# Levi-Civita Ricci-Flat Doubly Warped Product Hermitian Manifolds

Authors: Qihui Ni; Yong He; Jinhua Yang; Hui Zhang
Journal: Advances in Mathematical Physics
 (2022)

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## Abstract

LetM1,g and M2,h be two Hermitian manifolds. The doubly warped product (abbreviated as DWP) Hermitian manifold of M1,g and M2,h is the product manifold M1×M2 endowed with the warped product Hermitian metric G=f22g+f12h, where f1 and f2 are positive smooth functions on M1 and M2, respectively. In this paper, the formulae of Levi-Civita connection, Levi-Civita curvature, the first Levi-Civita Ricci curvature, and Levi-Civita scalar curvature of the DWP-Hermitian manifold are derived in terms of the corresponding objects of its components. We also prove that if the warped function f1 and f2 are holomorphic, then the DWP-Hermitian manifold is Levi-Civita Ricci-flat if and only if M1,g and M2,h are Levi-Civita Ricci-flat manifolds. Thus, we give an effective way to construct Levi-Civita Ricci-flat DWP-Hermitian manifold.


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## Body

## 1. Introduction

It is well-known that the classification of various Ricci-flat manifolds are important topics in differential geometry. In 1967, Tani [1] first proposed the concept of Ricci-flat space in Riemannian geometry. Alvarez-Gaume and Freedman [2] showed that Ricci-flat space is a kind of space with great significance in theoretical physics, which attracted many scholars’ research [3, 4]. In 1988, Bando and Kobayashi [5] characterized the Ricci-flat metric on Einstein-Ka¨hler manifold. In 2014, Liu and Yang [6] gave a sufficient and necessary condition for Hopf manifolds to be Levi-Civita Ricci-flat.Levi-Civita connection is one of the most natural and effective tools for studying Riemannian manifolds [7]. In the complex case, Hsiung et al. [8] studied the general sectional curvature, the holomorphic sectional curvature, and holomorphic bisectional curvature of almost Hermitian manifolds by Levi-Civita connection and showed the relevance of above sectional curvatures. In 2012, Liu and Yang [8] gave Ricci-type curvatures and scalar curvatures of Hermitian manifolds by Levi-Civita connection (resp. Chern connection and Bismut connection) and obtained the relevance of these curvatures.Warped product and twisted product are important methods used to construct manifold with special curvature properties in Riemann geometry and Finsler geometry. In Riemann geometry, Bishop and O’Neill [9] constructed Riemannian manifolds with negative curvature by warped product. Then, Brozos-Va’zquez et al. [10] used the warped product metrics to construct new examples of complete locally conformally flat manifolds with nonpositive curvature. After that, Leandro et al. [11] proved that an Einstein warped product manifold is a compact Riemannian manifold and its fibre is a Ricci-flat semi-Riemannian manifold.On the other hand, warped product was extended to real Finsler geometry by the work of Asanov [12, 13]. In 2016, He and Zhong [14] generalized the warped product to complex Finsler geometry and proved that if complex Finsler manifold M1,F1 and M2,F2 are projectively flat, then the DWP-complex Finsler manifold is projectively flat if and only if the warped functions are positive constants. Moreover, He and Zhang [15] extended the doubly warped product to Hermitian case and got the Chern curvature, Chern Ricci curvature, and Chern Ricci scalar curvature of DWP-Hermitian manifold. They also gave the necessary and sufficient condition for a compact nontrivial DWP-Hermitian manifold to be of constant holomorphic sectional curvature. Recently, Xiao et al. [16] systematically studied holomorphic curvatures of doubly twisted product complex Finsler manifolds, and they [17] gave the necessary and sufficient condition for doubly twisted product complex Finsler manifold to be locally dually flat.Thus, it is natural and interesting to ask the following question. LetM1,g and M2,h be two Levi-Civita Ricci-flat Hermitian manifolds, whether the DWP-Hermitian manifold is also a Levi-Civita Ricci-flat Hermitian manifold. Our purpose of doing this is to study the possibility of constructing Levi-Civita Ricci-flat manifold.The structure of this paper is as follows. In Section2, we briefly recall some basic concepts and notations which we need in this paper. In Section 3, we derive formulae of Levi-Civita connection, Levi-Civita curvature, the first Levi-Civita Ricci curvature, and Levi-Civita scalar curvature of DWP-Hermitian manifolds. In Section 4, we show that if the warped function f1 and f2 are holomorphic, then the DWP-Hermitian manifold is Levi-Civita Ricci-flat if and only if M1,g and M2,h are Levi-Civita Ricci-flat manifolds.


## 2. Preliminary

LetM,J,G be a Hermitian manifold with dimℂM=n; here, J is the complex structure, and G is a Hermitian metric. For a point p∈M, the complexified tangent bundle TpℂM=TpM⊗C is decomposed as
(1)TpℂM=Tp1,0M⊕Tp0,1M,where Tp1,0M and Tp0,1M are the eigenspaces of J corresponding to the eigenvalues −1 and −−1, respectively.In this paper, we set∂α=∂/∂zα and ∂α¯=∂/∂zα¯. Let z=z1,⋯,zn be the local holomorphic coordinates on M; then, the vector fields ∂1,⋯,∂n form a basis for Tp1,0M. Levi-Civita connection ∇LC on the holomorphic tangent bundle Tp1,0M is defined by [18]
(2)∇LC=π∘∇:ΓM,T1,0M⟶∇ΓM,TpM⊗TpM⟶πΓM,TpM⊗T1,0M.In local coordinate system, its connection is as follows [18]:
(3)∇∂/∂zαLC∂∂zβ£°=Γαβγ∂∂zγ,∇∂/∂z¯εLC∂∂zβ£°=Γε¯βγ∂∂zγ,where the Levi-Civita connection coefficients Γαβγ and Γα¯βγ are given by [18]
(4)Γαβγ=12Gγε¯∂αGβε¯+∂βGαε¯,(5)Γα¯βγ=12Gγε¯∂α¯Gβε¯−∂ε¯Gβα¯.LetK∈ΓM,Λ2TpM⊗T∗1,0M⊗T1,0M be the Levi-Civita curvature tensor such as
(6)KX,Ys=∇XLC∇YLCs−∇YLC∇XLCs−∇X,YLCs,where X,Y∈TpM,s∈T1,0M. In the local coordinate system, the coefficients of K are given by
(7)Kαβ¯γε=−∂β¯Γαγε−∂αΓβ¯γε+ΓαγλΓβ¯λε−Γβ¯γλΓλαε.Definition 1 (see [6]).
The first Levi-Civita Ricci curvatureK1 on the Hermitian manifold M,J,G is defined by
(8)K1=−1Kαβ¯1dzα∧dz¯β,where
(9)Kαβ¯1=Gγδ¯Kαβ¯γδ¯,(10)Kαβ¯γδ¯=Gεδ¯Kαβ¯γε.
Levi-Civita Ricci scalar curvatureSLC on T1,0M is given by
(11)SLC=Gαβ¯Kαβ¯1.Definition 2 (see [6]).
Hermitian metricG on M is called Levi-Civita Ricci-flat if
(12)K1G=0.
LetM1,g and M2,h be two Hermitian manifolds with dimℂM1=m and dimℂM2=n; then, M=M1×M2 is a Hermitian manifold with dimℂM=m+n.
Denoteπ1:M⟶M1 and π2:M⟶M2 the natural projections. Note that π1z=z1 and π2z=z2 for every z=z1,z2∈M with z1=z1,⋯,zm∈M1 and z2=zm+1,⋯,zm+n∈M2.
Denotedπ1:T1,0M⟶T1,0M1,dπ2:T1,0M⟶T1,0M2 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∈Tz1,0M with v1=v1,⋯,vm∈Tz11,0M1 and v2=vm+1,⋯,vm+n∈Tz21,0M2.Definition 3 (see [15]).
LetM1,g and M2,h be two Hermitian manifolds. f1:M1⟶0,+∞ and f2:M2⟶0,+∞ be two positive smooth functions. The doubly warped product (abbreviated as DWP) Hermitian manifold f2M1×f1M2,G is the product Hermitian manifold M=M1×M2 endowed with the Hermitian metric G:M⟶ℝ+ defined by
(13)Gz,v=f2∘π22zgπ1z,dπ1v+f1∘π12zhπ2z,dπ2v,for z=z1,z2∈M and v=v1,v2∈Tz1,0M. f1 and f2 are warped functions; the DWP-Hermitian manifold of M1,g and M2,h is denoted by f2M1×f1M2,G.If eitherf1=1 or f2=1, then f2M1×f1M2,G becomes a warped product of Hermitian manifolds M1,g and M2,h. If f1≡1 and f2≡1, then f2M1×f1M2,G becomes a product of Hermitian manifolds M1,g and M2,h. If neither f1 nor f2 is constant, then we call f2M1×f1M2,G a nontrivial DWP-Hermitian manifolds of M1,g and M2,h.Notation 4.
Lowercase Greek indices such asα, β, and γ will run from 1 to m+n, lowercase Latin indices such as i, j, and k will run from 1 to m, and lowercase Latin indices with a prime, such as i′, j′, and k′, will run from m+1 to m+n. Quantities associated to M1,g and M2,h are denoted with upper indices 1 and 2, respectively, such as Γjki1 and Γj′k′i′2 are Levi-Civita connection coefficients of M1,g and M2,h, respectively.
Denote(14)gij¯=∂2g∂vi∂vj¯,hi′j¯′=∂2h∂vi′∂vj′¯.
The fundamental tensor matrix ofG is given by
(15)Gαβ¯=∂2G∂vα∂vβ¯=f22gij¯00f12hi′j¯′,and its inverse matrix Gβ¯α is given by
(16)Gβ¯α=f2−2gj¯i00f1−2hj¯′i′.Proposition 5.
Letf2M1×f1M2,G be a DWP-Hermitian manifold of M1,g and M2,h. Then, the Levi-Civita connection coefficients Γαβγ associated to G are given by
(17)Γijk=Γijk1,Γi′jk=f2−1∂f2∂zi′δjk,Γij′k=f2−1∂f2∂zj′δik,Γi′j′k′=Γi′j′k′2,Γij′k′=f1−1∂f1∂ziδj′k′,Γi′jk′=f1−1∂f1∂zjδi′k′,Γi′j′k=Γijk′=0.Proof.
Substituting (15) and (16) into (4), we obtain
(18)Γijk=12Gkε¯∂iGjε¯+∂jGiε¯+12Gkε′¯∂iGjε′¯+∂jGiε′¯=12Gkl¯∂Gjl¯∂zi+∂Gil¯∂zj+12Gkl′¯∂Gjl′¯∂zi+∂Gil′¯∂zj=12f2−2gkl¯2f2∂f2∂zigjl¯+f22∂gjl¯∂zi+2f2∂f2∂zjgil¯+f22∂gil¯∂zj=12gkl¯∂gjl¯∂zi+∂gil¯∂zj=Γijk1.
Similarly, we can obtain other equations of Proposition5.Plugging (15) and (16) into (5), we have the following proposition.Proposition 6.
Letf2M1×f1M2,G be a DWP-Hermitian manifold of M1,g and M2,h. Then, the Levi-Civita connection coefficients Γα¯βγ associated to G are given by
(19)Γi¯jk=Γi¯jk1,Γi′¯j′k=−f2−2f1gkl¯∂f1∂z¯lhj′i′¯,Γi′¯jk=f2−1∂f2∂z¯i′δjk,Γi′¯j′k′=Γi′¯j′k′2,Γi¯jk′=−f1−2f2hk′l′¯∂f2∂z¯l′gji¯,Γi¯j′k′=f1−1∂f1∂z¯iδj′k′,Γi¯j′k=Γi′¯jk′=0.


## 3. Levi-Civita Ricci Scalar Curvature of Doubly Warped Product Hermitian Manifolds

In this section, we derive formulae of Levi-Civita curvature, Levi-Civita Ricci curvature, and Levi-Civita Ricci scalar curvature of DWP-Hermitian manifold.Proposition 7.
Letf2M1×f1M2,G be a DWP-Hermitian manifold of M1,g and M2,h. Then, the coefficients of Levi-Civita curvature tensor Kαβ¯γε are given by
(20)Kkj¯st=Kkj¯st1+f1−2hi′l′¯∂f2∂z¯l′∂f2∂zi′gsj¯δkt,(21)Kk′j′¯s′t′=Kk′j′¯s′t′2+f2−2gil¯∂f1∂z¯l∂f1∂zihs′j′¯δk′t′,(22)Kk′j′¯st=f2−2gtl¯∂f1∂z¯l∂f1∂zshk′j′¯,(23)Kkj¯s′t′=f1−2ht′l′¯∂f2∂z¯l′∂f2∂zs′gkj¯,(24)Kkj¯s′t=f1−1f2−1∂f1∂z¯j∂f2∂zi′δs′i′δkt−12f2−1∂f2∂zs′gtl¯∂gil¯∂z¯j−∂gij¯∂z¯lδki,(25)Kk′j′¯st′=f1−1f2−1∂f1∂zi∂f2∂z¯j′δsiδk′t′−12f1−1∂f1∂zshi′l′¯∂hi′l′¯∂z¯j′−∂hi′l′¯∂z¯l′δk′i′,(26)Kk′j′¯s′t=f2−3f1gtl¯∂f1∂z¯l∂f2∂zk′hs′j′¯+f2−2f1gtl¯∂f1∂z¯l12∂hk′l′¯∂zs′+∂hs′l′¯∂zk′δj′¯l′¯+∂hs′j′¯∂zk′.(27)Kkj¯st′=f1−3f2ht′l′¯∂f2∂z¯l′∂f1∂zkgsj¯+f1−2f2ht′l′¯∂f2∂z¯l′12∂gkl¯∂zs+∂gsl¯∂zkδj¯l¯+∂gsj¯∂zk,(28)Kkj′¯s′t=∂2lnf2∂zs′∂z¯j′δkt+12f2−1hi′l′¯∂f2∂zi′∂hs′l′¯∂z¯j′−∂hs′j′¯∂z¯l′δkt+f2−2f1hs′j′¯∂gtl¯∂zk∂f1∂z¯l−gtl¯∂2f1∂zk∂z¯l−12gil¯gtl¯∂f1∂z¯l∂gkl¯∂zi+∂gil¯∂zk,(29)Kk′j¯st′=∂2lnf1∂zs∂z¯jδk′t′+12f1−1gil¯∂f1∂zi∂gsl¯∂z¯j−∂gsj¯∂z¯lδk′t′+f1−2f2gsj¯∂ht′l′¯∂zk′∂f2∂z¯l′−ht′l′¯∂2f2∂zk′∂z¯l′−12hi′l′¯ht′l′¯∂f2∂z¯l′∂hk′l′¯∂zi′+∂hi′l′¯∂zk′.(30)Kk′j¯st=Kk′j¯s′t=Kkj′¯st=Kkj′¯st′=Kk′j¯s′t′=Kkj′¯s′t′=0.Proof.
Using (7), we have
(31)Kkj¯st=−∂j¯Γkst−∂kΓj¯st+ΓksλΓj¯λt−Γj¯sλΓλkt+Γksλ′Γj¯λ′t−Γj¯sλ′Γλ′kt.
Taking the formulae of Proposition5 and Proposition 6 into (31), we obtain
(32)Kkj¯st=−∂Γkst∂z¯j1−∂Γj¯st∂zk1+Γksi1Γj¯it1−Γj¯si1Γikt1+Γksi′Γj¯i′t−Γj¯si′Γi′kt=Kkj¯st1+f1−2hi′l′¯∂f2∂z¯l′∂f2∂zi′gsj¯δkt.
Similarly, we can obtain other equations of Proposition7.Proposition 8.
Letf2M1×f1M2,G be a DWP-Hermitian manifold of M1,g and M2,h. Then,
(33)Kkj¯sp¯=f22Kkj¯sp¯1+f1−2f22hi′l′¯∂f2∂z¯l′∂f2∂zi′gsj¯gkp¯,Kk′j′¯s′p′¯=f12Kk′j′¯s′p′¯2+f2−2f12gil¯∂f1∂z¯l∂f1∂zihs′j′¯hk′p′¯,Kkj¯s′p′¯=∂f2∂z¯l′∂f2∂zs′gkj¯δp′¯l′¯,Kk′j′¯sp¯=∂f1∂z¯l∂f1∂zshk′j′¯δp¯l¯,Kkj¯s′p¯=f1−1f2gkp¯∂f1∂z¯j∂f2∂zi′δs′i′−12f2∂f2∂zs′∂gil¯∂z¯j−∂gij¯∂z¯lδkiδp¯l¯,Kk′j′¯sp′¯=f1f2−1hk′p′¯∂f1∂zi∂f2∂z¯j′δsi−12f1∂f1∂zs∂hi′l′¯∂z¯j′−∂hi′j′¯∂z¯l′δk′i′δp′¯l′¯,Kkj¯sp′¯=f1−3f2∂f2∂z¯l′∂f1∂zkgsj¯δp′¯l′¯+f2∂f2∂z¯l′12∂gkl¯∂zs+∂gsl¯∂zkδj¯l¯+∂gsj¯∂zkδp′¯l′¯,Kk′j′¯s′p¯=f2−3f1∂f1∂z¯l∂f2∂zk′hs′j′¯δp¯l¯+f1∂f1∂z¯l12∂hk′l′¯∂zs′+∂hs′l′¯∂zk′δj′¯l′¯+∂hs′j′¯∂zk′δp¯l¯,Kk′j¯sp′¯=f12hk′p′¯∂2lnf1∂zs∂z¯j+12f1hk′p′¯gil¯∂f1∂zi∂gsl¯∂z¯j−∂gsj¯∂z¯l+f2gsj¯∂ht′l′¯∂zk′∂f2∂z¯l′ht′p′¯+δp′¯l′¯∂2f2∂zk′∂z¯l′+12hi′l′¯δp′¯l′¯∂f2∂z¯l′∂hk′l′¯∂zi′+∂hi′l′¯∂zk′,Kkj′¯s′p¯=f22gkp¯∂2lnf2∂zs′∂z¯j′+12f2gkp¯hi′l′¯∂f2∂zi′∂hs′l′¯∂z¯j′−∂hs′j′¯∂z¯l′+f1hs′j′¯∂gtl¯∂zk∂f1∂z¯lgtp¯+δp¯l¯∂2f1∂zk∂z¯l+12gil¯δp¯l¯∂f1∂z¯l∂gkl¯∂zi+∂gil¯∂zk.Kk′j¯s′p¯=Kk′j¯sp¯=Kk′j¯s′p′¯=Kkj′¯sp¯=Kkj′¯sp′¯=Kkj′¯s′p′¯=0.Proof.
According to (10), we get
(34)Kkj¯sp¯=Gεp¯Kkj¯sε=Gtp¯Kkj¯st+Gt′p¯Kkj¯st′.
Substituting (20), (27), and (15) into (34), we have
(35)Kkj¯sp¯=Gtp¯Kkj¯st1+f1−2hi′l′¯∂f2∂z¯l′∂f2∂zi′gsj¯δkt=f22Kkj¯sp¯1+f1−2f22hi′l′¯∂f2∂z¯l′∂f2∂zi′gsj¯gkp¯.
Similarly, we can obtain other equations of Proposition8.Proposition 9.
Letf2M1×f1M2,G be a DWP-Hermitian manifold of M1,g and M2,h. Then, the coefficients of the first Levi-Civita Ricci curvature Kαβ¯1 are given by
(36)Kkj¯1=Kkj¯11+2f1−2∂f2∂z¯l′∂f2∂zi′gkj¯hi′l′¯,Kk′j′¯1=Kk′j′¯12+2f2−2∂f1∂z¯l∂f1∂zshk′j′¯gsl¯,Kk′j¯1=0,Kkj′¯1=0.where Kkj¯11 and Kk′j′¯12 are coefficients of the first Levi-Civita Ricci curvature of g and h, respectively.Proof.
From (9) and (16), we get
(37)Kkj¯1=Gγδ¯Kkj¯γδ¯=Gsp¯Kkj¯sp¯+Gs′p′¯Kkj¯s′p′¯.
According to (16) and the first equation of proposition 8, we have
(38)Gsp¯Kkj¯sp¯=f2−2gsp¯f22Kkj¯sp¯1+f1−2f22hi′l′¯∂f2∂z¯l′∂f2∂zi′gsj¯gkp¯=Kkj¯11+f1−2∂f2∂z¯l′∂f2∂zi′gkj¯hi′l′¯.
Similarly, by using (16) and the third equation of proposition 8, we can get
(39)Gs′p′¯Kkj¯s′p′¯=f1−2∂f2∂z¯l′∂f2∂zs′gkj¯hs′l′¯.
Replacing the summation indexi′ on the right side of (38) with s′ and then taking it and (39) into (37), we can obtain
(40)Kkj¯1=Kkj¯11+2f1−2∂f2∂z¯l′∂f2∂zs′gkj¯hs′l′¯.
Similarly, we can obtain(41)Kk′j′¯1=Kk′j′¯12+2f2−2∂f1∂z¯l∂f1∂zshk′j′¯gsl¯,Kk′j¯1=0,Kkj′¯1=0.
This completes the proof.Theorem 10.
Letf2M1×f1M2,G be a DWP-Hermitian manifold of M1,g and M2,h. Then, the Levi-Civita Ricci scalar curvature of G along a nonzero vector v=vi,vi′∈Tz1,0M is given by
(42)SLCv=f2−2Sgv1+f1−2Shv2+2f1−2f2−2gsl¯∂f1∂z¯l∂f1∂zs+2f1−2f2−2hs′l′¯∂f2∂z¯l′∂f2∂zs′,where Sgv1 and Shv2 are Levi-Civita Ricci scalar curvatures of g and h, respectively.Proof.
According to (11), the Levi-Civita Ricci scalar curvature of G is given by
(43)SLC=Gαβ¯Kαβ¯1=Gkj¯Kkj¯1+Gk′j′¯Kk′j′¯1+Gk′j¯Kk′j¯1+Gkj′¯Kkj′¯1.
Combining (16) and (40), we have
(44)Gkj¯Kkj¯1=f2−2gkj¯Kkj¯11+2f1−2∂f2∂z¯l′∂f2∂zs′gkj¯hs′l′¯=f2−2Sgv1+2f1−2f2−2hs′l′¯∂f2∂z¯l′∂f2∂zs′.
Similarly, we can get(45)Gk′j′¯Kk′j′¯1=f1−2Shv2+2f1−2f2−2gsl¯∂f1∂z¯l∂f1∂zs,(46)Gk′j¯Kk′j¯1=0,(47)Gkj′¯Kkj′¯1=0.
Taking (44)–(47) into (43), we obtain (42).Theorem 11.
Letf2M1×f1M2,G be a DWP-Hermitian manifold of M1,g and M2,h. If f1 and f2 are holomorphic functions on M1 and M2, respectively, then SLCv=f2−2Sgv1+f1−2Shv2.Proof.
Iff1 and f2 are holomorphic functions on M1 and M2, respectively, i.e.,
(48)∂f1∂z¯l=0,∂f2∂z¯l′=0.
Thus,(49)2f1−2f2−2gsl¯∂f1∂z¯l∂f1∂zs=0,(50)2f1−2f2−2hs′l′¯∂f2∂z¯l′∂f2∂zs′=0.
Substituting (49) into (42), we have SLCv=f2−2Sgv1+f1−2Shv2.


## 4. Levi-Civita Ricci-Flat Doubly Warped Product Hermitian Manifolds

LetM1,g and M2,h be two Levi-Civita Ricci-flat Hermitian manifolds; one may want to know whether the DWP-Hermitian manifold f2M1×f1M2,G is also a Levi-Civita Ricci-flat Hermitian manifold. We shall give an answer to this question in this section.Theorem 12.
Letf2M1×f1M2,G be a DWP-Hermitian manifold of M1,g and M2,h. If f1 and f2 are holomorphic functions on M1 and M2, respectively, then f2M1×f1M2,G is Levi-Civita Ricci-flat if and only if M1,g and M2,h are Levi-Civita Ricci-flat.Proof.
Iff1 and f2 are holomorphic functions on M1 and M2, respectively, i.e.,
(51)∂f1∂z¯l=0,∂f2∂z¯l′=0.
Taking above equations into the first formula and second formula of (36), we get
(52)2f1−2∂f2∂z¯l′∂f2∂zi′gkj¯hi′l′¯=0,(53)2f2−2∂f1∂z¯l∂f1∂zshk′j′¯gsl¯=0.
Firstly, we assumef2M1×f1M2,G be Levi-Civita Ricci-flat; using Definition 2 and (36), we have
(54)Kkj¯1=Kkj¯11+2f1−2∂f2∂z¯l′∂f2∂zi′gkj¯hi′l′¯=0,Kk′j′¯1=Kk′j′¯12+2f2−2∂f1∂z¯l∂f1∂zshk′j′¯gsl¯=0,Kk′j¯1=0,Kkj′¯1=0.
Substituting (52) and (53) into the first formula and second formula of (54), respectively, we get
(55)Kkj¯1=Kkj¯11=0,Kk′j′¯1=Kk′j′¯12=0,Kk′j¯1=0,Kkj′¯1=0.
Obviously,(56)Kkj¯11=0,Kk′j′¯12=0.
According to Definition2, these mean that M1,g and M2,h are Levi-Civita Ricci-flat.
Conversely, we assumeM1,g and M2,h are Levi-Civita Ricci-flat; according to Definition 2, we know that
(57)Kkj¯11=0,(58)Kk′j′¯12=0.
Sincef1 and f2 are holomorphic, thus (52) and (53) are established. Then, taking (52), (53), (57), and (58) into (36), we obtain
(59)Kkj¯1=0,Kk′j′¯1=0,Kk′j¯1=0,Kkj′¯1=0.
By Definition2, (59) indicates that f2M1×f1M2,G is Levi-Civita Ricci-flat.Notation 13.
Theorem12 implies that when warped functions to be holomorphic, then the DWP-Hermitian manifold is a Levi-Civita Ricci-flat Hermitian manifold if and only if its component manifolds are Levi-Civita Ricci-flat. Thus, this theorem provides us an effective way to construct Levi-Civita Ricci-flat DWP-Hermitian manifold.


## 5. Conclusions

In this paper, we derived formulae of Levi-Civita connection, Levi-Civita curvature, the first Levi-Civita Ricci curvature, and Levi-Civita scalar curvature of the DWP-Hermitian manifold and proved that if the warped functionf1 and f2 are holomorphic, then the DWP-Hermitian manifold is Levi-Civita Ricci-flat if and only if M1,g and M2,h are Levi-Civita Ricci-flat manifolds. Thus, we gave an effective way to construct Levi-Civita Ricci-flat DWP-Hermitian manifold.



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*Source: 2077040-2022-07-05.xml*