# Locally Conformally Flat Doubly Twisted Product Complex Finsler Manifolds

Authors: Wei Xiao; Yong He; Shuwen Li; Qihui Ni
Journal: Journal of Mathematics
 (2022)

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## Abstract

LetM1×λ1,λ2M2,F be a doubly twisted product manifold of two strongly pseudoconvex complex Finsler manifolds M1,F1 and M2,F2. In this study, we give a characterization of locally conformally flat doubly twisted product complex Finsler manifold. We also obtain a necessary and sufficient condition for a doubly twisted product of two locally conformally flat complex Finsler manifolds to be locally conformally flat.


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## Body

## 1. Introduction

Warped product and twisted product are important methods used to construct manifolds with special curvature properties in Riemann geometry. The warped product manifold was first introduced in 1969 by O’Neill and Bishop to construct Riemannian manifolds with negative curvature [1]. Then, it was extended to real Finsler geometry by the work of Hushmandi et al. (see [2–4]). Later, it was studied by many authors to construct new examples of real Finlser manifold [5–7]. They consider warped product Finsler manifold with scalar flag curvature and some well-known non-Riemannian curvature properties such as Berwald, Landsberg, and S-curvature [8–11].In [12], He and Zhong extended the warped product to complex Finsler geometry and gave a possible way to construct complex Finsler metric (e.g., weakly complex Berwald metric and complex locally Minkowski metric).The twisted product manifold, as a generalization of warped product manifold, was mentioned first by Chen [13]. Then, the notion of twisted product was generalized for the pseudo-Riemannian manifold by Ponge and Reckziegel [14]. In [15], Kozma et al. extended the twisted product to real Finsler manifolds. Later, twisted product of Finsler manifolds was studied by many authors [16–19]. In 2021, Deng et al. obtained the necessary and sufficient conditions that the doubly twisted product of real Finsler manifolds is a Berwald manifold [20].Recently, we extended the twisted product to complex Finsler geometry and gave characterization of a doubly twisted product complex Finsler manifold to be Kähler Finsler manifold (resp. weakly Kähler Finsler manifold, complex Berwald manifold, weakly complex Berwald manifold, and complex Landsberg manifold) [21]. Later, we obtained necessary and sufficient conditions for a doubly twisted product complex Finsler manifold to be locally dually flat [22]. In [23], we gave a characterization for doubly twisted product complex Finsler manifold to be a complex Einstein–Finsler manifold.The Weyl theorem states that the projective and conformal properties of a Finsler space determine the metric properties uniquely [24, 25]. In Finsler geometry, it is important subject to study conformal transformations of Finsler manifolds. In [26], Aldea proved that the scale function of a conformal transformation between two complex Finsler metrics depends only on the position of the base manifold. Recently, many author studied conformal transformation in complex Finsler geometry [27–30]. In 2021, Li studied the locally conformal pseudo-Kähler Finsler manifolds [31].Particularly, a Finsler manifold which is conformally related to a Minkowski manifold is called locally conformally flat Finsler manifold. One of the most important problems in Finsler geometry is to study and characterize conformally flat Finsler manifold. In 2012, Matsumoto and Pripathi systematically studied locally conformally flat twisted product Riemann manifolds and proved its component manifolds are locally conformally flat [18]. Later, He also gave a necessary and sufficient condition for a doubly warped product complex Finsler manifolds to be locally conformally flat in [32].Thus, it is very natural and interesting to ask the following question. Under what condition does a doubly twisted product complex Finsler manifold be locally conformally flat? Under what condition does a doubly twisted product of two locally conformally flat complex Finsler manifolds be locally conformally flat?Inspired by the above questions, we will investigate the necessary and sufficient conditions for a doubly twisted product complex Finsler manifold to be locally conformally flat. We also give a necessary and sufficient condition that the doubly twisted product of two locally conformally flat complex Finsler manifolds is locally conformally flat.


## 2. Preliminary

In this section, we briefly recall some basic concepts and notations which we need in this study.LetM be a complex manifold of complex dimension n. We denote T1,0M as the holomorphic tangent bundle of M and M˜ as the complement of zero section in T1,0M. We denote z=z1,…,zn as the local holomorphic coordinates on M and z,v=z1,…,zn,v1,…,vn as the induced local holomorphic coordinates on the holomorphic tangent bundle T1,0M.Definition 1.
(see [33]). A strongly pseudoconvex complex Finsler metric F on a complex manifold M is a continuous function F:T1,0M⟶ℝ+ satisfying(i)
G=F2 is smooth on M˜.(ii)
Fp,v>0, for all p,v∈M˜.(iii)
Fp,ζv=ζFp,v, for all p,v∈T1,0M and ζ∈ℂ.(iv)
The Levi matrix (or complex Hessian matrix),(1)Gαβ¯=∂2G∂vα∂vβ¯,is positive definite onM˜.In this study, we denoteGν¯β the inverse matrix of Gαν¯ such that Gν¯βGαν¯=δαβ. We also use the notion in [33], that is, the derivatives of G with respect to the v-coordinates and z-coordinates are separated by semicolon; for instance,(2)Gμ¯;ν=∂2G∂zν∂vμ¯,Gα;ν¯=∂2G∂zν¯∂vα.In the following, we denote∂μ as the partial derivative with respect to the local coordinates zμ on M and ∂μ. as the partial derivative with respect to the fiber coordinates vμ.The Chern–Finsler connection was first constructed in [34] and systemically studied in [33]. Let D:XV1,0⟶XTℂ∗M˜⊗V1,0 be the Chern–Finsler connection associated to a strongly pseudoconvex complex Finsler metric F. The Chern–Finsler complex nonlinear connection Γ;μα associated to F is given by(3)Γ;μα≕Gν¯αGν¯;μ.The connection 1-formsωβα of D are given by(4)ωβα=Γβ;μαdzμ+Γβμαψμ,where(5)Γβ;μα=Gν¯αδμGβν¯=∂˙βΓ;μα,Γβμα=Gν¯α∂˙μGβν¯,and(6)δμ=∂μ−Γ;μα∂˙α,ψμ=dvμ+Γ;αμdzα.The complex Rund connection associated to a strongly pseudoconvex complex Finsler metricF was first introduced in [35] and were systemically studied in [36, 37]. Let D^:XV1,0⟶XTℂ∗M˜⊗V1,0 be the complex Rund connection; then, the connection 1-forms ω^ of D^ are given by(7)ω^βα=Γβ;μαdzμ,where Γβ;μα are defined by equation (5). It is clearly from equation (4) that ω^βα are just the horizontal part of ωβα.Definition 2.
(see [37, 38]). A complex Finsler manifold M,F is said to be modeled on a complex Minkowski space if the horizontal connection coefficients Γβ;μα of the Chern–Finsler connection or complex Rund connection coefficients depend only on the coordinates z of the base manifold M, i.e., Γβ;μαz,v=Γβ;μαz.Definition 3.
(see [36]). Let F be a complex Finsler metric on M. If there exits an open cover U,XU such that, on each πT−1U, the function F is a function of the fiber coordinate only, then the complex Finsler metric F will be a complex locally Minkowski metric.The necessary and sufficient condition of a complex Finsler metricF on M to be a complex locally Minkowski metric is that it is modeled on a complex Minkowski metric and the complex Rund connection coefficients on M,F is holomorphic. That is, the horizontal coefficients Γβ;μαz,v of the Chern–Finsler connection or complex Rund connection coefficients satisfy the following conditions:(8)Γβ;μαz,v=Γβ;μαz,∂Γβ;μα∂zν¯=0.Definition 4.
(see [38]). Let F and F˜ be two strongly pseudoconvex complex Finsler metric on complex manifold M. A conformal change of F is the change F⟶F˜=eσzF for a smooth function σz on M.Definition 5.
(see [38]). A complex Finsler manifold M,F is said to be locally conformally flat if it is locally conformal to a locally Minkowski space; that is, there exhibits an open covering Uα with a family σα of locally defined smooth functions σα:Uα⟶ℝ such that the metric F˜=eσαzF is a complex locally Minkowski metric on Uα.


## 3. Doubly Twisted Product of Complex Finsler Manifolds

LetM1,F1 and M2,F2 be two strongly pseudoconvex complex Finsler manifolds with dimℂM1\\=m and dimℂM2=n; then, M=M1×M2 is a strongly pseudoconvex complex Finsler manifold with dimℂM=m+n.LetT1,0M1 and T1,0M2 be the holomorphic tangent bundles of M1 and M2, respectively. Let π1:M1×M2⟶M1 and π2:M1×M2⟶M2 be natural projection maps; then, dπ1:T1,0M1×M2⟶T1,0M1 and dπ2:T1,0M1×M2⟶T1,0M2 be the holomorphic tangent maps induced by π1 and π2, respectively. Note that dπ1z,v=z1,v1 and dπ2z,v=z2,v2 for every v=v1,v2∈Tz1,0M1×M2 with v1=v1,…,vm∈Tz11,0M1 and v2=vm+1,…,vm+n∈Tz21,0M2.Definition 6.
(see [21]). Let M1,F1 and M2,F2 be two strongly pseudoconvex complex Finsler manifolds and λi:M1×M2⟶0,+∞i=1,2 be smooth functions. The doubly twisted product (abbreviated as DTP) complex Finsler manifold of M1,F1 and M2,F2 is the product complex manifold M=M1×M2 endowed with the complex Finsler metric F:M˜⟶ℝ+ given by(9)F2z,v=λ12zF12π1z,dπ1v+λ22zF22π2z,dπ2v,for z=z1,z2∈M and v=v1,v2∈Tz1,0M−zero section. The functions λ1 and λ2 are called twisted functions. The DTP-complex Finsler manifold of M1,F1 and M2,F2 is denoted by M1×λ1,λ2M2,F.In the case ofλ1≡1 or λ2≡1, the corresponding DTP-complex Finsler manifold is called twisted product complex Finsler manifold. If λ1≡λ2≡1, M,F becomes the product complex Finsler manifold. If neither λ1 nor λ2 is a constant, then we call M,F a nontrivial (proper) DTP-complex Finsler manifold of M1,F1 and M2,F2.Notation: lowercase Greek indices such asα,β,andγ will run from 1 to m+n, whereas lowercase Latin indices such as i,j,andk will run from 1 to m, and lowercase Latin indices with a prime such as i′,j′,andk′ will run from m+1 to m+n. Quantities associated to F1 and F2 are denoted with upper indices 1 and 2, respectively, such as Γj;ki1,Γj′;k′i′2.Denoteg=F12,h=F22, and(10)gij¯=∂2g∂vi∂vj¯,hi′j′¯=∂2h∂vi′∂vj′¯,so equation (9) is equal to(11)G=F2=λ12g+λ22h.Proposition 1.
(see [21]). Let M1×λ1,λ2M2,F be a doubly twisted product manifold of complex Finsler manifolds M1,F1 and M2,F2. Then, the fundamental tensor matrix of F is given by(12)Gαβ¯=∂2G∂vα∂vβ¯=λ12gij¯00λ22hi′j′¯,with its inverse matrix Gβ¯α given by(13)Gβ¯α=λ1−2gj¯i00λ2−2hj′¯i′.Lemma 1.
(see [21]). Let M1×λ1,λ2M2,F be a doubly twisted product manifold of two strongly pseudoconvex complex Finsler manifolds M1,F1 and M2,F2. Then, the Chern–Finsler complex nonlinear connection coefficients associated to F are given by(14)Γ;ki=Γ;ki1+2λ1−1∂λ1∂zkvi,Γ;ki′=2λ2−1∂λ2∂zkvi′,Γ;k′i=2λ1−1∂λ1∂zk′vi,Γ;k′i′=Γ;k′i′2+2λ2−1∂λ2∂zk′vi′.Proposition 2.
(see [21]). Let M1×λ1,λ2M2,F be a doubly twisted product manifold of two strongly pseudoconvex complex Finsler manifolds M1,F1 and M2,F2; the horizontal coefficients of the Chern–Finsler connection associated to F are given by(15)Γj;ki=Γj;ki1+2λ1−1∂λ1∂zkδji,Γj;k′i=2λ1−1∂λ1∂zk′δji,Γj′;k′i′=Γj′;k′i′2+2λ2−1∂λ2∂zk′δj′i′,Γj′;ki′=2λ2−1∂λ2∂zkδj′i′,Γj′;ki=Γj′;k′i=Γj;ki′=Γj;k′i′=0.


## 4. Locally Conformally Flat Doubly Twisted Product Complex Finsler Manifolds

LetF be a complex Finsler metric on M. A conformal transformation of F is a change F⟶eσzF, where σz:M⟶ℝ is a smooth real function. We denote eσzF by F˜ and use the symbol “∼” to mark the geometric objects associated to the complex Finsler metric F˜, e.g., Γ˜β;μα are the horizontal coefficients of Chern–Finsler connection associate to F˜.LetM1,F1 and M2,F2 be locally conformally flat complex Finsler manifolds. In this section, we shall give the necessary and sufficient conditions for a doubly twisted product manifold M1×λ1,λ2M2,F of M1,F1 and M2,F2 to be locally conformally flat.Proposition 3.
LetM1×λ1,λ2M2,F be a doubly twisted product manifold of two strongly pseudoconvex complex Finsler manifolds M1,F1 and M2,F2, and F˜ is conformal transformation of F. Then, the coefficients Γ˜;μα of Chern–Finsler connection associated to F˜ are given by(16)Γ˜;ki=Γ;ki1+2λ1−1∂λ1∂zk+∂σ∂zkvi,Γ˜;ki′=2λ2−1∂λ2∂zk+∂σ∂zkvi′,Γ˜;k′i=2λ1−1∂λ1∂zk′+∂σ∂zk′vi,Γ˜;k′i′=Γ;k′i′2+2λ2−1∂λ2∂zk′+∂σ∂zk′vi′,where σz:M⟶ℝ is a smooth real function.Proof.
Using equation (3), we obtain(17)Γ˜;μα=G˜ν¯αG˜ν¯;μ=Γ;μα+2∂σ∂zμvα.
According to Lemma1, we have(18)Γ˜;ki=Γ;ki+2∂σ∂zkvi=Γ;ki1+2λ1−1∂λ1∂zk+∂σ∂zkvi.
Similarly, we can get other equations of Proposition3.Proposition 4.
LetM1×λ1,λ2M2,F be a doubly twisted product manifold of two strongly pseudoconvex complex Finsler manifolds M1,F1 and M2,F2, and F˜ is conformal transformation of F. Then, the horizontal coefficients Γ˜β;μα of the Chern–Finsler connection associated to F˜ are given by(19)Γ˜j;ki=Γj;ki1+2λ1−1∂λ1∂zk+∂σ∂zkδji,Γ˜j;k′i=2λ1−1∂λ1∂zk′+∂σ∂zk′δji,Γ˜j′;k′i′=Γj′;k′i′2+2λ2−1∂λ2∂zk′+∂σ∂zk′δj′i′,Γ˜j′;ki′=2λ2−1∂λ2∂zk+∂σ∂zkδj′i′,Γ˜j′;ki=Γ˜j′;k′i=Γ˜j;ki′=Γ˜j;k′i′=0,where σz:M⟶ℝ is a smooth real function.Proof.
According to equation (5) and Proposition 2, by straightforward computation, we obtain(20)Γ˜j;ki=∂˙jΓ˜;ki=Γj;ki+2∂σ∂zkδji=Γj;ki1+2λ1−1∂λ1∂zk+∂σ∂zkδji.
Similarly, we can obtain other equations of Proposition4.Theorem 1.
LetM1×λ1,λ2M2,F be a doubly twisted product manifold of two strongly pseudoconvex complex Finsler manifolds M1,F1 and M2,F2; then, M1×λ1,λ2M2,F is locally conformally flat if and only if there exists an open covering Uα and a family of smooth functions σα:Uα⟶ℝ such that(21)Γj;ki1z,v=Γj;ki1z,Γj′;k′i′2z,v=Γj′;k′i′2z,∂Γj;ki1∂zs¯+2∂2lnλ1∂zk∂zs¯+∂2σα∂zk∂zs¯δji=0,∂Γj′;k′i′2∂zs′¯+2∂2lnλ2∂zk′∂zs′¯+∂2σα∂zk′∂zs′¯δj′i′=0,∂2lnλ1∂zk∂zs′¯+∂2σα∂zk∂zs′¯=∂2lnλ1∂zk′∂zs¯+∂2σα∂zk′∂zs¯=∂2lnλ1∂zk′∂zs′¯+∂2σα∂zk′∂zs′¯=0,∂2lnλ2∂zk∂zs¯+∂2σα∂zk∂zs¯=∂2lnλ2∂zk∂zs′¯+∂2σα∂zk∂zs′¯=∂2lnλ2∂zk′∂zs¯+∂2σα∂zk′∂zs¯=0.Proof.
According to Definition5, M,F is locally conformally flat if and only if there exists an open covering Uα and a family of smooth functions σα:Uα⟶ℝ on M such that F˜α=eσαz is a locally Minkowski metric on Uα. Notice that the complex Finsler metric F˜α is a complex locally Minkowski metirc if and only if it is modeled on a complex Minkowski metric and the complex Rund connection coefficients of F˜α is holomorphic. Thus, we obtain(22)Γ˜β;μαz,v=Γ˜β;μαz,(23)∂Γ˜β;μα∂zν¯=0.
Submit the equations of Proposition4 into equations (22) and (23); after a straightforward computation, we have(24)Γ˜j;ki1z,v=Γ˜j;ki1z,Γ˜j′;k′i′2z,v=Γ˜j′;k′i′2z,∂Γ˜j;ki∂zs¯=∂Γj;ki1∂zs¯+2∂2lnλ1∂zk∂zs¯+∂2σα∂zk∂zs¯δji=0,∂Γ˜j;ki∂zs′¯=2∂2lnλ1∂zk∂zs′¯+∂2σα∂zk∂zs′¯δji=0,∂Γ˜j′;k′i′∂zs¯=2∂2lnλ2∂zk′∂zs¯+∂2σα∂zk′∂zs¯δj′i′=0,∂Γ˜j′;k′i′∂zs′¯=∂Γj′;k′i′2∂zs′¯+2∂2lnλ2∂zk′∂zs′¯+∂2σα∂zk′∂zs′¯δj′i′=0,∂Γ˜j;k′i∂zs¯=2∂2lnλ1∂zk′∂zs¯+∂2σα∂zk′∂zs¯δji=0,∂Γ˜j;k′i∂zs′¯=2∂2lnλ1∂zk′∂zs′¯+∂2σα∂zk′∂zs′¯δji=0,∂Γ˜j′;ki′∂zs¯=2∂2lnλ2∂zk∂zs¯+∂2σα∂zk∂zs¯δj′i′=0,∂Γ˜j′;ki′∂zs′¯=2∂2lnλ2∂zk∂zs′¯+∂2σα∂zk∂zs′¯δj′i′=0.
Noticing thatΓ˜j;ki1=Γj;ki1+2∂σα/∂zkδji, so Γ˜j;ki1z,v=Γ˜j;ki1z if and only if Γj;ki1z,v=Γj;ki1z. Similarly, we know that Γ˜j′;k′i′2z,v=Γ˜j′;k′i′2z if and only if Γj′;k′i′2z,v=Γj′;k′i′2z.
Thus, we can obtain equation (21).Corollary 1.
LetM1×λ1,1M2,F be a twisted product manifold of two strongly pseudoconvex complex Finsler manifolds M1,F1 and M2,F2; then, M1×λ1,1M2,F is locally conformally flat if and only if M2,F2 is locally conformally flat, and there exists an open covering Uα and a family of smooth functions σα:Uα⟶ℝ such that(25)Γj;ki1z,v=Γj;ki1z,∂Γj;ki1∂zs¯+2∂2lnλ1∂zk∂zs¯δji=0,∂2lnλ1∂zk′∂zs′¯+∂2σα∂zk′∂zs′¯=0,∂2lnλ1∂zkzs′¯=∂2lnλ1∂zk′zs¯=0,∂2σα∂zk∂zs¯=∂2σα∂zk∂zs′¯=∂2σα∂zk′∂zs¯=0.Proof.
SinceM1×λ1,1M2,F be a twisted product complex Finsler manifold, so(26)∂λ2∂zα=0.
By plunging equation (26) into equation (21), we obtain(27)Γj;ki1z,v=Γj;ki1z,Γj′;k′i′2z,v=Γj′;k′i′2z,(28)∂Γj;ki1∂zs¯+2∂2lnλ1∂zk∂zs¯+∂2σα∂zk∂zs¯δji=0,(29)∂Γj′;k′i′2∂zs′¯+2∂2σα∂zk′∂zs′¯δj′i′=0,(30)∂2lnλ1∂zk∂zs′¯+∂2σα∂zk∂zs′¯=∂2lnλ1∂zk′∂zs¯+∂2σα∂zk′∂zs¯=∂2lnλ1∂zk′∂zs′¯+∂2σα∂zk′∂zs′¯=0,(31)∂2σα∂zk∂zs¯=∂2σα∂zk∂zs′¯=∂2σα∂zk′∂zs¯=0.
From equation (29) and noticing Γ˜j′;k′i′2z,v=Γ˜j′;k′i′2z, we obtain that M2,F2 is a locally conformally flat manifold.
Submit equations (31) into equations (28) and (30), respectively; we can obtain equation (25).Theorem 2.
LetM1×λ1,λ2M2,F be a doubly twisted product manifold of two locally conformally flat complex Finsler manifolds M1,F1 and M2,F2; then, M1×λ1,λ2M2,F is locally conformally flat if and only if lnλ1 and lnλ2 are pluriharmonic functions on M1 and M2, respectively, and there exists an open covering Uα and a family of smooth functions σα:Uα⟶ℝ such that(32)∂2lnλ1∂zk∂zs′¯+∂2σα∂zk∂zs′¯=∂2lnλ1∂zk′∂zs¯+∂2σα∂zk′∂zs¯=∂2lnλ1∂zk′∂zs′¯+∂2σα∂zk′∂zs′¯=0,∂2lnλ2∂zk∂zs¯+∂2σα∂zk∂zs¯=∂2lnλ2∂zk∂zs′¯+∂2σα∂zk∂zs′¯=∂2lnλ2∂zk′∂zs¯+∂2σα∂zk′∂zs¯=0.Proof.
According to Theorem1, M1×λ1,λ2M2,F is locally conformally flat if and only if equation (21) holds on Uα.
SinceM1,F1 and M2,F2 are locally conformally flat complex Finsler manifolds, thus,(33)Γ˜j;ki1z,v=Γ˜j;ki1z,∂Γj;ki1∂zs¯+2∂2σα∂zk∂zs¯δji=0,(34)Γ˜j′;k′i′2z,v=Γ˜j′;k′i′2z,∂Γj′;k′i′2∂zs′¯+2∂2σα∂zk′∂zs′¯δj′i′=0.
Plunging equations (33) and (34) into equation (21), equation (21) can be simplified to(35)∂2lnλ1∂zk∂zs¯=0,(36)∂2lnλ2∂zk′∂zs′¯=0,(37)∂2lnλ1∂zk∂zs′¯+∂2σα∂zk∂zs′¯=∂2lnλ1∂zk′∂zs¯+∂2σα∂zk′∂zs¯=∂2lnλ1∂zk′∂zs′¯+∂2σα∂zk′∂zs′¯=0,(38)∂2lnλ2∂zk∂zs¯+∂2σα∂zk∂zs¯=∂2lnλ2∂zk∂zs′¯+∂2σα∂zk∂zs′¯=∂2lnλ2∂zk′∂zs¯+∂2σα∂zk′∂zs¯=0.
Equations (35) and (36) mean that lnλ1 and lnλ2 are pluriharmonic functions on M1 and M2, respectively. Therefore, M1×λ1,λ2M2,F is locally conformally flat if and only if lnλ1 and lnλ2 are pluriharmonic functions on M1 and M2, respectively, and equation (32) holds on Uα.


## 5. Conclusion

This study considered about conformal transformations of a doubly twisted product complex Finsler manifolds and gave a characterization for doubly twisted product complex Finsler manifolds to be locally conformally flat.Locally conformally flat doubly warped product complex Finsler manifolds were studied by the work of He [32]. He also obtained the relations of locally conformally flatness between doubly warped product manifolds and its components. This study extended the work due to He, where we obtained the relations of locally conformally flatness between doubly twisted product complex Finsler manifolds and its components. Moreover, Theorem 2 gave an answer to question mentioned above in Section 1, that is, the doubly twisted product of two locally conformally flat complex Finsler manifolds is locally conformally flat when lnλ1 and lnλ2 are pluriharmonic functions on M1 and M2, respectively, and equation (32) holds. Our approach to this problem depends on the existence of solutions for system (32). We have been trying to find solutions of the PDE systems, but, unfortunately, we have not found a possible way to get some solutions up to now. Investigating possible solutions for PDE systems will be the subject matter of future works.



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