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 struc-ture reduces to a contact or to a Sasakian strucstruc-ture. 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 calleda 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 Ji
j(x). Let gij(x, y) = 12∂˙i∂˙jL
2(x, y) be the associated Finsler
metric tensor field [11, 12, 14]. We say that the fundamental functionL(x, y) satisfies
the Rizza condition, if
L(x, φθy) =L(x, y), ∀θ∈R, where φiθj=δijcosθ+Jjisinθ.
∗
Balkan Journal of Geometry and Its Applications, Vol.16, No.2, 2011, pp. 1-12. c
In this case, M is called an almost Hermitian Finsler manifold or simply, a Rizza manifold. The Rizza condition practically requires that, for ally∈TxM, the Finsler normLshould be constant on the orbity→φθ(y), θ∈R, which lives inside the fiber
TxM. We note that each such orbit is determined by the action on y ∈ TxM of a special elementφof the loop group ΛAut(TxM) ofid−automorphisms of the tangent
bundle, defined byθ∈S1−→φ
φθ=I·cosθ+J·sinθ∈Aut(TxM).
In [6], it was shown that the Rizza condition holds if the following equivalent conditions hold true:1
• gpq(x, φθy)φpθiφ q
θj=gij(x, y), ∀θ∈R;
• gij(x, y)Ji
m(x)ymyj = 0; (gim(x, y)−gpq(x, y)J p
i(x)Jmq(x))ym= 0;
• gim(x, y)Jjm(x) +gjm(x, y)Jim(x) + 2Cijm(x, y)Jrm(x)yr= 0, whereCijm 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 letM be a Rizza manifold. Then
(1.1) geij(x, y) =
1 2 ¡
gij(x, y) +gpq(x, y)Jip(x)J q j(x)
¢
,
is a homogeneous symmetric generalized metric onT M, which is called ageneralized Finsler metric; this satisfies the relation egpq(x, y)Jip(x)Jjq(x) =egij(x, y). Also, in a
Rizza manifold, the relationgij(x, y) = ˙∂i∂˙j
³
1
2egpq(x, y)y
pyq´holds true.
Conversely, forM a manifold admitting an almost complex structureJi
j(x), Ichijy¯o
showed ([6]) that M admits a Finsler metric which constructs a Rizza structure to-gether with Ji
j(x), if and only if M admits a generalized Finsler metric egij(x, y)
satisfying the following conditions:
• egjk(x, y) =egpq(x, y)Jjp(x)J q k(x);
• ∂k˙ egpq(x, y)ypyq = 0;
• (egjk(x, y) + ˙∂kegjm(x, y)ym)ζjζk 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), Jeij(x, y) =egim(x, y)Jjm(x).
Then we obtain the following relations ([6])
(1.3) Jije =−Jji,e JimJe jm=−egij, Jije = 1
2(Jij−Jji).
By using [6, Theorem, eq. (3)] and denotingyi:=gijyj, we infer
(1.4) egijyj = 1
2(gijy
j+gpqJp iJ
q
jyj) =yi,
1We shall assume that all the Latin indices run within the range 1, n, and that the Einstein
It follows that
(1.5) egijyiyj =L2.
Also, the relations (1.2), (1.3) and (1.4) lead to
(1.6) Jijye j =−Jjiye j=−gjmJe imyj =−Jimym=−Jijyj.
By a contact manifold we mean a differentiable manifold M2n+1 endowed with with a 1-formη, such thatη∧(dη)n 6= 0. It is well known ([2]) that given the form η, there exist a unique vector fieldξ, such thatdη(ξ, X) = 0 and η(ξ) = 1; this field is called thecharacteristic vector fieldor theReeb vector fieldof the contact formη.
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
f2
=−I+η⊗ξ anddη(X, Y) =g(f X, Y). Then we refer to (f, ξ, η, g) as a contact metric structureand to M2n+1
with such a structure as acontact metric manifold. Analmost contact structure, (f, ξ, η), consists of a field of endomorphismsf, and a 1-formηsuch thatf2
=−I+η⊗ξandη(ξ) = 1 and analmost 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 M2n+1×
R carries a natural almost complex structure defined by
J(X, gdtd) = (f X −gξ, η(X)d
dt) and the underlying almost contact structure is said
to benormalifJ is integrable. The normality condition can be expressed asN = 0, where N is defined by N = Nf +dη⊗ξ and Nf is the Nijenhuis tensor of f. A
Sasakianmanifold 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: letN
be a (2n+s)-dimensional manifold endowed with an endomorphismf of rank 2nof the tangent bundle, satisfyingf3+f = 0. If there exists on N the vector fields (ξ
b)
and the 1-formsηa (a, b= 1,2, . . . , s) such that
ηa(ξb) =δab, f(ξa) = 0, ηa◦f = 0, f2=−I+ s
X
a=1
ηa⊗ξa,
whereI is the identity automorphism of the tangent bundle, thenN is said to bea framedf-manifold[1].
In this paper, by using an almost complex structure Ji
j(x) and the generalized
Finsler metricgije (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 T Mg = 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
T M
g
∂i = ∂
∂xi and Gmi are the components of an Ehresmann connection on M. Then we can globally define onT M an (1,1)-tensor fieldF, such that
(2.1) F(δi) =Jik∂k,˙ F( ˙∂i) =Jikδk.
SinceJi
kJjk =−δji, it is obvious thatF defines an almost complex structure onT M.
Moreover, we can introduce an inner producth ·, · isuch that
(2.2) hδi, δji=egij, hδi,∂j˙ i= 0, h∂i,˙ ∂j˙ i=egij.
Then the inner product gives onT M a globally defined Riemann metricG, as follows
(2.3) G=egijdxidxj+egijδyiδyj,
where (dxi, δyi) is the dual basis of (δ
i,∂˙i) andδyi=dyi+Gimdxm. Using (2.1) and
(2.3), we obtain
G(F(δi), F(δj)) =JikJjlG( ˙∂k,∂l˙) =JikJjlegkl=egij =G(δi, δj).
Similarly, we obtainG(F( ˙∂i), F( ˙∂j)) = egij =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 al-most Hermitian structure (G, F), where F and G are defined by (2.1) and (2.3), respectively.
Now we define the vector fieldsξ1, ξ2 and 1-formsη1, η2 onT Mg respectively by
(2.4) ξ1:=ymJmi δi, ξ2:=yi∂˙i
and
(2.5) η1
:= 1
L2y
mJe
imdxi, η2:=
1
L2yiδy
i.
Lemma 2.2. LetF be defined by (2.1),ξ1, ξ2 be defined by (2.4) andη1, η2be defined
by (2.5). Then we have
(2.6) F(ξ1) =−ξ2, F(ξ2) =ξ1,
and
(2.7) η1◦F =η2, η2◦F =−η1.
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
(η1
◦F)( ˙∂i) = Jikη
1 (δk) =
1
L2y
mJk iJekm=
1
L2egmiy
m= 1 L2yi=η
2 ( ˙∂i),
(η2
◦F)(δi) = Jikη
2 ( ˙∂k) =
1
L2J
k iyk =
1
L2J
k
iegkryr=−
1
L2Jeiry
r=−η1 (δi),
and (η1◦F)(δ
Lemma 2.3. LetGbe defined by (2.3),ξ1, ξ2be defined by (2.4) andη1, η2 be defined
by (2.5). Then we have
(2.8) η1
(X) = 1
L2G(X, ξ1), η 2
(X) = 1
L2G(X, ξ2),
and
(2.9) ηa(ξb) =δba, a, b= 1,2,
whereX∈ X(T Mg).
Proof. In the adapted basis (δi,∂i˙), we obtain
G(δi, ξ1) =G(δi, ykJkjδj) =ykJkjegij =ykJike =L 2
η1(δi).
Similarly, we inferG( ˙∂i, ξ2) =L2η2( ˙∂i),G( ˙∂i, ξ1) = 0 =L2η1( ˙∂i) andG(δi, ξ2) = 0 =
L2
η2
(δi), which completes the proof of (2.8). As well, using (1.3) and (1.5) we get
η1(ξ1) =η1(ykJkiδi) = 1
L2y
kJi
kymJime =
1
L2y
kymegmk = 1,
andη1(
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 structureF, we define a new tensor fieldf of type (1,1) onT Mg by
(2.10) f =F+η1⊗ξ2−η2⊗ξ1.
By using the above relation, (2.6) and (2.9) we infer thatf(ξ1) =f(ξ2) = 0. Similarly, from (2.7) and (2.9)we conclude that (η1◦
f)(X) = (η2◦
f)(X) = 0, where X ∈ X(T Mg). These relations and (2.6), (2.10) give us
f2(X) =F(f(X)) =F2(X) +η1(X)F(ξ2)−η2(X)F(ξ1) =−X+η1(X)ξ1+η2(X)ξ1.
Therefore we have
Lemma 2.4. The following properties hold true for tensor fieldf defined by (2.10):
f(ξa) = 0, ηa◦f = 0, a= 1,2,
(2.11)
f2
(X) = −X+η1
(X)ξ1+η2(X)ξ2, X∈ X(T Mg). (2.12)
Theorem 2.5. Let f,(ξa),(ηa), a= 1,2 be defined respectively by (2.10), (2.4) and (2.5). Then the triple(f,{ξa},{ηa}) provides a framedf-structure on T M.g
Proof. Considering Lemma 2.4 and relation (2.9), in order to complete the proof, we need to provef3
+f = 0 and to show thatf is of rank 2n−2. Sincef(ξ1) =f(ξ2) = 0, by using (2.11) we derive thatf3(X) =−f(X), for all X ∈ X(T Mg). Now we need to show that kerf = Span{ξ1, ξ2}. It is clear that Span{ξ1, ξ2} ⊆ kerf, because
f(ξ1) =f(ξ2) = 0. Now letX ∈kerf; thenf(X) = 0 implies thatF(X) +η1(X)ξ2−
η2
(X)ξ1= 0. Thus we infer thatF2(X) =η2(X)F(ξ1)−η1(X)F(ξ2).SinceF2=−I, it follows from Lemma 2.2 thatX=η1
Theorem 2.6. The Riemannian metricGdefined by (2.3) satisfies
Proof. By using (2.10), we get the local expression off as follows:
f(δi) =
By using (2.3), (2.13) we obtain
G(f(δi), f(δj)) =JikJjhegkh+ y
By using (1.5) and (1.6), the above equation leads to
(2.15) G(f(δi), f(δj)) =gije − 1
which allow to rewrite (2.15) as follows
G(f(δi), f(δj)) =G(δi, δj)−L2η1(δi)η1(δj)−L2η2(δi)η2(δj).
Proof. By using (1.4), (2.13) and (2.14), we obtain
SinceJije =−Jjie , then from (2.16) and (2.17), we derive that Ω(δi,∂j˙) =−Ω( ˙∂j, δi).
∂i, then these equations give us the relations
Xi(Jeji+
s. Transvecting (2.19) and (2.20) withJrjegrh and using these equations we derive
thatXi = 1
ThereforeSpan{ξ1, ξ2}contains the annihilator of Ω, and consequently the annihilator of Ω is Span{ξ1, ξ2}. Also we obtain [ξ1, ξ2] =yjJjiδi =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)∈T Mg|L(x, y) = 1},
which is a submanifold of dimension 2n−1 of T Mg. Note thatξ2 defined in (2.4) is a unit vector field onIM, sinceG(ξ2, ξ2) = 1. It is easy to show thatξ2 is a normal vector field onIM with respect to the metricG. Indeed, if the local equations ofIM
inT Mg are given by
SinceF is a horizontally covariant constant, i.e., ∂L
The natural frame field{ ∂
Thus by using (3.1) and the equality yk
L =
Thusξ2is orthogonal to any tangent toIM vector. Also, the vector fieldξ1is tangent toIM, sinceG(ξ1, ξ2) = 0.
Lemma 3.1. The hypersurfaceIM is invariant with respect tof, i.e.,f(Tu(IM))⊆ Tu(IM),∀u∈IM.
Proof. By using (2.8) and the second equation of (2.11), we get
G
Thus the hypersurfaceIM is invariant with respect to f. ¤
Lemma 3.2. Let the framed f-structure be given by Theorem 2.5. Then restricting this toIM, we have
defines an almost contact metric structure onIM.
If we put ˙δj = ¯f(δj), then we get nlocal vector fields which are tangent toIM, sinceIM is an invariant hypersurface. These, together with δi, i= 1, . . . , n, are all tangent toIMand they are not linearly independent. But, consideringδi, i= 1, . . . , n
It is known that ∇iym = 0, where ∇ means h-covariant derivative with respect to
the Cartan Finsler connection (Γi
jk, Gij). From this equation we derive that δiym =
−yrΓm
ri. Thus we get
dη¯(δi, δj) =ym(∇iJjme − ∇jJime ).
Also, we inferdη¯( ˙δi,δ˙j) = 0. On the other hand, the relation
¯
G( ¯f(X),f¯(Y)) = ¯G(X, Y)−η¯(X)¯η(Y)
gives us
Ω(δi,δ˙j) =geij−JeirJejsyrys, and Ω(δi, δj) = Ω( ˙δi,δ˙j) = 0.
These relations imply that Ω =dη¯if and only if
∇iJjme − ∇jJime = 0, and ym( ˙∂kJime )Jjk = 0.
Thus we have the following.
Theorem 3.4. Let M be a Rizza manifold endowed with the Cartan Finsler connec-tion. Then( ¯f ,ξ,¯η,¯ G¯)is a contact metric structure onIM if and only ifym∂j˙Jime = 0 and∇iJejm=∇jJeim.
It is known that a Rizza manifold satisfying∇kJji = 0 is said to be aK¨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, thenM is Landsberg space, and that the following relations hold true
∇kJji= 0, ∇kgij=∇kegij = 0, ∇kJije = 0
on K¨ahlerian Finsler manifolds. These relations and Theorem 3.4 lead to
Corollary 3.5. IfM is a K¨ahlerian Finslerian manifold, then( ¯f ,ξ,¯η,¯ G¯)is a contact Riemannian structure onIM if and only ifym∂˙
jJeim= 0.
Now, letgij be a Riemannian metric. Then we have ˙∂kgij = 0, and consequently, ˙
∂kegij = 0. Therefore we obtain ˙∂kJeim = 0, sinceJeim =egijJmj and Jmj is a function
depending on (xh) only. This leads to the following
Corollary 3.6. If M is a K¨ahlerian Riemann manifold, then( ¯f ,ξ,¯ η,¯ G¯)is a contact Riemannian structure onIM.
We further obtain sufficiency conditions for the contact metric structure ( ¯f ,ξ,¯η,¯ G¯) given by Theorem 3.4, to be Sasakian.
Theorem 3.7. Let M be a Rizza manifold endowed with the Cartan Finsler connec-tion. Then( ¯f ,ξ,¯ η,¯ G¯)on IM is Sasakian if and only if
∇iJjh+JjrLhir=∇jJih+JirLhjr,
(3.5)
Rkji= (JjkJeil−JikJejl)yl,
(3.6)
ymJime hyhJhr∇rJjk+yh(Jlk+Jlse ysyk)∇jJhl +yhylykJhr∇rJjle
i = 0,
yrymh(Jire ∇jJmk −Jjre ∇iJmk) +JmhynJnkys(Jire ∇hJjse −Jjre ∇hJise )
+Jmh(Jjre ∇hJil−Jire ∇hJjl) +JmhLshlJsk(JjrJe il−JirJe jl)i= 0,
(3.8)
whereLs
hl=ym∇mChls is the Landsberg tensor.
Proof. The Nijenhuis tensor field of ¯f is defined by
(3.9) Nf¯(δi, δj) = [ ˙δi,δ˙j]−f¯[ ˙δi, δj]−f¯[δi,δ˙j] + ¯f2[δi, δj].
By using (2.13), (2.14) and the relationyh( ˙∂hJire ) = 0, we obtain
(3.10) [ ˙δi,δ˙j] =
h
Jih( ˙∂hJejl)ylyk+JikJejlyl−Jjh( ˙∂hJeil)ylyk−JjkJeilyl
i ˙
∂k.
Also, (2.13), (2.14) and the relationδjyl=−yrΓlrj give us
¯
f[ ˙δi, δj] + ¯f[δi,δj˙] =h∇iJjh− ∇jJih+ylyh(∇iJjle − ∇jJile )
+JjrLhir−JirLhjriJhkδk.
(3.11)
Similarly, we infer
(3.12) f¯2[δi, δj] =−Rkij∂k˙ =Rkji∂k,˙
whereRkij =δjGki −δiGkj. Setting (3.10), (3.11) and (3.12) into (3.9), we get
Nf¯(δi, δj) =
h
∇jJih− ∇iJjh+ylyh(∇jJeil− ∇iJejl) +JirLhjr−JjrLhir
i
Jhkδk
+hRkji+Jih( ˙∂hJejl)ylyk+JikJejlyl−Jjh( ˙∂hJeil)ylyk−JjkJeilyl
i ˙
∂k.
(3.13)
By consideration ofdη¯(δi, δj) =ym(∇
iJjme − ∇jJime ), we obtain
(dη¯⊗ξ¯)(δi, δj) =yl(∇iJejl− ∇jJeil)yhJhkδk.
Therefore we get
N(δi, δj) =h(∇jJih− ∇iJjh+JirLhjr−JjrLhir)Jhk
i
δk+hRkji+Jih( ˙∂hJjle )ylyk
+JikJjlye l−Jjh( ˙∂hJile )ylyk−JjkJilye li∂k˙ .
(3.14)
Similarly, we obtain
N( ˙δi, δj) = −f N¯ f¯(δi, δj) +
h
ymyrJrk(Jih∂˙hJejm−Jjh∂˙hJeim) +ymJeim(δjk
−JejsysJhkyh−yrJrsRhsjJhk)
i
δk+
h
ymJeimyh(Jhr∇rJjk
+Jlk∇jJhl +Jlsye syk∇jJhl +ylykJhr∇rJjle )
i ˙
∂k,
and
N( ˙δi,δ˙j) =−Nf¯(δi, δj) +
h
yrymJkm(∇jJeir− ∇iJejr) +yrym(Jeir∇jJmk −Jejr∇iJmk)
+yrymJmhynJnkys(Jeir∇hJejs−Jejr∇hJeis) +yrymJmh(Jejr∇hJil−Jeir∇hJjl)
+yrymJmhLshlJsk(JejrJil−JeirJjl)
i
δk+
h
yrymJms(JeirRkjs−JejrRkis)
+yrJeirysJejsymJml ynJnhRklh−yrJeirJjk+yrJejrJik
i ˙
∂k.
(3.16)
Since ( ¯f ,ξ,¯η,¯ G¯) is a contact structure, then from Theorem 3.4 we haveym∂˙
jJeim= 0
and∇iJjme =∇jJime . Setting these equations into (3.14), (3.15) and (3.16), we imply
that ( ¯f ,ξ,¯η,¯ G¯) is Sasakian if and only if
Jhk(∇iJjh+JjrLhir) =Jhk(∇jJih+JirLhjr),
(3.17)
Rkji= (JjkJeil−JikJejl)yl,
(3.18)
ymJimδe jk−Jimye mJjsye sJhkyh−ymJimye rJrsRhsjJhk= 0,
(3.19)
ym[Jimye hJhr∇rJjk+Jimye h(Jlk+Jlsye syk)∇jJhl +Jimye hylykJhr∇rJjle ] = 0,
(3.20)
yrymh(Jire ∇jJmk −Jjre ∇iJmk) +JmhynJnkys(Jire ∇hJjse −Jjre ∇hJise )
+Jmh(Jjre ∇hJil−Jire ∇hJjl) +JmhLshlJsk(JjrJe il−JirJe jl)i= 0,
(3.21)
yrymJms(JirRe kjs−JjrRe kis) +yrJirye sJjsye mJml ynJnhRklh
−yrJeirJjk+yrJejrJik= 0.
(3.22)
It is easy to check that if (3.18) holds, then (3.19) and (3.22) hold true as well. This
completes the proof. ¤
Now, ifM is a K¨ahlerian Finsler manifold, then we have∇iJjk =∇iJejk= 0 and Lk
ij= 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 connection. Then( ¯f ,ξ,¯η,¯ G¯) onIM is Sasakian if and only if the following relation holds:
Rkji= (JjkJile −JikJjle )yl.
References
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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,
