# On Twisted Products Finsler Manifolds

Authors: E. Peyghan; A. Tayebi; L. Nourmohammadi Far
Journal: ISRN Geometry
 (2013)

---

## Abstract

On the product of two Finsler manifoldsM1×M2, we consider the twisted metric F which is constructed by using Finsler metrics F1 and F2 on the manifolds M1 and M2, respectively. We introduce horizontal and vertical distributions on twisted product Finsler manifold and study C-reducible and semi-C-reducible properties of this manifold. Then we obtain the Riemannian curvature and some of non-Riemannian curvatures of the twisted product Finsler manifold such as Berwald curvature, mean Berwald curvature, and we find the relations between these objects and their corresponding objects on M1 and M2. Finally, we study locally dually flat twisted product Finsler manifold.


---

## Body

## 1. Introduction

Twisted and warped product structures are widely used in geometry to construct new examples of semi-Riemannian manifolds with interesting curvature properties (see [1–3]). Twisted product metric tensors, as a generalization of warped product metric tensors, have also been useful in the study of several aspects of submanifold theory, namely, in hypersurfaces of complex space forms [4], in Lagrangian submanifolds [5], in decomposition of curvature netted hypersurfaces [6], and so forth.The notion of twisted product of Riemannian manifolds was mentioned first by Chen in [7] and was generalized for the pseudo-Riemannian case by Ponge and Reckziegel [8]. Chen extended the study of twisted product for CR-submanifolds in Kähler manifolds [9].On the other hand, Finsler geometry is a natural extension of Riemannian geometry without the quadratic restriction. Therefore, it is natural to extend the construction of twisted product manifolds for Finsler geometry. In [10], Kozma-Peter-Shimada extended the construction of twisted product for the Finsler geometry.Let(M1,F1) and (M2,F2) be two Finsler manifolds with Finsler metrics F1 and F2, respectively, and let f:M1×M2→R+ be a smooth function. On the product manifold M1×M2, we consider the metric
(1)F(v1,v2)=F12(v1)+f2(x,y)F22(v2)
for all (x,y)∈M1×M2 and (v1,v2)∈TM1∘×TM2∘, where TM1∘ is the slit tangent manifold TM1∘=TM1∖{∘}. The manifold M1×M2 endowed with this metric, we call the twisted product of the manifolds M1 and M2 and denote it by M1×fM2. The function f will be called the twisted function. In particular, if f is constant on M2, then M1×fM2 is called warped product manifold.Let(M,F) be a Finsler manifold. The second and third order derivatives of (1/2)Fx2 at y∈TxM0 are the symmetric trilinear forms gy and Cy on TxM, which called the fundamental tensor and Cartan torsion, respectively. A Finsler metric is called semi-C-reducible if its Cartan tensor is given by
(2)Cijk=p1+n{hijIk+hjkIi+hkiJj}+qC2IiIjIk,
where p=p(x,y) and q=q(x,y) are scalar function on TM, hij is the angular metric, and C2=IiIi [11]. If q=0, then F is called C-reducible Finsler metric, and if p=0, then F is called C2-like metric.The geodesic curves of a Finsler metricF on a smooth manifold M are determined by the system of second-order differential equations c¨i+2Gi(c˙)=0, where the local functions Gi=Gi(x,y) are called the spray coefficients. F is called a Berwald metric, if Gi are quadratic in y∈TxM for any x∈M. Taking a trace of Berwald curvature yields mean Berwald curvature E. Then F is said to be isotropic mean Berwald metric if E=((n+1)/2)cF-1h, where h=hijdxi⊗dxj is the angular metric and c=c(x) is a scalar function on M [12].The second variation of geodesics gives rise to a family of linear mapsRy=Rikdxk⊗(∂/∂xi)|x:TxM→TxM at any point y∈TxM. Ry is called the Riemann curvature in the direction y. A Finsler metric F is said to be of scalar flag curvature, if for some scalar function K on TM0 the Riemann curvature is in the form Rki=KF2hji. If K=constant, then F is said to be of constant flag curvature.In this paper, we introduce the horizontal and vertical distributions on tangent bundle of a doubly warped product Finsler manifold and construct the Finsler connection on this manifold. Then, we study some geometric properties of this product manifold such as C-reducible and semi-C-reducible. Then, we introduce the Riemmanian curvature of twisted product Finsler manifold(M1×fM2,F) and find the relation between it and Riemmanian curvatures of its components (M1,F1) and (M2,F2). In the cases that (M1×fM2,F) is flat or it has the scalar flag curvature, we obtain some results on its components. Then, we study twisted product Finsler metrics with vanishing Berwald curvature and isotropic mean Berwald curvature, respectively. Finally, we study locally dually flat twisted product Finsler manifold. We prove that there is not exist any locally dually flat proper twisted product Finsler manifold.


## 2. Preliminary

LetM be an n-dimensional C∞ manifold. Denote by TxM the tangent space at x∈M, by TM=∪x∈MTxM the tangent bundle of M, and by TM∘=TM∖{0} the slit tangent bundle on M [13]. A Finsler metric on M is a function F:TM→[0,∞) which has the following properties:(i)
Fis C∞ on TM∘;(ii)
F is positively 1-homogeneous on the fibers of tangent bundle TM;(iii)
for eachy∈TxM, the following quadratic form gy on TxM is positive definite:
(3)gy(u,v)≔12∂2∂s∂t[F2(y+su+tv)]∣s,t=0,u,v∈TxM.Let x∈M and Fx:=F∣TxM. To measure the non-Euclidean feature of Fx, define Cy:TxM⊗TxM⊗TxM→ℝ by
(4)Cy(u,v,w):=12ddt[gy+tw(u,v)]∣t=0,u,v,w∈TxM.
The family C:={Cy}y∈TM∘ is called the Cartan torsion. It is well known that C=0 if and only if F is Riemannian [14].Fory∈TxM∘, define mean Cartan torsion Iy by Iy(u):=Ii(y)ui, where Ii:=gjkCijk, Cijk=(1/2)(∂gij/∂yk) and u=ui(∂/∂xi)∣x. By Deicke's theorem, F is Riemannian if and only if Iy=0.Let(M,F) be a Finsler manifold. For y∈TxM∘, define the Matsumoto torsion My:TxM⊗TxM⊗TxM→ℝ by My(u,v,w):=Mijk(y)uivjwk, where
(5)Mijk:=Cijk-1n+1{Iihjk+Ijhik+Ikhij},
where hij:=FFyiyj is the angular metric. In [15], it is proved that a Finsler metric F on a manifold M of dimension n≥3 is a Randers metric if and only if  My=0, for all y∈TM0. A Randers metric F=α+β on a manifold M is just a Riemannian metric α=aijyiyj perturbed by a one form β=bi(x)yi on M such that ∥β∥α<1.A Finsler metric is called semi-C-reducible if its Cartan tensor is given by(6)Cijk=p1+n{hijIk+hjkIi+hkiIj}+qC2IiIjIk,
where p=p(x,y) and q=q(x,y) are scalar function on TM and C2=IiIi with p+q=1. In [11], Matsumoto-Shibata proved that every (α,β) metric on a manifold M of dimension n≥3 is semi-C-reducible.Given a Finsler manifold(M,F), then a global vector field G is induced by F on TM∘, which in a standard coordinate (xi,yi) for TM∘ is given by G=yi(∂/∂xi)-2Gi(x,y)(∂/∂yi), where
(7)Gi:=14gil{∂2F2∂xk∂ylyk-∂F2∂xl},y∈TxM.G is called the spray associated to (M,F). In local coordinates, a curve c(t) is a geodesic if and only if its coordinates (ci(t)) satisfy c¨i+2Gi(c˙)=0 [16].A Finsler metricF=F(x,y) on a manifold M is said to be locally dually flat if at any point there is a coordinate system (xi) in which the spray coefficients are in the following form:
(8)Gi=-12gijHyj,
where H=H(x,y) is a C∞ scalar function on TM∘ satisfying H(x,λy)=λ3H(x,y) for all λ>0. Such a coordinate system is called an adapted coordinate system. In [17], Shen proved that the Finsler metric F on an open subset U⊂ℝn is dually flat if and only if it satisfies (F2)xkylyk=2(F2)xl.For a tangent vectory∈TxM∘, define By:TxM⊗TxM⊗TxM→TxM and Ey:TxM⊗TxM→ℝ by By(u,v,w):=Bijkl(y)ujvkwl(∂/∂xi)∣x and Ey(u,v):=Ejk(y)ujvk, where
(9)Bijkl:=∂3Gi∂yj∂yk∂yl,Ejk:=12Bmjkm.B and E are called the Berwald curvature and mean Berwald curvature, respectively. Then F is called a Berwald metric and weakly Berwald metric if B=0 and E=0, respectively [14]. It is proved that on a Berwald space, the parallel translation along any geodesic preserves the Minkowski functionals [18].A Finsler metricF is said to be isotropic Berwald metric and isotropic mean Berwald metric if its Berwald curvature and mean Berwald curvature are in the following form, respectively:
(10)Bijkl=c{Fyjykδil+Fykylδij+Fylyjδik+Fyjykylyi},Eij=12(n+1)cF-1hij,
where c=c(x) is a scalar function on M [19].The Riemann curvatureRy=Rikdxk⊗(∂/∂xi)∣x:TxM→TxM is a family of linear maps on tangent spaces defined by
(11)Rik=2∂Gi∂xk-yj∂2Gi∂xj∂yk+2Gj∂2Gi∂yj∂yk-∂Gi∂yj∂Gj∂yk.
The flag curvature in Finsler geometry is a natural extension of the sectional curvature in Riemannian geometry was first introduced by L. Berwald [20]. For a flag P=span{y,u}⊂TxM with flagpole y, the flag curvature K=K(P,y) is defined by
(12)K(P,y):=gy(u,Ry(u))gy(y,y)gy(u,u)-gy(y,u)2.
We say that a Finsler metric F is of scalar curvature if for any y∈TxM, the flag curvature K=K(x,y) is a scalar function on the slit tangent bundle TM∘. If K=constant, then F is said to be of constant flag curvature.


## 3. Nonlinear Connection

Let(M1,F1) and (M2,F2) be two Finsler manifolds. Then the functions
(13)(i)gij(x,y)=12∂2F12(x,y)∂yi∂yj,(ii)gαβ(u,v)=12∂2F22(u,v)∂vα∂vβ
define a Finsler tensor field of type (0,2) on TM1∘ and TM2∘, respectively. Now let (M1×fM2,F) be a doubly warped Finsler manifold, x=(x,u)∈M, y=(y,v)∈TxM, M=M1×M2, and TxM=TxM1⊕TuM2. Then by using (13), we conclude that
(14)(gab(x,u,y,v))=(12∂2F2(x,u,y,v)∂yayb)=[gij00f2gαβ],
where ya=(yi,vα), gij=gij, gαβ=f2gαβ, giβ=gαj=0, i,j,…∈{1,…,n1}, α,β,…∈{1,…,n2}, and a,b,…∈{1,…,n1+n2}.Now we consider spray coefficients ofF1, F2, and F as
(15)Gi(x,y)=14gih(∂2F12∂yh∂xjyj-∂F12∂xh)(x,y),(16)Gα(u,v)=14gαγ(∂2F22∂vγ∂uβvβ-∂F22∂uγ)(u,v),(17)Ga(x,y)=14gab(∂2F2∂yb∂xcyc-∂F2∂xb)(x,y).
Taking into account the homogeneity of both F12 and F22 and using (15) and (16), we can conclude that Gi and Gα are positively homogeneous of degree two with respect to (yi) and (vα), respectively. Hence from Euler theorem for homogeneous functions, we infer that
(18)∂Gi∂yjyj=2Gi,∂Gα∂vβvβ=2Gα.
By setting a=i in (17), we have
(19)Gi(x,u,y,v)=14gih(∂2F2∂yh∂xjyj+∂2F2∂yh∂uαvα-∂F2∂xh).
Direct calculations give us
(20)∂F2∂xh=∂F12∂xh+∂f2∂xhF22,∂2F2∂yh∂xj=∂2F12∂yh∂xj,∂2F2∂yh∂uα=0.
Putting these equations together gih=gih in the previous equation and using (15) imply that
(21)Gi(x,u,y,v)=Gi(x,y)-12ffiF22.
Similarly, by setting a=α in (17) and using (16), we obtain
(22)Gα(x,u,y,v)=Gα(u,v)+f-1(fjvαyj+fλvαvλ-12fγgαγF22),
where fi=∂f/∂xi, fγ=∂f/∂uγ, fi=gihfh, and fγ=gλγfλ. Therefore we have Ga=(Gi,Gα), where Ga, Gi, and Gα are given by (17), (21), and (22), respectively.Now, we put(23)(i)Gba:=∂Ga∂yb,(ii)Gji:=∂Gi∂yj,(iii)Gβα:=∂Gα∂vβ.
Then we have the following.Lemma 1.
The coefficientsGba defined by (23) satisfy in the following:
(24)(Gba(x,u,y,v))=[Gji(x,u,y,v)Gjα(x,u,y,v)Gβi(x,u,y,v)Gβα(x,u,y,v)],
where
(25)Gji(x,u,y,v):=∂Gi∂yj=Gji+CjihffhF22,(26)Gβi(x,u,y,v):=∂Gi∂vβ=-ffivβ,(27)Gjα(x,u,y,v):=∂Gα∂yj=f-1fjvα,(28)Gβα(x,u,y,v):=∂Gα∂vβ=Gβα+f-1(CβαγfγF22+fjyjδβαGβα+f-1aa-fαvβ+fβvα+fγvγδβα).Next,VTM∘ kernel of the differential of the projection map
(29)π:=(π1,π2):TM1∘⊕TM2∘⟶M1×M2,
which is a well-defined subbundle of TTM∘, is considered. Locally, Γ(VTM∘) is spanned by the natural vector fields {∂/∂y1,…,∂/∂yn1,∂/∂v1,…,∂/∂vn2}, and it is called the twisted vertical distribution on TM∘. Then, using the functions given by (25)–(28), the nonholonomic vector fields are defined as follows:
(30)δtδtxi:=∂∂xi-Gij∂∂yj-Giβ∂∂vβ,(31)δtδtuα:=∂∂uα-Gαj∂∂yj-Gαβ∂∂vβ,
which make it possible to construct a complementary vector subbundle HTM∘ to VTM∘ in TTM∘ as follows:
(32)HTM∘:=span{δtδtx1,…,δtδtxn1,δtδtu1,…,δtδtun2}.HTM∘ is called the twisted horizontal distribution on TM∘. Thus the tangent bundle of TM∘ admits the decomposition
(33)TTM∘=HTM∘⊕VTM∘.
It is shown that G:=(Gba) is a nonlinear connection on TM=TM1⊕TM2. In the following, we compute the nonlinear connection of a twisted product Finsler manifold.Proposition 2.
If(M1×fM2,F) is a twisted product Finsler manifold, then G=(Gba) is the nonlinear connection on TM. Further, one has
(34)∂Gji∂ykyk+∂Gji∂vγvγ=Gji,∂Gβi∂ykyk+∂Gβi∂vγvγ=Gβi,∂Gjα∂ykyk+∂Gjα∂vγvγ=Gjα,∂Gβα∂ykyk+∂Gβα∂vγvγ=Gβα.Definition 3.
Using decomposition (33), the twisted vertical morphism vt:TTM∘→VTM∘ is defined by
(35)vt:=∂∂yi⊗δtyi+∂∂vα⊗δtvα,
where
(36)δtyi:=dyi+Gjidxj+Gβiduβ,δtvα:=dvα+Gjαdxj+Gβαduβ.For this projective morphism, the following hold:(37)vt(∂∂yi)=∂∂yi,vt(∂∂vα)=∂∂vα,vt(δtδtxi)=0,vt(δtδtui)=0.
From the previous equations, we conclude that
(38)(vt)2=vt,ker(vt)=HTM∘.
This mapping is called the twisted vertical projective.Definition 4.
Using decomposition (33), the doubly warped horizontal projective ht:TTM∘→HTM∘ is defined by
(39)ht=id-vt
or
(40)ht=δtδtxi⊗dxi+δtδtuα⊗duα.For this projective morphism, the following hold:(41)ht(δtδtxi)=δtδtxi,ht(δtδtuα)=δtδtuα,ht(∂∂yi)=0,ht(∂∂vα)=0.
Thus we result that
(42)(ht)2=ht,ker(ht)=VTM∘.Definition 5.
Using decomposition (33), the twisted almost tangent structure Jt:HTM∘→VTM∘ is defined by
(43)Jt:∂∂yi⊗dxi+∂∂vα⊗duα,
or
(44)Jt(δtδtxi)=∂∂yi,Jt(δtδtuα)=∂∂vα,Jt(∂∂yi)=Jt(∂∂vα)=0.Thus we result that(45)(Jt)2=0,kerJt=ImJt=VTM∘.
Here, we introduce some geometrical objects of twisted product Finsler manifold. In order to simplify the equations, we rewritten the basis of HTM∘ and VTM∘ as follows:
(46)δtδtxa=δtδtxiδai+δtδtuαδaα,∂∂ya=∂∂yiδai+∂∂vαδaα.
Thus
(47)TTM∘=span{δtδtxa,∂∂ya}.
The Lie brackets of this basis is given by
(48)[δtδtxa,δtδtxb]=Rabc∂∂yc,[δtδtxa,∂∂yb]=Gabc∂∂yc,[∂∂ya,∂∂yb]=0,
where
(49)(i)Rabc=δtGacδtxb-δtGbcδtxa,(50)(ii)Gabc=∂Gac∂yb.Therefore, we have the following.Corollary 6.
Let(M1×fM2,F) be a twisted product Finsler manifold. Then
(51)Rabc=(Rijk,Riβk,Rαjk,Rαβk,Rijγ,Riβγ,Rαjγ,Rαβγ),
where
(52)Rijk:=δtGikδtxj-δtGjkδtxi,Riβk:=δtGikδtuβ-δtGβkδtxi,Rαjk:=δtGαkδtxj-δtGjkδtuα,Rαβk:=δtGαkδtuβ-δtGβkδtuα,Rijγ:=δtGiγδtxj-δtGjγδtxi,Riβγ:=δtGiγδtuβ-δtGβγδtxi,Rαjγ:=δtGαγδtxj-δtGjγδtuα,Rαβγ:=δtGαγδtuβ-δtGβγδtuα.With a simple calculation, we have the following.Corollary 7.
Let(M1×fM2,F) be a twisted product Finsler manifold. Then
(53)Gabc=(Gijk,Giβk,Gαjk,Gαβk,Gijγ,Giβγ,Gαjγ,Gαβγ),
where
(54)Gαβγ=∂Gαγ∂vβ=Gαβγ+f-1(Cα;βγλfλF22+2Cαγλfλvβ+2Cβγλfλvα=Gαβγ+f-1dd-fγgαβ+fβδαγ+fαδβγ)=Gβαγ,Gijk=∂Gik∂yj=Gijk+Ci;jkhffhF22=Gjik,Giβk=∂Gik∂vβ=2Cikhffhvβ=Gβik,Gαβk=∂Gαk∂vβ=-ffkgαβ=Gβαk,Giβγ=∂Giγ∂vβ=f-1fiδβγ=Gβiγ,Gijγ=∂Giγ∂yj=Gjiγ=0,
where Ci;jkh=∂Cikh/∂yj. Apart from Gabc, the functions Fabc are given by
(55)Fabc=12gce(δtgeaδtxb+δtgebδtxa-δtgabδtxe).Corollary 8.
Let(M1×fM2,F) be a twisted product Finsler manifold. Then
(56)Fabc=(Fijk,Fiβk,Fαjk,Fαβk,Fijγ,Fiβγ,Fαjγ,Fαβγ),
where
(57)Fijk=Fijk-(MjrCirk+MirCjrk-MhrCijrgkh),(58)Fiβk=-GβrCirk=Fβik,(59)Fαβk=-ffkgαβ+f2gkhGhλCαβλ,(60)Fijγ=f-2gγλGλrCijr,(61)Fiβγ=f-1fiδβγ-GiαCαβγ=Fβiγ,(62)Fαβγ=Fαβγ+Nαβγ-(MβμCαμγ+MαμCβμγ-MλμCαβμgγλ),(63)Fijk=12gkh(δghiδxj+δghjδxi-δgijδxh),Fαβγ=12gγλ(δgλαδuβ+δgλβδuα-δgαβδuλ),Mir=CirhffhF22,Mαμ=f-1(CαμγfγF22+fryrδαμ+fγvγδαμ-gμγfγvα+fαvμ),Nαβγ=f-1(fβδαγ+fαδβγ-fλgγλgαβ).Proof.
By using (55), we have
(64)Fijk=12gkh(δtghiδtxj+δtghjδtxi-δtgijδtxh).
Since gij is a function with respect to (x,y), then by (25) and (30) we obtain
(65)δtghiδtxj=δghiδxj-2MjrChir.
Interchanging i, j, and h in the previous equation gives us
(66)δtghjδtxi=δghjδxi-2MirChjr,δtgijδtxh=δgijδxh-2MhrCijr.
Putting these equation in (64) give us (57). In the similar way, we can prove the another relation.By using (i) of (23) and (57)–(62), we can conclude the following.Lemma 9.
Let(M1×fM2,F) be a twisted product Finsler manifold. Then ycFbca=Gba, where Fbca and Gba are defined by (55) and (i) of (23), respectively.The Cartan torsion is one of the most important non-Riemannian quantity in Finsler geometry, and it is first introduced by Finsler and emphasized by Cartan which measures a departure from a Riemannian manifold. More precisely, a Finsler metric reduces to a Riemannian metric if and only if it has vanishing Cartan torsion. The local components of Cartan tensor field of the twisted Finsler manifold(M1×fM2,F) is defined by
(67)Cbca=12gae∂gbe∂yc.
From this definition, we conclude the following.Lemma 10.
LetCijk and Cαβγ be the local components of Cartan tensor field on M1 and M2, respectively. Then one has
(68)Cabc=(Cijk,Ciβk,Cαjk,Cαβk,Cijγ,Ciβγ,Cαjγ,Cαβγ),
where
(69)Cijk=12gkh∂gij∂yh=Cijk,Cαβγ=12gγλ∂gαβ∂vλ=Cαβγ,
and Ciβk=Cαjk=Cαβk=Cijγ=Ciβγ=Cαjγ=0.By using the Lemma10, we can get the following.Corollary 11.
Let(M1×fM2,F) be a twisted product Finsler manifold. Then (M1×fM2,F) is a Riemannian manifold if and only if (M1,F1) and (M2,F2) are Riemannian manifold.Various interesting special forms of Cartan tensors have been obtained by some Finslerians [11]. The Finsler spaces having such special forms have been called C-reducible, C2-like, semi-C-reducible, and so forth. In [21], Matsumoto introduced the notion of C-reducible Finsler metrics and proved that any Randers metric is C-reducible. Later on, Matsumoto-Ho-jo- proves that the converse is true too [15].Here, we define the Matsumoto twisted tensorMabc for a twisted product Finsler manifold (M1×fM2,F) as follows:
(70)Mabc=Cabc-1n+1{Iahbc+Ibhac+Ichab},
where Ia=gbcCabc, Cabc=gcdCabd, and hab=gab-(1/F2)yayb. By attention to the previous equation and relations
(71)Cijk=Cijk,Cαβγ=f2Cαβγ,
we obtain
(72)Mαjk=-1n+1{Iα(gjk-1F2yjyk)-f2F2vα(Ijyk+Ikyj)}.
Contracting the previous equation in yjyk gives us
(73)yjykMαjk=-f2F12F22(n+1)F2Iα.
Similarly, we obtain
(74)vλvβMiβλ=-f2F12F22(n+1)F2Ii.
Therefore if Miβλ=Mαjk=0, then we get Ii=Iα=0; that is, (M1,F1) and (M2,F2) are Riemannian manifolds. Thus we have the following.Theorem 12.
There is not exist any C-reducible twisted product Finsler manifold.Now, we are going to consider semi-C-reducible twisted product Finsler manifold(M1×fM2,F). Let (M1×fM2,F) be a semi-C-reducible twisted product Finsler manifold. Then we have
(75)Cabc=pn+1{Iahbc+Ibhac+Ichab}+qC2IaIbIc,
where C2=IaIa and p and q are scalar function on M1×fM2 with p+q=1. This equation gives us
(76)0=Cαjk=pn+1{Iα(gjk-1F2yjyk)-f2F2vα(Ijyk+Ikyj)}+qC2IαIjIk.
Contractiing the previous equation with yjyk implies that
(77)pf2F12F22Iα=0.
Therefore we have p=0 or Iα=0. If p=0, then F is C2-like metric. But if p≠0, then Iα=0; that is, F2 is Riemannian metric. In this case, with similar way, we conclude that F1 is Riemannian metric. But, definition F cannot be a Riemannian metric. Therefore we have the following.Theorem 13.
Every semi-C-reducible twisted product Finsler manifold(M1×fM2,F) is a C2-like manifold.


## 4. Riemannian Curvature

The Riemannian curvature of twisted product Finsler manifold(M1×fM2,F) with respect to Berwald connection is given by
(78)Rbcda=δtFbcaδtxd-δtFbdaδtxc+FdeaFbce-FceaFbde.Lemma 14.
Let(M1×fM2,F) be a twisted product Finsler manifold. Then one has
(79)Racd=ybRbcda,
where Rcda and ybRbcda are given by (50) and (78).Proof.
By using (78), we have
(80)ybRbkli=ybδtFbkiδtxl-ybδtFbliδtxk+ybFleiFbke.-ybFkeiFble.
By using Corollary 8 and Lemma 9, we obtain
(81)ybδtFbkiδtxl=δtGkiδtxl+FjkiGlj+FβkiGlβ,ybFleiFbke=FlhiGkh+FlγiGkγ.
Interchanging i and j in the previous equation implies that
(82)ybδtFbliδtxk=δtGliδtxk+FjliGkj+FβliGkβ,ybFkeiFble=FkhiGlh+FkγiGlγ.
Setting (81) and (82) in (80) gives us ybRbkli=Rikl. In the similar way, we can obtain this relation for another indices.Using (78), we can compute the Riemannian curvature of a twisted product Finsler manifold.Lemma 15.
Let(M1×fM2,F) be a twisted product Finsler manifold. Then the coefficients of Riemannian curvature are as follows:
(83)Rjkli=Rjkli-{{Mlr∂Fjki∂yr+δtMjkiδtxl+FlhiMjkh+MlhiFjkh-MlhiMjkhdddd+f-2gαγGαrGγmClriCjkm∂Fjki∂yr}-ℭlk∂Fjki∂yr},(84)Rαkli={-δtδtxl(GαrCkri)-(Frli-Mrli)GαmCkmr-f-1GβrClrifkδαβ+GβrGkμClriCαμβδtδtxl}-ℭlk,(85)Rjβλi={-δtδtuλ(GβrCjri)+GλmGβlCrmiCjlr-(figαλ-fGhμgihCαλμ)(fjδβα-fGjνCβνα)δtδtxl}-ℭλβ,(86)Rαβli=δtδtuβ(GαrClri)-δtδtxlf(figαβ-fGhλgihCαβλ)-GβmGαsCrmiClsr+(figμβ-fGhλgihCμβλ)×(flδαμ-fGlνCανμ)-fgrh(Frli-Mrli)×(fhgαβ-fGhλCαβλ)-GμrClri(Fαβμ+Nαβμ-Mαβμ),(87)Rjβli=-δtδtxl(GβrCjri)-δtδtuβ(Fjli-Mjli)-(Flri-Mlri)GβsCjsr-f-1GαrClri×(fjδβα-fGjμCβμα)+GβsCrsi(Fjlr-Mjlr)+f-1GμrCjlr(fiδβμ-fGhλgihCβλμ),(88)Rαβλi={-δtδtuλ(ffigαβ-f2gihGhμCαβμ)+fGλsCrsi×(frgαβ-fGlμCαβμgrl)-f(figλμ-fgihGhκCλμκ)×(Fαβμ+Nαβμ-Mαβμ)δtδtuλ}-ℭλβ,(89)Rjklγ={δtδtxl(f-2gγλGλrCjkr)+f-2gγλGλsClrs×(Fjkr-Mjkr)+f-3GμrCjkr(flgγμ-fGlαCαγμ)δtδtuλ}-ℭlk,(90)Rjβlγ=δtδtxl(f-1fjδβγ-fGjαCαβγ)-δtδtuβ(f-2gγλGλrCjlr)-f-2gγλGλsGβmClshChjm+f-2(flδμγ-fGlαCμαγ)(fjδβμ-fGjνCβνμ)-f-1(frδβγ-fGrαCβαγ)(Fjlr-Mjlr)-f-2gμλGλrCjlr(Fβμγ+Nβμγ-Mβμγ),(91)Rαβlγ=δtδtxl(Fαβγ+Nαβγ-Mαβγ)-δtδtuβ(f-1flδαγ-GlμCαμγ)-f-1gγλGλsClsh(fhgαβ-fGhμCαβμ)+f-1(flδμγ-fGlκCμκγ)(Fαβμ+Nαβμ-Mαβμ)+f-1GαsClsr(frδβγ-fGrκCβκγ)-f-1(flδαμ-fGlκCακμ)(Fβμγ+Nβμγ-Mβμγ),(92)Rjβλγ={δtδtuλ(f-1fjδβγ-GjαCαβγ)+f-1(Fαλγ+Nαλγ-Mαλγ)(fjδβα-fGjνCβνα)-f-1GβmCjmr(frδλγ-fGrαCλαγ)δtδtuλ}-ℭλβ,(93)Rαklγ={δtδtxl(f-1fkδαγ-GkμCαμγ)+f-2(flδβγ-fGlκCβκγ)(fkδαβ-fGkνCανβ)-f-2GμsGαmgγμClshChkmδtδtuλ}-ℭlk,(94)Rαβλγ=Rαβλγ-{{Mλκ∂Fαβγ∂vκ+δtMαβγδtuλ+FλμγMαβμ+MλμγFαβμRαβλγ-aaa-MλμγMαβμ+δtNαλγδtuβ+FβμγNαλμRαβλγ-aaa+NβμγFαλμ+NβμγNαλμ+NλμγMαβμRαβλγ-aaa+NαβμMλμγ+(grsδλγfs-fgrsGsκCλκγ)Rαβλγ-aaa×(gαβfr-fGrμCαβμ)∂Fαβγ∂vκ}-ℭλβ∂Fαβγ∂vκ},
where
(95)Mjki=MkrCjri+MjrCkri-MhrgihCjkr,Mαβγ=MβμCαμγ+MαμCβμγ-MνμgγνCαβμ,Nαβγ=f-1(fβδαγ+fαδβγ-fλgγλgαβ)
and ℭji denotes the interchange of indices i, j, and subtraction.By Theorem18, we have the following.Theorem 16.
Let(M1×fM2,F) be a flat twisted product Finsler manifold, and let (M1,F1) be Riemannian. If f is a function on M2, only, then (M1,F1) is locally flat.Similarly, we get the following.Theorem 17.
Let(M1×fM2,F) be a flat twisted product Finsler manifold, and let (M2,F2) be Riemannian. If f is a function on M1, only, then (M2,F2) is a space of positive constant curvature ||gradf||2.Proof.
SinceM2 is Riemannain and f is a function on M1, then by (94), we obtain
(96)Rαβλγ=Rαβλγ+∥gradf∥2(δλγgαβ-δβγgαλ).
Since (M1×fM2,F) is flat, then Rαβλγ=0. Thus the proof is complete.Theorem 18.
Let(M1×fM2,F) be a twisted product Riemannian manifold, and let f be a function on M2, only. Then (M1×fM2,F) is flat, if and only if (M1,F1) is flat, and the Riemannian curvature of (M2,F2) satisfies in the following equation:
(97)Rαβλγ={δtNαλγδtuβ+FβμγNαλμ+NβμγFαλμ+NβμγNαλμ}-ℭλβ.


## 5. Twisted Product Finsler Manifolds with Non-Riemannian Curvature Properties

There are several important non-Riemannian quantities such as the Berwald curvatureB, the mean Berwald curvature E, and the Landsberg curvature L [22]. They all vanish for Riemannian metrics, hence they are said to be non-Riemannian. In this section, we find some necessary and sufficient conditions under which a twisted product Riemannian manifold are Berwaldian, of isotropic Berwald curvature, of isotropic mean Berwald curvature. First, we prove the following.Lemma 19.
Let(M1×fM2,F) be a twisted product Finsler manifold. Then the coefficients of Berwald curvature are as follows:
(98)Bαβλγ=Bαβλγ+f-1(Cλ;α;βγνfνF22+2Cα;βγνfνvλBαβλγ+f-1aa+2Cα;λγνfνvβ+2CαγνfνgλβBαβλγ+f-1aa+2Cλ;βγνfνvα+2CβγνfνgλαBαβλγ+f-1aa+2Cλγνfνgαβ-2Cαβλfγ),(99)Bijlk=Bijlk+fCl;j;ikhfhF22,(100)Biβlk=2fCi;lkhfhvβ,(101)Bαβlk=2fgαβClkhfh,(102)Bαβλk=-2fCαβλfk,(103)Biβλγ=Bijλγ=Bijkγ=0.Let(M1×fM2,F) is a Berwald manifold. Then we have Babcd=0. By using (102), we get
(104)Cαβλfk=0.
Multiplying this equation in gkr, we obtain
(105)Cαβλfr=0.
Thus if f is not constant on M1, then we have Cαβλ=0. Also, from (101), we result that
(106)Clkhfh=0.
Differentiating this equation with respect to yj gives us
(107)Cl;jkhfh=0.
Similarly we obtain
(108)Cl;j;ikhfh=0.
Setting the last equation in (99) implies that Bijlk=0; that is, (M1,F1) is Berwaldian. These explanations give us the following theorem.Theorem 20.
Let(M1×fM2,F) be a twisted product Finsler manifold, and let f be not constant on M1. Then (M1×fM2,F) is Berwaldian if and only if (M1,F1) is Berwaldian, (M2,F2) is Riemannian, and the equation Clkhfh=0 is hold.But iff is constant on M1, that is, fi=0, then we get the following.Theorem 21.
Let(M1×fM2,F) be a twisted product Finsler manifold, and f is constant on M1. Then (M1×fM2,F) is Berwaldian if and only if (M1,F1) is Berwaldian and the Berwald curvature of (M2,F2) satisfies in the following equation:
(109)Bαβλγ=-f-1(Cβ;α;λγνfνF22+2Cβ;αγνfνvλ+2Cλ;αγνfνvβ-f-1aaa+2Cαγνfνgλβ+2Cβ;λγνfνvα-f-1aaa+2Cβγνfνgλα+2Cλγνfνgαβ-f-1aaa-2gγνCαβλfν).Here, we consider twisted product Finsler manifold(M1×fM2,F) of isotropic Berwald curvature.Theorem 22.
Every isotropic Berwald twisted product Finsler manifold(M1×fM2,F) is a Berwald manifold.Proof.
Let(M1×fM2,F) be an isotropic Berwald manifold. Then we have
(110)Babcd=cF-1{hadhbc+hbdhac+hcdhab+2Cabcyd},
where c=c(x) is a function on M. Setting a=j, b=k, c=l, and d=γ and using (103) imply that
(111)cF-1{3F2yjykylvγ-vγ(yjgkl+ykgjl+ylgjk)}=0.
Multiplying the previous equation in yjyk, we derive that cf2F12F22=0. Thus we have c=0; that is, (M1×fM2) is Berwaldian.Now, we are going to study twisted product Finsler manifold of isotropic mean Berwald curvature. For this work, we must compute the coefficients of mean Berwald curvature of a twisted product Finsler manifold.Lemma 23.
Let(M1×fM2,F) be a twisted product Finsler manifold. Then the coefficients of mean Berwald curvature are as follows:
(112)Eαβ=Eαβ+fgαβIhfh+12fI;α;βνfνF22+f-1fν×(Cα;βγνvγ+I;ανvβ+I;βνvα+Cαβν+Iνgαβ),(113)Eij=Eij+12fI;j;ihfhF22,(114)Eiβ=fI;ihfhvβ,
where Eij and Eαβ are the coefficients of mean Berwald curvature of (M1,F1) and (M2,F2), respectively.Proof.
By definition and Lemma19, we get the proof.Theorem 24.
The twisted product Finsler manifold(M1×fM2,F) is weakly Berwald if and only if (M1,F1) is weakly Berwald, Ihfh=0, and the following hold:
(115)Eαβ=-12fI;α;βνfνF22-f-1fν×(Cα;βγνvγ+I;ανvβ+I;βνvα+Cαβν+Iνgαβ).Proof.
If(M1×fM2) be a weakly Berwald manifold, then we have
(116)Eαβ=Eij=Eiβ=0.
Thus by using (114), we result that I;ihfh=0. This equation implies that
(117)I;j;ihfh=0,Ihfh=0.
By setting these equations in (112) and (113), we conclude that Eij=0 and Eαβ satisfies in (115).Now, iff is constant on M2, then (115) implies that Eαβ=0. Thus we conclude the following.Corollary 25.
Let(M1×fM2,F) be a twisted product Finsler manifold, and let f be a function on M1, only. Then (M1×fM2,F) is weakly Berwald if and only if (M1,F1) and (M2,F2) are weakly Berwald manifolds and Ihfh=0.Now, we consider twisted product Finsler manifolds with isotropic mean Berwald curvature. It is remarkable that as a consequence of Lemma23, we have the following.Lemma 26.
Twisted product Finsler manifold(M1×fM2,F) is isotropic mean Berwald manifold if and only if
(118)Eαβ+fgαβIhfh+12fI;α;βνfνF22+f-1fν×(Cα;βγνvγ+I;ανvβ+I;βνvα+Cαβν+Iνgαβ)-n+12cf2F-1(gαβ-f2F2vαvβ)=0,(119)Eij+12fI;j;ihfhF22-n+12cF-1(gij-1F2yiyj)=0,(120)c(n+1)F-3yi+fI;ihfh=0,
where c=c(x) is a scalar function on M.Theorem 27.
Every twisted product Finsler manifold(M1×fM2,F) with isotropic mean Berwald curvature is a weakly Berwald manifold.Proof.
Suppose thatF is isotropic mean Berwald twisted product Finsler metric. Then differentiating (120) with respect to vγ gives us
(121)c(n+1)f2F-5vγyi=0.
Thus, we conclude that c=0. This implies that F reduces to a weakly Berwald metric.


## 6. Locally Dually Flat Twisted Product Finsler Manifolds

In [23], Amari and Nagaoka introduced the notion of dually flat Riemannian metrics when they study the information geometry on Riemannian manifolds. Information geometry has emerged from investigating the geometrical structure of a family of probability distributions and has been applied successfully to various areas including statistical inference, control system theory, and multiterminal information theory. In Finsler geometry, Shen extends the notion of locally dually flatness for Finsler metrics [17]. Dually flat Finsler metrics form a special and valuable class of Finsler metrics in Finsler information geometry, which play a very important role in studying flat Finsler information structure [24, 25].In this section, we study locally dually flat twisted product Finsler metrics. It is remarkable that a Finsler metricF=F(x,y) on a manifold M is said to be locally dually flat if at any point there is a standard coordinate system (xa,ya) in TM such that it satisfies
(122)∂2F2∂xb∂yayb=2∂F2∂xa.
In this case, the coordinate (xa) is called an adapted local coordinate system. By using (122), we can obtain the following lemma.Lemma 28.
Let(M1×fM2,F) be a twisted product Finsler manifold. Then F is locally dually flat if and only if F1 and F2 satisfy in the following equations:
(123)∂2F12∂xk∂ylyk=2∂F12∂xl+4fflF22,(124)4fkvβyk+f∂2F22∂uα∂vβvα+4fαvβvα=2f∂F22∂uβ+4fβF22.Now, letF be a locally dually flat Finsler metric. Taking derivative with respect to vγ from (123) yields fl=0, which means that f is a constant function on M1. In this case, the relations (123) and (124) reduce to the following:
(125)∂2F12∂xk∂ylyk=2∂F12∂xl,(126)f∂2F22∂uα∂vβvα+4fαvβvα=2f∂F22∂uβ+4fβF22.
By (125), we deduce that F1 is locally dually flat.Now, we assume thatF1 and F2 are locally dually flat Finsler metrics. Then we have
(127)∂2F12∂xk∂ylyk=2∂F12∂xl,∂2F22∂uα∂vβvα=2∂F22∂uβ.
By (127), we derive that (123) and (124) are hold if and only if the following hold:
(128)fl=0,fαvβvα=fβF22.
Therefore we can conclude the following.Theorem 29.
Let(M1×fM2,F) be a twisted product Finsler manifold.(i)
IfF is locally dually flat, then F1 is locally dually flat, f is a function with respect to (uα) only, and F2 satisfies in (126).(ii)
IfF1 and F2 are locally dually flat, then F is locally dually flat if and only if f is a function with respect (uα) only and F2 satisfies in (128).By Theorem29, we conclude the following.Corollary 30.
There is not exist any locally dually flat proper twisted product Finsler manifold.



---

*Source: 732432-2013-07-10.xml*