZ, øX, øW ) = UøY,øZ{(Koλo(øY ) +∇βøYKo)T (øX, øZ, øW )}.
As S(øY, øZ, øX, øW ) is skew-symmetric in the arguments øX and øW while the
right-hand side is symmetric in the same arguments, we obtain
K2oS(øY, øZ, øX, øW ) = 0, (3.23)
UøY,øZ{(Koλo(øY ) +∇βøYKo)T (øZ, øX, øW )} = 0. (3.24)
It follows from (3.23) and () that
P (øX, øY, øZ, øW ) = λo(øZ)T (øX, øY, øW )− λo(øW )T (øX, øY, øZ).
On the other hand, if Ko 6= 0, then the v-curvature tensor S vanishes from (3.23).
Next, it is seen from (3.24) that, if V(øY ) := Koλo(øY ) +∇βøYKo 6= 0, then there
exists a scalar function Υ =
T (øX,øZ,øW )T (øX,øY,øZ)T (øY,øZ,øW )
(T (øX,øY,øW ))2(V(øZ))3
such that
T (øX, øY, øW ) = ΥV(øX)V(øY )V(øW ).
Summing up, we have
Theorem 3.38. Let (M,L) be a Finsler manifold of dimensions n ≥ 3. If (M,L) is
h-isotropic and Ch-recurrent, then
(a) the recurrence vector λo satisfies : (∇βøXλo)(øY ) = (∇βøY λo)(øX),
(b) the h-curvature tensor R = 0 and the (v)h-torsion tensor R̂ = 0,
(c) the hv-curvature tensor P has the property that
P (øX, øY, øZ, øW ) = λo(øZ)T (øX, øY, øW )− λo(øW )T (øX, øY, øZ),
(d) the (v)hv-torsion tensor P̂ (øX, øY ) = KoT (øX, øY ).
Moreover, if Ko 6= 0, then
(e) the v-curvature tensor S vanishes,
(f) the (h)hv-torsion tensor T satisfies : T (øX, øY, øW ) = ΥV(øX)V(øY )V(øW ).
By Definition 2.10 and Theorem 3.38, we immediately have :
Corollary 3.39. A Finsler manifold (M,L) of dimension n ≥ 3 which is h-isotropic
and Ch-recurrent is necessarily P2-like.
Now, we define an operator P which aids us to investigate the R3-like manifolds.
Definition 3.40.
(a) If ω is a π-tensor field of type (1,p), then P · ω is a π-tensor field of the same
type defined by :
(P · ω)(øX1, ..., øXp) := φ(ω(φ(øX1), ..., φ(øXp))),
where φ is the vector π-form defined by (3.1).
(b) If ω is a π-tensor field of type (0,p), then P · ω is a π-tensor field of the same
type defined by :
(P · ω)(øX1, ..., øXp) := ω(φ(øX1), ..., φ(øXp)).
Remark 3.41. Since φ(φ(øX)) = φ(øX) for every øX ∈ X(π(M)) (Lemma 3.2),
then the operator P is a projector (i.e. P · (P · ω) = P · ω).
Definition 3.42. A π-tensor field ω is said to be indicatory if it satisfies the
condition : P · ω = ω.
The following result gives a characterization of the indicatory property for certain
types of π-tensor fields :
Lemma 3.43.
(a) A vector (2)π-form ω is indicatory if, and only if, ω(øX, øη) = 0 = ω(øη, øX)
and g(ω(øX, øY ), øη) = 0.
(b) A scaler (2) π-form ω is indicatory if, and only if, ω(øX, øη) = 0 = ω(øη, øX).
Proof.
(a) Let ω be a vector (2)π-form. By Definition 3.40(a) and taking into account (3.1),
we get
(P · ω)(øX, øY ) = φ(ω(φ(øX), φ(øY )))
= φ{ω(øX − L−1ℓ(øX)øη, øY − L−1ℓ(øY )øη)}
= φ{ω(øX, øY )− L−1ℓ(øY )ω(øX, øη)−
−L−1ℓ(øX)ω(øη, øY ) + L−2ℓ(øX)ℓ(øY )ω(øη, øη)}
= ω(øX, øY )− L−2g(ω(øX, øY ), øη)øη − φ{L−1ℓ(øY )ω(øX, øη)+
+L−1ℓ(øX)ω(øη, øY )− L−2ℓ(øX)ℓ(øY )ω(øη, øη)}
(3.25)
Now, if ω(øX, øη) = 0 = ω(øη, øX) and g(ω(øX, øY ), øη) = 0, then (3.25) implies
that (P · ω)(øX, øY ) = ω(øX, øY ) and hence ω is indicatory.
On the other hand, if ω is indicatory, then ω(øX, øY ) = φ(ω(φ(øX), φ(øY ))).
From which, setting øX = øη (resp. øY = øη) and taking into account the fact that
φ(øη) = 0 (Lemma 3.2), we get ω(øη, øY ) = 0 (resp. ω(øX, øη) = 0). From this, to-
gether with (P·ω)(øX, øY ) = ω(øX, øY ), Equation (3.25) implies that L−2g(ω(øX, øY ), øη)øη =
0. Consequently, g(ω(øX, øY ), øη) = 0.
(b) The proof is similar to that of (a) and we omit it. �
Proposition 3.44. For a Finsler manifold (M,L), the following tensors are
indicatory :
(a) The π-tensor field φ,
(b) The mixed torsion tensor T ,
(c) The v-curvature tensor S,
(d) The angular metric tensor ~,
(e) The π-tensor field P · ω for every π-tensor field ω.
Now, we define the following π-tensor fields:
F : F (X, Y ) := 1
{Rich(X, Y )−
Schg(X,Y )
2(n−1)
Fo : g(Fo(øX), øY ) := F (øX, øY ),
F a : F a(øX) := F (øη, øX),
F b : F b(øX) := F (øX, øη),
m : m(øX, øY ) := (P · F )(øX, øY ),
mo : g(mo(øX), øY ) := m(øX, øY ),
a : a(øX) := L−1(P · F a)(øX),
øa : g(øa, øY ) := a(øX),
b : b(øX) := L−1(P · F b)(øX),
øb : g(øb, øX) := b(øX),
c : c := L−2F (øη, øη),
R̂ : R̂(øX, øY ) := R(øX, øY )øη,
H : H(øX) := R(øη, øX)øη = R̂(øη, øX).


(3.26)
Remark 3.45. One can show that m, mo, a and b are indicatory and H(øη) = 0.
Proposition 3.46. If (M,L) is an R3-like Finsler manifold, then the π-tensor field
F can be written in the form
F (øX, øY ) = m(øX, øY ) + ℓ(øX)a(øY ) + ℓ(øY )b(øX) + c ℓ(øX)ℓ(øY ). (3.27)
Proof. The proof follows from Definitions 2.14 and 3.40(b), taking into account
Equations (3.1) and (3.26). In more details :
(P · F )(øX, øY ) = F (φ(øX), φ(øY ))
= F (øX − L−1ℓ(øX)øη, øY − L−1ℓ(øY )øη)
= F (øX, øY )− L−1ℓ(øY )F (øX, øη)−
−L−1ℓ(øX)F (øη, øY ) + L−2ℓ(øX)ℓ(øY )F (øη, øη)
= F (øX, øY )− L−1ℓ(øY ){(P · F b)(øX) + L−1ℓ(øX)F (øη, øη)}−
−L−1ℓ(øX){(P · F a)(øY ) + L−1ℓ(øY )F (øη øη)}+ L−2ℓ(øX)ℓ(øY )F (øη, øη)
= F (øX, øY )− ℓ(øX)a(øY )− ℓ(øY )b(øX)− c ℓ(øX)ℓ(øY ). �
Remark 3.47. One can show that the π-tensor fields a and b satisfy the following
relations
F a(øX) = L{a(øX) + c ℓ(øX)},
F b(øX) = L{b(øX) + c ℓ(øX)}.
(3.28)
Proposition 3.48. In an R3-like Finsler manifold (M,L), we have :
(a) R(øX, øY )øZ = g(øX, øZ)Fo(øY )+F (øX, øZ)øY−g(øY, øZ)Fo(øX)−F (øY, øZ)øX.
(b) R̂(øX, øY ) = g(øX, øη)Fo(øY )+F (øX, øη)øY −g(øY, øη)Fo(øX)−F (øY, øη)øX.
(c) H(øY ) = L2Fo(øY ) + c L
2øY − g(øY, øη)Fo(øη)− F (øY, øη)øη.
(d) Fo(øX) = mo(øX) + øa ℓ(øX) + L
−1b(øX)øη + c L−1ℓ(øX)øη.
Consequently,
(e) R̂(øX, øY ) = L{ℓ(øX)(mo(øY ) + c φ(øY )) + b(øX)φ(øY )}−
− L{ℓ(øY )(mo(øX) + c φ(øX)) + b(øY )φ(øX)}.
(f) H(øY ) = L2{mo(øY ) + c φ(øY )}.
Proof.
(a) Since (M,L) is an R3-like manifold, then by Definition 2.14, we have
R(X, Y , Z,W ) =g(X,Z)F (Y ,W )− g(Y , Z)F (X,W )+
+ g(Y ,W )F (X,Z)− g(X,W )F (Y , Z).
From which, using the fact that g(Fo(øX), øY ) = F (øX, øY ) and that the Finsler
metric g is non-degenerate, the result follows.
(b) Follows from (a) by setting øZ = øη.
(c) Follows from (b) by setting øX = øη.
(d) By (3.27) and (3.26), we get
g(Fo(øX), øY ) = g(mo(øX), øY )+g(øa, øY ) ℓ(øX)+L
−1b(øX)g(øη, øY )+c L−1ℓ(øX)g(øη, øY ).
Hence, the result follows, from the non-degeneracy of g.
(e) Follows by substituting Fo(øX) (from (d)) and F
b(øX) (from (3.28)) into (b).
(f) Follows from (e) by setting øX = øη, taking into account Remark 3.45 and the
fact that ℓ(øη) = L. �
Remark 3.49. In view of (3.26) and Lemma 3.2, Definition 2.13(a) can be reformu-
lated as follows:
A Finsler manifold (M,L) is of scaler curvature if the π-tensor field H satisfies the
relation H(øX) = L2κφ(øX), where κ is a scalar function on TM.
Definition 3.50. A Finsler manifold (M,L) is said to be of perpendicular scalar
(or of p-scalar) curvature if the h-curvature tensor R satisfies the condition
(P · R)(øX, øY, øZ, øW ) = Ro{~(øX, øZ)~(øY, øW )− ~(øX, øW )~(øY, øZ)}, (3.29)
where Ro is a function called the perpendicular scalar curvature.
Definition 3.51. A Finsler manifold (M,L) is said to be of s-ps curvature if (M,L)
is both of scalar curvature and of p-scalar curvature.
Proposition 3.52. If mo(øX) = t φ(øX), then an R3-like Finsler manifold is a
Finsler manifold of s-ps curvature.
Proof. Under the given assumption and taking into account Proposition 3.48(f), we
H(øX) = L2κφ(øX), with κ = t + c.
Thus, the considered manifold is of scalar curvature.
Now, we prove that the given manifold is of p-scalar curvature. Applying the
projection P on the h-curvature tensor R of an R3-like manifold, we get
(P · R)(øX, øY, øZ, øW ) = R(φ(øX), φ(øY ), φ(øZ), φ(øW ))
= g(φ(øX), φ(øZ))(P · F )(øY, øW ) + g(φ(øY ), φ(øW ))(P · F )(øX, øZ)−
−g(φ(øY ), φ(øZ))(P · F )(øX, øW )− g(φ(øX), φ(øW ))(P · F )(øY, øZ)
= g(φ(øX), φ(øZ))m(øY, øW ) + g(φ(øY ), φ(øW ))m(øX, øZ)−
−g(φ(øY ), φ(øZ))m(øX, øW )− g(φ(øX), φ(øW ))m(øY, øZ).
(3.30)
Since
g(φ(øX), φ(øY )) = g(φ(øX), øY − L−1ℓ(øY )øη) = g(φ(øX), øY )− L−1ℓ(øY )g(φ(øX), øη)
= ~(øX, øY )− L−1ℓ(øY )~(øX, øη) = ~(øX, øY ),
then, by using again the given assumption (mo = t φ =⇒ m = t~), Equation (3.30)
reduces to
(P · R)(øX, øY, øZ, øW ) = ~(øX, øZ)m(øY, øW ) + ~(øY, øW )m(øX, øZ)−
−~(øY, øZ)m(øX, øW )− ~(øX, øW )m(øY, øZ)
= 2t{~(øX, øZ)~(øY, øW )− ~(øY, øZ)~(øX, øW )}.
Therefore, by taking Ro = 2t, we have
(P · R)(øX, øY, øZ, øW ) = Ro{~(øX, øZ)~(øY, øW )− ~(øY, øZ)~(øX, øW )}.
Consequently, the given manifold is of p-scalar curvature. �
Theorem 3.53. If an R3-like Finsler manifold (M,L) is of p-scalar curvature, then
it is of s-ps curvature.
Proof. Since the considered manifold is R3-like, then, by the same procedure as in
the proof of Proposition 3.52, we have
(P · R)(øX, øY, øZ, øW ) = ~(øX, øZ)m(øY, øW ) + ~(øY, øW )m(øX, øZ)−
−~(øY, øZ)m(øX, øW )− ~(øX, øW )m(øY, øZ).
(3.31)
On the other hand, since the considered manifold is of p-scalar curvature, then the
h-curvature tensor satisfies
(P · R)(øX, øY, øZ, øW ) = Ro{~(øX, øZ)~(øY, øW )− ~(øY, øZ)~(øX, øW )}. (3.32)
Now, from Equations (3.31) and (3.32), we obtain
UøX,øY {Ro~(øX, øZ)~(øY, øW )− ~(øX, øZ)m(øY, øW )− ~(øY, øW )m(øX, øZ)} = 0.
Using (3.26) and the non-degeneracy of the metric tensor g, the above equation
reduces to
UøX,øY {Ro~(øX, øZ)φ(øY )− ~(øX, øZ)mo(øY )−m(øX, øZ)φ(øY )} = 0. (3.33)
Since the π-tensor fields φ,m and mo are indicatory, then
Tr{øY 7−→ ~(øX, øY )φ(øZ)} = g(øX, φ(øZ)) = ~(øX, øZ),
Tr{øY 7−→ ~(øX, øY )mo(øZ)} = m(øX, øZ),
Tr{øY 7−→ m(øX, øY )φ(øZ)} = m(øX, øZ).
Consequently, if we take the trace of both sides of Equation (3.33), making use of
Lemma 3.43, we get
(n− 2)Ro~(øX, øZ)− (n− 3)m(øX, øZ)− (n− 1)t ~(øX, øZ) = 0,
where t := 1
Tr(mo). From which, using (3.26) and Lemma 3.2, we get
(n− 2)Roφ− (n− 3)mo − (n− 1)t φ = 0. (3.34)
Again, taking the trace of the above equation, we obtain
(n− 1)(n− 2)(Ro − 2t) = 0.
Substituting the above relation into (3.34), we get mo = t φ. Hence, by Proposition
3.52, the result follows. �
Theorem 3.54. If an R3-like Finsler manifold (M,L) is of scalar curvature, then it
is of s-ps curvature.
Proof. Since the given manifold is R3-like, then the π-tensor H is given by (cf.
Proposition 3.48):
H(øX) = L2{mo(øX) + c φ(øX)}. (3.35)
And since the considered manifold is of scalar curvature, then
H(øX) = L2κφ(øX). (3.36)
From Equations (3.35) and (3.36), we deduce thatmo(øX) = (κ−c)φ(øX) =: tφ(øX).
Hence, by Proposition 3.52, the result follows. �
Now, let us define the π-tensor field
Ψ(øX, øY, øZ, øW ) = R(øX, øY, øZ, øW )− 1
UøX,øY {g(øX, øZ)Ric
h(øY, øW )+
+g(øY, øW )Rich(øX, øZ)− rg(øX, øZ)g(øY, øW )},
(3.37)
where r = 1
Sch. From Definition 2.14 and (3.37), we immediately obtain
Theorem 3.55. An R3-like Finsler manifold is characterized by
Ψ(øX, øY, øZ, øW ) = 0.
The tensor field Ψ in the above theorem being of the same form as the Weyl
conformal tensor in Riemannian geometry, we draw the following
Theorem 3.56. An R3-like Riemannian manifold is conformally flat.
Remark 3.57. It should be noted that some important results of [8], [9], [11], [13],
[19], [20],...,etc. (obtained in local coordinates) are retrieved from the above mentioned
global results (when localized).
Appendix. Local formulae
For the sake of completeness, we present in this appendix a brief and concise
survey of the local expressions of some important geometric objects and the local
definitions of the special Finsler manifolds treated in the paper.
Let (U, (xi)) be a system of local coordinates on M and (π−1(U), (xi, yi)) the
associated system of local coordinates on TM . We use the following notations :
(∂i) := (
): the natural basis of TxM, x ∈M ,
(∂̇i) := (
): the natural basis of Vu(TM), u ∈ TM ,
(∂i, ∂̇i): the natural basis of Tu(TM),
(ø∂i): the natural basis of the fiber over u in π
−1(TM) (ø∂i is the lift of ∂i at u).
To a Finsler manifold (M,L), we associate the geometric objects :
gij :=
∂̇i∂̇jL
2 = ∂̇i∂̇jE: the Finsler metric tensor,
Cijk :=
∂̇k gij : the Cartan tensor,
~ij := gij − ℓiℓj (ℓi := ∂L/∂y
i): the angular metric tensor,
Gh: the components of the canonical spray,
Ghi := ∂̇iG
Ghij := ∂̇jG
i = ∂̇j ∂̇iG
(δi) := (∂i −G
i ∂̇h): the basis of Hu(TM) adapted to G
(δi, ∂̇i): the basis of Tu(TM) = Hu(TM)⊕ Vu(TM) adapted to G
We have :
γ(ø∂i) = ∂̇i,
ρ(∂i) = ø∂i, ρ(∂̇i) = 0, ρ(δi) = ø∂i,
β(ø∂i) = δi,
J(∂i) = ∂̇i, J(∂̇i) = 0, J(δi) = ∂̇i,
h := βoρ = dxi ⊗ ∂i −G
j ⊗ ∂̇i v := γoK = dy
i ⊗ ∂̇i +G
j ⊗ ∂̇i.
We define :
γhij :=
ghℓ(∂i gℓj + ∂j giℓ − ∂ℓ gij),
Chij :=
ghℓ(∂̇i gℓj + ∂̇j giℓ − ∂̇ℓ gij) =
ghℓ ∂̇i gjℓ = g
hℓCijℓ,
Γhij :=
ghℓ(δi gℓj + δj giℓ − δℓ gij) .
Then, we have :
• The canonical spray G: Gh = 1
γhij y
• The Barthel connection Γ: Ghi = ∂̇iG
h = Γhijy
j = Ghijy
• The Cartan connection CΓ: ( Γhij, G
i , C
The associated h-covariant (resp. v-covariant) derivative is denoted by p (resp. |),
where Ki
j|k := δkK
mk −K
jk and K
j |k := ∂̇kK
mk −K
• The Berwald connection BΓ: ( Ghij, G
i , 0).
The associated h-covariant (resp. v-covariant) derivative is denoted by
p(resp.
where Ki
:= δkK
mk −K
jk and K
:= ∂̇kK
We also have Ghij = Γ
ij + C
ij |k y
k = Γhij + C
ij |o, where C
ij |o = C
ij |k y
For the Cartan connection, we have :
(v)h-torsion : Rijk = δkG
j − δjG
k = Ujk{δkG
(v)hv-torsion : P ijk = G
jk − Γ
jk = C
jk|my
m = C i
jk|0,
(h)hv-torsion : C ijk = 1/2{g
ri∂̇rgjk},
h-curvature : Rihjk = Ujk{δkΓ
hj + Γ
mk} − C
hv-curvature : P ihjk = ∂̇kΓ
hj − C
+ C ihmP
v-curvature : Sihjk = C
mj − C
mk = Ujk{C
For the Berwald connection, we have :
(v)h-torsion : R∗ijk = δkG
j − δjG
k = Ujk{δkG
h-curvature : R∗ihjk = Ujk{δkG
hj +G
hv-curvature : P ∗ihjk = ∂̇kG
hj =: G
In the following, we give the local definitions of the special Finsler spaces treated
in the paper. For each special Finsler space (M,L), we set its name, its defining
property and a selected reference in which the local definition is located:
• Rimaniann manifold [22] : gij(x, y) ≡ gij(x) ⇐⇒ Cijk = 0 ⇐⇒ Ci := C
ik = 0
(Deicke’s theorem [4]).
• Minkowaskian manifold [22]: gij(x, y) ≡ gij(y) ⇐⇒ C
= 0 and Rhijk = 0.
• Berwald manifold [22]: Γhij(x, y) ≡ Γ
ij(x) (i.e. ∂̇kΓ
ij = 0) ⇐⇒ C
• Ch-recurrent manifold [13]: Chij|k = µkChij ,
where µj is a covariant vector field.
• P ∗-Finsler manifold [7]: Ch
ij|0 = λ(x, y)C
where λ(x, y) = PiC
; Pi := P
ik = C
ik|0 = Ci|0 and C
2 = CiC
i 6= 0.
• Cv-recurrent manifold [13]: C ijk|l = λl C
jk or Cijk|l = λl Cijk.
• C0-recurrent manifold [13]: C ijk
= λl C
jk or Cijk
= λl Cijk.
• Semi-C-reducible manifold (dimM ≥ 3) [18]:
Cijk =
(n+ 1)
(~ijCk + ~jkCi + ~kiCj) +
CiCjCk, C
2 6= 0,
where µ and τ are scalar functions satisfying µ+ τ = 1.
• C-reducible manifold (dimM ≥ 3) [15]: Cijk =
(~ijCk + ~jkCi + ~kiCj).
• C2-like manifold (dimM ≥ 2) [17]: Cijk =
CiCjCk, C
2 6= 0.
• quasi-C-reducible manifold (dimM ≥ 3) [23]: Cijk = AijCk + AjkCi + AkiCj ,
where Aij(x, y) is a symmetric tensor field satisfying Aijy
i = 0.
• S3-like manifold (dimM ≥ 4) [6]: Slijk =
(n−1)(n−2)
{~ik~lj − ~ij~lk},
where S is the vertical scalar curvature.
• S4-like manifold (dimM ≥ 5) [6]: Slijk = ~ljFik − ~lkFij + ~ikFlj − ~ijFlk,
where Fij :=
{Sij −
2(n−2)
S~ij}; Sij being the vertical Ricci tensor.
• Sv-recurrent manifold [20], [11]: Shijk|m = λmShijk,
where λj(x, y) is a covariant vector field.
• Second order Sv-recurrent manifold [20], [11]: Shijk|m|n = ΘmnShijk,
where Θij(x, y) is a covariant tensor field.
• Landsberg manifold [7]: P hkji y
k = 0 ⇐⇒ (∂̇iΓ
k = 0 ⇐⇒ Ch
yk = 0.
• General Landsberg manifold [10]: P rijry
i = 0 ⇐⇒ Cj|o = 0.
• P -symmetric manifold [19]: Phijk = Phikj.
• P2-like manifold (dimM ≥ 3) [14]: Phijk = αhCijk − αiChjk,
where αk(x, y) is a covariant vector field.
• P -reducible manifold (dimM ≥ 3) [19]: Pijk =
(~ij Pk + ~jk Pi + ~ki Pj),
where Pijk = ghiP
• h-isotropic manifold (dimM ≥ 3) [13]: Rhijk = ko{ghjgik − ghkgij},
for some scalar ko, where Rhijk = gilR
• Manifold of scalar curvature [21]: Rijkl y
iyk = kL2~jl,
for some function k : TM −→ R .
• Manifold of constant curvature [21]: the function k in the above definition is
constant.
• Manifold of perpendicular scalar (or of p-scalar ) curvature [8], [9]:
P ·Rhijk := ~
k Rlmnr = Ro{~ik~hj − ~ij~hk},
where Ro is a function called a perpendicular scalar curvature.
• Manifold of s-ps curvature [8], [9]: (M,L) is both of scalar curvature and of
p-scalar curvature.
• R3-like manifold (dimM ≥ 4) [8]: Rhijk = ghjFik − ghkFij + gikFhj − gijFhk,
where Fij :=
{Rij −
r gij}; Rij := R
ijh, r :=
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