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Structure of the indicatrix bundle
of Finsler-Rizza manifolds
V. Balan, E. Peyghan and A. Tayebi

Abstract. In this paper, we construct a framed f -structure on the slit
tangent space of a Rizza manifold. This induces on the indicatrix bundle
an almost contact metric. We find the conditions under which this structure reduces to a contact or to a Sasakian structure. Finally we study
these structures on K¨ahlerian Finsler manifolds.
M.S.C. 2010: 53B40, 53C60, 32Q60, 53C15.
Key words: Contact structures; framed f -manifolds; K¨ahlerian structures; Rizza
manifolds.

1

Introduction

In [15, 16], G. B. Rizza introduced on almost complex manifolds (M, J), the so-called
Rizza condition. For Finsler structures (M, L) - where L is the Finsler metric, a
triple (M, J, L) which satisfies the Rizza condition is called a Finsler-Rizza manifold.
It was shown by Heil ([4]) that if the fundamental tensor gij of the Finsler metric
L is compatible with the almost complex structure J, then the Finsler structure is

Riemannian. This leads to considering a weaker assumption on the Finsler metric,
like the Rizza condition. The notion of Rizza manifolds was developed further in
Finslerian framework by Y. Ichijy¯
o, who showed that every tangent space to a Rizza
manifold is a complex Banach space and that Rizza condition does not necessarily
reduce the Finsler metric to a Riemannian one ([7, 8]). He also defined a notion
of K¨ahler Finsler metric and showed that for a K¨ahler Finsler manifold, the almost
complex structure is integrable. Recently, several mathematicians studied Rizza and
K¨ahler Finsler manifolds ([5, 9, 10, 13, 17]).
Let (M, L) be an n-dimensional Finsler manifold (n even), admitting an almost
complex structure Jji (x). Let gij (x, y) = 21 ∂˙i ∂˙j L2 (x, y) be the associated Finsler
metric tensor field [11, 12, 14]. We say that the fundamental function L(x, y) satisfies
the Rizza condition, if
L(x, φθ y) = L(x, y),

∀θ ∈ R,

where

φiθj = δji cos θ + Jji sin θ.

∗
Balkan
Journal of Geometry and Its Applications, Vol.16, No.2, 2011, pp. 1-12.
c Balkan Society of Geometers, Geometry Balkan Press 2011.
°

2

V. Balan, E. Peyghan and A. Tayebi

In this case, M is called an almost Hermitian Finsler manifold or simply, a Rizza
manifold. The Rizza condition practically requires that, for all y ∈ Tx M , the Finsler
norm L should be constant on the orbit y → φθ (y), θ ∈ R, which lives inside the fiber
Tx M . We note that each such orbit is determined by the action on y ∈ Tx M of a
special element φ of the loop group ΛAut(Tx M ) of id−automorphisms of the tangent
φ

bundle, defined by θ ∈ S 1 −→ φθ = I · cos θ + J · sin θ ∈ Aut(Tx M ).
In [6], it was shown that the Rizza condition holds if the following equivalent

conditions hold true:1
• gpq (x, φθ y)φpθi φqθj = gij (x, y),
•
•

∀θ ∈ R;

q
i
(x)) y m = 0;
gij (x, y)Jm
(x)y m y j = 0; (gim (x, y) − gpq (x, y)Jip (x)Jm
gim (x, y)Jjm (x) + gjm (x, y)Jim (x) + 2Cijm (x, y)Jrm (x)y r = 0, where Cijm

is the

Cartan tensor of the Finsler metric.
As well, it was shown that if the Finsler metric L satisfies gpq (x, y)Jip (x)Jjq (x) =
gij (x, y), then it reduces to a Riemannian metric and accordingly, (J, g) reduces to
an almost Hermitian structure.

Now let M be a Rizza manifold. Then
¢
1¡
gij (x, y) + gpq (x, y)Jip (x)Jjq (x) ,
(1.1)
geij (x, y) =
2

is a homogeneous symmetric generalized metric on T M , which is called a generalized
Finsler metric; this satisfies the relation gepq (x, y)Jip (x)Jjq (x) = geij (x, y). Also, in a
³
´
Rizza manifold, the relation gij (x, y) = ∂˙i ∂˙j 1 gepq (x, y)y p y q holds true.
2

Conversely, for M a manifold admitting an almost complex structure Jji (x), Ichijy¯
o
showed ([6]) that M admits a Finsler metric which constructs a Rizza structure together with Jji (x), if and only if M admits a generalized Finsler metric geij (x, y)
satisfying the following conditions:
• gejk (x, y) = gepq (x, y)Jjp (x)Jkq (x);

• ∂˙k gepq (x, y)y p y q = 0;
• (e
gjk (x, y) + ∂˙k gejm (x, y)y m )ζ j ζ k is positive definite.

In the following, for a given Rizza manifold (M, J, L) we shall denote
(1.2)

Jij (x, y) = gim (x, y)Jjm (x),

Then we obtain the following relations ([6])
(1.3)

Jeij = −Jeji ,

Jeij (x, y) = geim (x, y)Jjm (x).

Jeim Jjm = −e
gij ,

1
Jeij = (Jij − Jji ).
2

By using [6, Theorem, eq. (3)] and denoting yi := gij y j , we infer
(1.4)

geij y j =

1
(gij y j + gpq Jip Jjq y j ) = yi ,
2

1 We shall assume that all the Latin indices run within the range 1, n, and that the Einstein
summation rule for indices applies.

3

Structure of the indicatrix bundle of Finsler-Rizza manifolds

It follows that
geij y i y j = L2 .

(1.5)

Also, the relations (1.2), (1.3) and (1.4) lead to
(1.6)

Jeij y j = −Jeji y j = −e
gjm Jim y j = −Jim ym = −Jij yj .

By a contact manifold we mean a differentiable manifold M 2n+1 endowed with
with a 1-form η, such that η ∧ (dη)n 6= 0. It is well known ([2]) that given the form
η, there exist a unique vector field ξ, such that dη(ξ, X) = 0 and η(ξ) = 1; this field
is called the characteristic vector field or the Reeb vector field of the contact form η.
We say that a Riemannian metric g is an associated metric for a contact form
η if (i) η(X) = g(X, ξ) and (ii) there exist a field of endomorphisms f , such that
f 2 = −I + η ⊗ ξ and dη(X, Y ) = g(f X, Y ). Then we refer to (f, ξ, η, g) as a contact
metric structure and to M 2n+1 with such a structure as a contact metric manifold.
An almost contact structure, (f, ξ, η), consists of a field of endomorphisms f , and a

1-form η such that f 2 = −I +η⊗ξ and η(ξ) = 1 and an almost contact metric structure
owns a Riemannian metric which satisfies the compatibility condition g(f X, f Y ) =
g(X, Y ) − η(X)η(Y ).
The product M 2n+1 × R carries a natural almost complex structure defined by
d
d
) = (f X − gξ, η(X) dt
) and the underlying almost contact structure is said
J(X, g dt
to be normal if J is integrable. The normality condition can be expressed as N = 0,
where N is defined by N = Nf + dη ⊗ ξ and Nf is the Nijenhuis tensor of f . A
Sasakian manifold is a normal contact metric manifold.
A framed f -structure is a natural generalization of an almost contact structure.
It was introduced by S. I. Goldberg and K. Yano [3]. We recall its definition: let N
be a (2n + s)-dimensional manifold endowed with an endomorphism f of rank 2n of
the tangent bundle, satisfying f 3 + f = 0. If there exists on N the vector fields (ξb )
and the 1-forms η a (a, b = 1, 2, . . . , s) such that
η a (ξb ) = δba ,

f (ξa ) = 0,

η a ◦ f = 0,

f 2 = −I +

s
X

η a ⊗ ξa ,

a=1

where I is the identity automorphism of the tangent bundle, then N is said to be a
framed f -manifold [1].
In this paper, by using an almost complex structure Jji (x) and the generalized
Finsler metric geij (x, y) defined by (1.1) we introduce an almost Hermitian structure
(G, F ) on T M . Then we obtain a framed f -structure on Tg
M = T M \{0} and by
its restriction to the indicatrix bundle IM we introduce an almost contact metric
structure on IM . We further find the conditions under which this structure is a

contact metric structure and a Sasakian structure. Also, we study these structures
on K¨ahlerian Finsler manifolds.

2

A framed f - structure on Tg
M

Let M be a Rizza manifold, and let T M be the tangent bundle over M . We shall
∂
˙ ˙
further use a local frame (δi , ∂˙i ) of T M , where we put δi = ∂i − Gm
i ∂m , ∂i = ∂y i ,

4

V. Balan, E. Peyghan and A. Tayebi

∂
m

∂i = ∂x
are the components of an Ehresmann connection on M . Then we
i and Gi
can globally define on T M an (1,1)-tensor field F , such that

(2.1)

F (δi ) = Jik ∂˙k ,

F (∂˙i ) = Jik δk .

Since Jki Jjk = −δji , it is obvious that F defines an almost complex structure on T M .
Moreover, we can introduce an inner product h · , · i such that
(2.2)

hδi , δj i = geij ,

hδi , ∂˙j i = 0,

h∂˙i , ∂˙j i = geij .

Then the inner product gives on T M a globally defined Riemann metric G, as follows
(2.3)

G = geij dxi dxj + geij δy i δy j ,

where (dxi , δy i ) is the dual basis of (δi , ∂˙i ) and δy i = dy i + Gim dxm . Using (2.1) and
(2.3), we obtain
G(F (δi ), F (δj )) = Jik Jjl G(∂˙k , ∂˙l ) = Jik Jjl gekl = geij = G(δi , δj ).

Similarly, we obtain G(F (∂˙i ), F (∂˙j )) = geij = G(∂˙i , ∂˙j ) and G(F (δi ), F (∂˙j )) = 0 =
G(δi , ∂˙j ), which ultimately lead to
Theorem 2.1. On every Rizza manifold M , its tangent bundle T M admits an almost Hermitian structure (G, F ), where F and G are defined by (2.1) and (2.3),
respectively.
Now we define the vector fields ξ1 , ξ2 and 1-forms η 1 , η 2 on Tg
M respectively by

(2.4)

i
ξ1 := y m Jm
δi ,

ξ2 := y i ∂˙i

and
(2.5)

η 1 :=

1 me
y Jim dxi ,
L2

η 2 :=

1
yi δy i .
L2

Lemma 2.2. Let F be defined by (2.1), ξ1 , ξ2 be defined by (2.4) and η 1 , η 2 be defined
by (2.5). Then we have
(2.6)

F (ξ1 ) = −ξ2 ,

F (ξ2 ) = ξ1 ,

and
(2.7)

η1 ◦ F = η2 ,

η 2 ◦ F = −η 1 .

Proof. Relation (2.6) immediately follows from (2.1) and (2.4). To prove (2.7), we
use the adapted basis (δi , ∂˙i ) and (1.3), which infer
1 m ke
1
1
y Ji Jkm = 2 gemi y m = 2 yi = η 2 (∂˙i ),
L2
L
L
1
1
1
(η 2 ◦ F )(δi ) = Jik η 2 (∂˙k ) = 2 Jik yk = 2 Jik gekr y r = − 2 Jeir y r = −η 1 (δi ),
L
L
L

(η 1 ◦ F )(∂˙i ) = Jik η 1 (δk ) =

and (η 1 ◦ F )(δi ) = 0 = η 2 (δi ), (η 2 ◦ F )(∂˙i ) = 0 = η 1 (∂˙i ), which lead to (2.7).

¤

Structure of the indicatrix bundle of Finsler-Rizza manifolds

5

Lemma 2.3. Let G be defined by (2.3), ξ1 , ξ2 be defined by (2.4) and η1 , η2 be defined
by (2.5). Then we have
η 1 (X) =

(2.8)

1
G(X, ξ1 ),
L2

η 2 (X) =

1
G(X, ξ2 ),
L2

and
η a (ξb ) = δba , a, b = 1, 2,

(2.9)
where X ∈ X (Tg
M ).

Proof. In the adapted basis (δi , ∂˙i ), we obtain
G(δi , ξ1 ) = G(δi , y k Jkj δj ) = y k Jkj geij = y k Jeik = L2 η 1 (δi ).

Similarly, we infer G(∂˙i , ξ2 ) = L2 η 2 (∂˙i ), G(∂˙i , ξ1 ) = 0 = L2 η 1 (∂˙i ) and G(δi , ξ2 ) = 0 =
L2 η 2 (δi ), which completes the proof of (2.8). As well, using (1.3) and (1.5) we get
η 1 (ξ1 ) = η 1 (y k Jki δi ) =

1
1 k i me
y Jk y Jim = 2 y k y m gemk = 1,
2
L
L

and η 1 (ξ2 ) = 0, which prove the first relation of (2.9). In a similar way, one can prove
the second relation of (2.9).
¤
Using the almost complex structure F , we define a new tensor field f of type (1, 1)
on Tg
M by
f = F + η 1 ⊗ ξ2 − η 2 ⊗ ξ1 .

(2.10)

By using the above relation, (2.6) and (2.9) we infer that f (ξ1 ) = f (ξ2 ) = 0. Similarly,
from (2.7) and (2.9)we conclude that (η 1 ◦ f )(X) = (η 2 ◦ f )(X) = 0, where X ∈
X (Tg
M ). These relations and (2.6), (2.10) give us
f 2 (X) = F (f (X)) = F 2 (X) + η 1 (X)F (ξ2 ) − η 2 (X)F (ξ1 ) = −X + η 1 (X)ξ1 + η 2 (X)ξ1 .

Therefore we have
Lemma 2.4. The following properties hold true for tensor field f defined by (2.10):
(2.11)
(2.12)

f (ξa ) = 0,

η a ◦ f = 0, a = 1, 2,

f 2 (X) = −X + η 1 (X)ξ1 + η 2 (X)ξ2 , X ∈ X (Tg
M ).

Theorem 2.5. Let f, (ξa ), (η a ), a = 1, 2 be defined respectively by (2.10), (2.4) and
(2.5). Then the triple (f, {ξa }, {η a }) provides a framed f -structure on Tg
M.

Proof. Considering Lemma 2.4 and relation (2.9), in order to complete the proof, we
need to prove f 3 +f = 0 and to show that f is of rank 2n−2. Since f (ξ1 ) = f (ξ2 ) = 0,
by using (2.11) we derive that f 3 (X) = −f (X), for all X ∈ X (Tg
M ). Now we need
to show that ker f = Span{ξ1 , ξ2 }. It is clear that Span{ξ1 , ξ2 } ⊆ ker f , because
f (ξ1 ) = f (ξ2 ) = 0. Now let X ∈ ker f ; then f (X) = 0 implies that F (X) + η 1 (X)ξ2 −
η 2 (X)ξ1 = 0. Thus we infer that F 2 (X) = η 2 (X)F (ξ1 )−η 1 (X)F (ξ2 ). Since F 2 = −I,
it follows from Lemma 2.2 that X = η 1 (X)ξ1 + η 2 (X)ξ2 , i.e., X ∈ Span{ξ1 , ξ2 }. ¤

6

V. Balan, E. Peyghan and A. Tayebi

Theorem 2.6. The Riemannian metric G defined by (2.3) satisfies
G(f (X), f (Y )) = G(X, Y ) − L2 η 1 (X)η 1 (Y ) − L2 η 2 (X)η 2 (Y ),
for X, Y ∈ X (Tg
M ).

Proof. By using (2.10), we get the local expression of f as follows:
´
³
1
(2.13)
f (δi ) = Jik + 2 Jeir y r y k ∂˙k ,
L
³
´
1
k
f (∂˙i ) = Ji − 2 Jrk y r yi δk .
(2.14)
L
By using (2.3), (2.13) we obtain
G(f (δi ), f (δj )) = Jik Jjh gekh +

1 e e l k h i
yr h e k h
h k
e
J
J
y
g
e
+
J
J
y
g
e
+
Jir Jjl y y y gekh .
jr
kh
ir
kh
i
j
L2
L2

By using (1.5) and (1.6), the above equation leads to
(2.15)

G(f (δi ), f (δj )) = geij −

But we have the relations
G(δi , δj ) = geij ,

η 1 (δi )η 2 (δj ) =

which allow to rewrite (2.15) as follows

1 e e r k
Jir Jjk y y .
L2

1 re ke
y Jir y Jjk
L4

η 2 (δi )η 2 (δj ) = 0,

G(f (δi ), f (δj )) = G(δi , δj ) − L2 η 1 (δi )η 1 (δj ) − L2 η 2 (δi )η 2 (δj ).
We similarly obtain
1
G(f (∂˙i ), f (∂˙j )) = geij − 2 yi yj = G(∂˙i , ∂˙j ) − L2 η 1 (∂˙i )η 1 (∂˙j ) − L2 η 2 (∂˙i )η 2 (∂˙j ),
L

and

G(f (δi ), f (∂˙j )) = 0 = G(δi , ∂˙j ) − L2 η 1 (δi )η 1 (∂˙j ) − L2 η 2 (δi )η 2 (∂˙j ),

which completes the proof.
Let us set Ω(X, Y ) = G(f X, Y ) for X, Y ∈ X (Tg
M ). Then we have

¤

Theorem 2.7. The map Ω is a 2-form on Tg
M . Further, the annihilator of Ω is
Span{ξ1 , ξ2 }, which is an integrable distribution.
Proof. By using (1.4), (2.13) and (2.14), we obtain
1 re k
1
y Jir y gekj = Jeji + 2 Jeir y r yj ,
L2
L
1
1
(2.17) Ω(∂˙i , δj ) = G(f (∂˙i ), δj ) = (Jik − 2 yi y r Jrk )e
gkj = Jeji − 2 yi y r Jejr ,
L
L
(2.18) Ω(δi , δj ) = Ω(∂˙i , ∂˙j ) = 0.
(2.16) Ω(δi , ∂˙j ) = G(f (δi ), ∂˙j ) = Jik gekj +

Structure of the indicatrix bundle of Finsler-Rizza manifolds

7

Since Jeij = −Jeji , then from (2.16) and (2.17), we derive that Ω(δi , ∂˙j ) = −Ω(∂˙j , δi ).
Then, using (2.16) we obtain Ω(X, Y ) = −Ω(Y, X), ∀X, Y ∈ X (Tg
M ). Thus Ω is
a 2-form on Tg
M . We further show that the annihilator of Ω is Span{ξ1 , ξ2 }. Let
¯
X = X i δi + X i ∂˙i ∈ X (Tg
M ). By using (1.4) and (2.16)-(2.16) we get
¯

j e
Ω(X, ξ1 ) = X i y m Jm
(Jji −

1
1
¯
yi y r Jejr ) = X i y m (e
gim − 2 yi y r germ ) = 0.
2
L
L

We similarly obtain Ω(X, ξ2 ) = 0. Therefore the annihilator of Ω contains Span{ξ1 , ξ2 }.
Now let X belong to the annihilator of Ω. Then we have Ω(X, δj ) = 0 and Ω(X, ∂˙j ) =
¯
0. If we assume X = X i δi + X i ∂˙i , then these equations give us the relations
1 e m
Jim y yj ) = 0,
L2
1
¯
X i (Jeji − 2 Jejm y m yi ) = 0.
L
X i (Jeji +

(2.19)
(2.20)

Since Jeij = geim Jjm , then by direct calculation we obtain Jeji Jrj gerh = δih and Jesr gerh =
−Jsh . Transvecting (2.19) and (2.20) with Jrj gerh and using these equations we derive
¯
¯
that X i = L12 X h Jehm y m y s Jsi and X i = L12 X h yh y i . Thus we deduce that
X=

1 ¯
1
1 he
1 ¯
X Jhm y m y s Jsi δi + 2 X h yh y i ∂˙i = 2 X h Jehm y m ξ1 + 2 X h yh ξ2 .
2
L
L
L
L

Therefore Span{ξ1 , ξ2 } contains the annihilator of Ω, and consequently the annihilator
of Ω is Span{ξ1 , ξ2 }. Also we obtain [ξ1 , ξ2 ] = y j Jji δi = ξ1 . Hence the distribution
Span{ξ1 , ξ2 } is integrable.
¤

3

Almost contact structure on the indicatrix bundle

Let (M, J, L) be a Finsler-Rizza manifold, and let IM be its indicatrix bundle of
(M, L), i.e.,
IM = {(x, y) ∈ Tg
M |L(x, y) = 1},

which is a submanifold of dimension 2n − 1 of Tg
M . Note that ξ2 defined in (2.4) is
a unit vector field on IM , since G(ξ2 , ξ2 ) = 1. It is easy to show that ξ2 is a normal
vector field on IM with respect to the metric G. Indeed, if the local equations of IM
in Tg
M are given by
xi = xi (uγ ),

then we have

y i = y i (uγ ),

γ ∈ {1, . . . , 2n − 1},

∂L ∂xi
∂L ∂y i
+
= 0.
∂xi ∂uγ
∂y i ∂uγ

Since F is a horizontally covariant constant, i.e.,
(3.1)

³

Nik

∂L
∂xi

∂L
= Nik ∂y
k , it follows that

∂xi
∂y k ´ ∂L
+
= 0.
∂uγ
∂uγ ∂y k

8

V. Balan, E. Peyghan and A. Tayebi

The natural frame field { ∂u∂ γ } on IM is given by
(3.2)

³
i
∂xi ∂
∂y i ∂
∂xi
∂y k ´ ∂
∂
k ∂x
=
+
=
δ
+
N
+
.
i
i
∂uγ
∂uγ ∂xi
∂uγ ∂y i
∂uγ
∂uγ
∂uγ ∂y k

Thus by using (3.1) and the equality
(3.3)

G

yk
L

=

∂L
,
∂y k

we obtain

´
³
³ ∂
i
∂y k ´ ∂L
k ∂x
,
ξ
+
= 0.
=
L
N
2
i
∂uγ
∂uγ
∂uγ ∂y k

Thus ξ2 is orthogonal to any tangent to IM vector. Also, the vector field ξ1 is tangent
to IM , since G(ξ1 , ξ2 ) = 0.
Lemma 3.1. The hypersurface IM is invariant with respect to f , i.e., f (Tu (IM )) ⊆
Tu (IM ), ∀u ∈ IM .
Proof. By using (2.8) and the second equation of (2.11), we get
µ µ
µ
¶ ¶
¶
∂
∂
2
G f
=
(η
◦
f
)
,
ξ
= 0., ∀γ = 1, 2, . . . , 2n − 1.
2
∂uγ
∂uγ
Thus the hypersurface IM is invariant with respect to f .

¤

Lemma 3.2. Let the framed f -structure be given by Theorem 2.5. Then restricting
this to IM , we have
η 1 = y m Jeim dxi ,

η 2 = 0,

f (X) = F (X) + η 1 (X)ξ2 ,

∀X ∈ X (IM ).

Proof. Since L2 = 1 on IM and η 2 (X) = G(X, ξ2 ) = 0, the claim follows.

¤

¯ = G|IM , then from Theorem 2.6
Denoting η¯ = η 1 |IM , ξ¯ = ξ|IM , f¯ = f |IM and G
¯ f¯(X), f¯(Y )) = G(X,
¯
we get G(
Y ) − η¯(X)¯
η (Y ). Therefore Theorem 2.5 and Lemma
3.2 imply that
¯ η¯, G)
¯
Theorem 3.3. Let the framed f -structure be given by Theorem 2.5. Then (f¯, ξ,
defines an almost contact metric structure on IM .
If we put δ˙j = f¯(δj ), then we get n local vector fields which are tangent to IM ,
since IM is an invariant hypersurface. These, together with δi , i = 1, . . . , n, are all
tangent to IM and they are not linearly independent. But, considering δi , i = 1, . . . , n
and δ˙j with j = 1, . . . , n − 1, we obtain a set (δi , δ˙j ) of local vector fields which form
a local bases in the fibers of the tangent bundle to IM . By using (2.13) and the
definition of η¯, we obtain
³
´
(3.4)
d¯
η (δi , δ˙j ) = − Jjk Jeik + y m (∂˙k Jeim )Jjk + y r y k Jejr Jeik + y r Jejr y m y k ∂˙k Jeim .
It is easy to see that Jeim is positive homogenous of degree 0. Thus we have y k ∂˙k Jeim =
0. Also, we have Jjk Jeik = −e
gij . By using these relations and (3.4), we obtain
d¯
η (δi , δ˙j ) = geij − Jeik Jejr y r y k − y m (∂˙k Jeim )Jjk .

Structure of the indicatrix bundle of Finsler-Rizza manifolds

9

It is known that ∇i y m = 0, where ∇ means h-covariant derivative with respect to
the Cartan Finsler connection (Γijk , Gij ). From this equation we derive that δi y m =
−y r Γm
ri . Thus we get
d¯
η (δi , δj ) = y m (∇i Jejm − ∇j Jeim ).
Also, we infer d¯
η (δ˙i , δ˙j ) = 0. On the other hand, the relation

¯ f¯(X), f¯(Y )) = G(X,
¯
G(
Y ) − η¯(X)¯
η (Y )
gives us
Ω(δi , δ˙j ) = geij − Jeir Jejs y r y s ,

and Ω(δi , δj ) = Ω(δ˙i , δ˙j ) = 0.

These relations imply that Ω = d¯
η if and only if
∇i Jejm − ∇j Jeim = 0,

Thus we have the following.

and y m (∂˙k Jeim )Jjk = 0.

Theorem 3.4. Let M be a Rizza manifold endowed with the Cartan Finsler connec¯ η¯, G)
¯ is a contact metric structure on IM if and only if y m ∂˙j Jeim = 0
tion. Then (f¯, ξ,
and ∇i Jejm = ∇j Jeim .

It is known that a Rizza manifold satisfying ∇k Jji = 0 is said to be a K¨
ahlerian
Finsler manifold. Further, if gij is a Riemannian metric, then we call it K¨
ahlerian
Riemann manifold. It was proved (Ichijy¯
o, [6]) that if M be a K¨ahlerian Finsler
manifold, then M is Landsberg space, and that the following relations hold true
∇k Jji = 0, ∇k gij = ∇k geij = 0, ∇k Jeij = 0

on K¨ahlerian Finsler manifolds. These relations and Theorem 3.4 lead to
¯ η¯, G)
¯ is a contact
Corollary 3.5. If M is a K¨
ahlerian Finslerian manifold, then (f¯, ξ,
m ˙ e
Riemannian structure on IM if and only if y ∂j Jim = 0.

Now, let gij be a Riemannian metric. Then we have ∂˙k gij = 0, and consequently,
j
j
is a function
∂˙k geij = 0. Therefore we obtain ∂˙k Jeim = 0, since Jeim = geij Jm
and Jm
h
depending on (x ) only. This leads to the following
¯ η¯, G)
¯ is a contact
Corollary 3.6. If M is a K¨
ahlerian Riemann manifold, then (f¯, ξ,
Riemannian structure on IM .

¯ η¯, G)
¯
We further obtain sufficiency conditions for the contact metric structure (f¯, ξ,
given by Theorem 3.4, to be Sasakian.
Theorem 3.7. Let M be a Rizza manifold endowed with the Cartan Finsler connec¯ η¯, G)
¯ on IM is Sasakian if and only if
tion. Then (f¯, ξ,
(3.5)

∇i Jjh + Jjr Lhir = ∇j Jih + Jir Lhjr ,

(3.6)

Rkji = (Jjk Jeil − Jik Jejl )y l ,
h
i
y m Jeim y h Jhr ∇r Jjk + y h (Jlk + Jels y s y k )∇j Jhl + y h y l y k Jhr ∇r Jejl = 0,

(3.7)

10

(3.8)

V. Balan, E. Peyghan and A. Tayebi
h
h n k s e
k
k
y Jn y (Jir ∇h Jejs − Jejr ∇h Jeis )
) + Jm
− Jejr ∇i Jm
y r y m (Jeir ∇j Jm
i
h e
h s k e
+Jm
(Jjr ∇h Jil − Jeir ∇h Jjl ) + Jm
Lhl Js (Jjr Jil − Jeir Jjl ) = 0,

s
where Lshl = y m ∇m Chl
is the Landsberg tensor.

Proof. The Nijenhuis tensor field of f¯ is defined by
(3.9)

Nf¯(δi , δj ) = [δ˙i , δ˙j ] − f¯[δ˙i , δj ] − f¯[δi , δ˙j ] + f¯2 [δi , δj ].

By using (2.13), (2.14) and the relation y h (∂˙h Jeir ) = 0, we obtain
h
i
(3.10)
[δ˙i , δ˙j ] = Jih (∂˙h Jejl )y l y k + Jik Jejl y l − Jjh (∂˙h Jeil )y l y k − Jjk Jeil y l ∂˙k .

Also, (2.13), (2.14) and the relation δj y l = −y r Γlrj give us

(3.11)

h
f¯[δ˙i , δj ] + f¯[δi , δ˙j ] = ∇i Jjh − ∇j Jih + y l y h (∇i Jejl − ∇j Jeil )
i
+Jjr Lhir − Jir Lhjr Jhk δk .

Similarly, we infer
(3.12)

f¯2 [δi , δj ] = −Rkij ∂˙k = Rkji ∂˙k ,

where Rkij = δj Gki − δi Gkj . Setting (3.10), (3.11) and (3.12) into (3.9), we get
i
h
Nf¯(δi , δj ) = ∇j Jih − ∇i Jjh + y l y h (∇j Jeil − ∇i Jejl ) + Jir Lhjr − Jjr Lhir Jhk δk
i
h
(3.13)
+ Rkji + Jih (∂˙h Jejl )y l y k + Jik Jejl y l − Jjh (∂˙h Jeil )y l y k − Jjk Jeil y l ∂˙k .
By consideration of d¯
η (δi , δj ) = y m (∇i Jejm − ∇j Jeim ), we obtain
¯ i , δj ) = y l (∇i Jejl − ∇j Jeil )y h J k δk .
(d¯
η ⊗ ξ)(δ
h

Therefore we get
h
i
h
N (δi , δj ) = (∇j Jih − ∇i Jjh + Jir Lhjr − Jjr Lhir )Jhk δk + Rkji + Jih (∂˙h Jejl )y l y k
i
+Jik Jejl y l − Jjh (∂˙h Jeil )y l y k − Jjk Jeil y l ∂˙k .
(3.14)
Similarly, we obtain

h
N (δ˙i , δj ) = −f¯Nf¯(δi , δj ) + y m y r Jrk (Jih ∂˙h Jejm − Jjh ∂˙h Jeim ) + y m Jeim (δjk
i
h
−Jejs y s Jhk y h − y r Jrs Rhsj Jhk ) δk + y m Jeim y h (Jhr ∇r Jjk
i
(3.15)
+Jlk ∇j Jhl + Jels y s y k ∇j Jhl + y l y k Jhr ∇r Jejl ) ∂˙k ,

Structure of the indicatrix bundle of Finsler-Rizza manifolds

11

and
h
k
k
− Jejr ∇i Jm
)
N (δ˙i , δ˙j ) = −Nf¯(δi , δj ) + y r y m J k m(∇j Jeir − ∇i Jejr ) + y r y m (Jeir ∇j Jm

(3.16)

h e
h n k s e
(Jjr ∇h Jil − Jeir ∇h Jjl )
+y r y m Jm
y Jn y (Jir ∇h Jejs − Jejr ∇h Jeis ) + y r y m Jm
i
h
h s k e
s e
+y r y m Jm
Lhl Js (Jjr Jil − Jeir Jjl ) δk + y r y m Jm
(Jir Rkjs − Jejr Rkis )
i
l n h k
+y r Jeir y s Jejs y m Jm
y Jn R lh − y r Jeir Jjk + y r Jejr Jik ∂˙k .

¯ η¯, G)
¯ is a contact structure, then from Theorem 3.4 we have y m ∂˙j Jeim = 0
Since (f¯, ξ,
and ∇i Jejm = ∇j Jeim . Setting these equations into (3.14), (3.15) and (3.16), we imply
¯ η¯, G)
¯ is Sasakian if and only if
that (f¯, ξ,
(3.17) Jhk (∇i Jjh + Jjr Lhir ) = Jhk (∇j Jih + Jir Lhjr ),
(3.18) Rkji = (Jjk Jeil − Jik Jejl )y l ,

(3.19) y m Jeim δjk − Jeim y m Jejs y s Jhk y h − y m Jeim y r Jrs Rhsj Jhk = 0,

(3.20) y m [Jeim y h Jhr ∇r Jjk + Jeim y h (Jlk + Jels y s y k )∇j Jhl + Jeim y h y l y k Jhr ∇r Jejl ] = 0,
h
k
k
h n k s e
y r y m (Jeir ∇j Jm
y Jn y (Jir ∇h Jejs − Jejr ∇h Jeis )
− Jejr ∇i Jm
) + Jm
i
h e
h s k e
+Jm
(Jjr ∇h Jil − Jeir ∇h Jjl ) + Jm
(3.21)
Lhl Js (Jjr Jil − Jeir Jjl ) = 0,
(3.22)

l n h k
s e
y Jn R lh
y r y m Jm
(Jir Rkjs − Jejr Rkis ) + y r Jeir y s Jejs y m Jm

−y r Jeir Jjk + y r Jejr Jik = 0.

It is easy to check that if (3.18) holds, then (3.19) and (3.22) hold true as well. This
completes the proof.
¤
Lkij

Now, if M is a K¨ahlerian Finsler manifold, then we have ∇i Jjk = ∇i Jejk = 0 and
= 0. Thus, the relations (3.5), (3.7) and (3.8) hold true, and we have the following

Theorem 3.8. Let (M, F ) be a K¨
ahlerian Finsler manifold with the Cartan Finsler
¯ η¯, G)
¯ on IM is Sasakian if and only if the following relation
connection. Then (f¯, ξ,
holds:
Rkji = (Jjk Jeil − Jik Jejl )y l .

References
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V. Balan, E. Peyghan and A. Tayebi

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Authors’ addresses:
Vladimir Balan,
Univ. Politehnica of Bucharest, Faculty of Applied Sciences, Bucharest, Romania.
E-mail: vbalan@mathem.pub.ro, vladimir.balan@upb.ro
Esmaeil Peyghan,
Arak University, Faculty of Science, Department of Mathematics, Arak, Iran.
E-mail: epeyghan@gmail.com
Akbar Tayebi,
Qom University, Faculty of Science, Department of Mathematics, Qom, Iran.
E-mail: akbar.tayebi@gmail.com