Levi-Civita Ricci-Flat Doubly Warped Product Hermitian Manifolds

Abstract

Let and be two Hermitian manifolds. The doubly warped product (abbreviated as DWP) Hermitian manifold of and is the product manifold endowed with the warped product Hermitian metric , where and are positive smooth functions on and , 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 and are holomorphic, then the DWP-Hermitian manifold is Levi-Civita Ricci-flat if and only if and are Levi-Civita Ricci-flat manifolds. Thus, we give an effective way to construct Levi-Civita Ricci-flat DWP-Hermitian manifold.

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-Khler 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 and 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. Let and 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 Section 2, 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 and are holomorphic, then the DWP-Hermitian manifold is Levi-Civita Ricci-flat if and only if and are Levi-Civita Ricci-flat manifolds.

2. Preliminary

Let be a Hermitian manifold with ; here, is the complex structure, and is a Hermitian metric. For a point , the complexified tangent bundle is decomposed as where and are the eigenspaces of corresponding to the eigenvalues and , respectively.

In this paper, we set and . Let be the local holomorphic coordinates on ; then, the vector fields form a basis for . Levi-Civita connection on the holomorphic tangent bundle is defined by [18]

In local coordinate system, its connection is as follows [18]: where the Levi-Civita connection coefficients and are given by [18]

Let be the Levi-Civita curvature tensor such as where . In the local coordinate system, the coefficients of are given by

Definition 1 (see [6]). The first Levi-Civita Ricci curvature on the Hermitian manifold is defined by where Levi-Civita Ricci scalar curvature on is given by

Definition 2 (see [6]). Hermitian metric on is called Levi-Civita Ricci-flat if Let and be two Hermitian manifolds with and ; then, is a Hermitian manifold with .
Denote and the natural projections. Note that and for every with and .
Denote the holomorphic tangent maps induced by and , respectively. Note that and for every with and .

Definition 3 (see [15]). Let and be two Hermitian manifolds. and be two positive smooth functions. The doubly warped product (abbreviated as DWP) Hermitian manifold is the product Hermitian manifold endowed with the Hermitian metric defined by for and . and are warped functions; the DWP-Hermitian manifold of and is denoted by .

If either or , then becomes a warped product of Hermitian manifolds and . If and , then becomes a product of Hermitian manifolds and . If neither nor is constant, then we call a nontrivial DWP-Hermitian manifolds of and .

Notation 4. Lowercase Greek indices such as , , and will run from to , lowercase Latin indices such as , , and will run from to , and lowercase Latin indices with a prime, such as , , and , will run from to . Quantities associated to and are denoted with upper indices and , respectively, such as and are Levi-Civita connection coefficients of and , respectively.
Denote The fundamental tensor matrix of is given by and its inverse matrix is given by

Proposition 5. Let be a DWP-Hermitian manifold of and . Then, the Levi-Civita connection coefficients associated to are given by

Proof. Substituting (15) and (16) into (4), we obtain Similarly, we can obtain other equations of Proposition 5.

Plugging (15) and (16) into (5), we have the following proposition.

Proposition 6. Let be a DWP-Hermitian manifold of and . Then, the Levi-Civita connection coefficients associated to are given by

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. Let be a DWP-Hermitian manifold of and . Then, the coefficients of Levi-Civita curvature tensor are given by

Proof. Using (7), we have Taking the formulae of Proposition 5 and Proposition 6 into (31), we obtain Similarly, we can obtain other equations of Proposition 7.

Proposition 8. Let be a DWP-Hermitian manifold of and . Then,

Proof. According to (10), we get Substituting (20), (27), and (15) into (34), we have Similarly, we can obtain other equations of Proposition 8.

Proposition 9. Let be a DWP-Hermitian manifold of and . Then, the coefficients of the first Levi-Civita Ricci curvature are given by where and are coefficients of the first Levi-Civita Ricci curvature of and , respectively.

Proof. From (9) and (16), we get According to (16) and the first equation of proposition 8, we have Similarly, by using (16) and the third equation of proposition 8, we can get Replacing the summation index on the right side of (38) with and then taking it and (39) into (37), we can obtain Similarly, we can obtain This completes the proof.

Theorem 10. Let be a DWP-Hermitian manifold of and . Then, the Levi-Civita Ricci scalar curvature of along a nonzero vector is given by where and are Levi-Civita Ricci scalar curvatures of and , respectively.

Proof. According to (11), the Levi-Civita Ricci scalar curvature of is given by Combining (16) and (40), we have Similarly, we can get Taking (44)–(47) into (43), we obtain (42).

Theorem 11. Let be a DWP-Hermitian manifold of and . If and are holomorphic functions on and , respectively, then .

Proof. If and are holomorphic functions on and , respectively, i.e., Thus, Substituting (49) into (42), we have .

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

Let and be two Levi-Civita Ricci-flat Hermitian manifolds; one may want to know whether the DWP-Hermitian manifold is also a Levi-Civita Ricci-flat Hermitian manifold. We shall give an answer to this question in this section.

Theorem 12. Let be a DWP-Hermitian manifold of and . If and are holomorphic functions on and , respectively, then is Levi-Civita Ricci-flat if and only if and are Levi-Civita Ricci-flat.

Proof. If and are holomorphic functions on and , respectively, i.e., Taking above equations into the first formula and second formula of (36), we get Firstly, we assume be Levi-Civita Ricci-flat; using Definition 2 and (36), we have Substituting (52) and (53) into the first formula and second formula of (54), respectively, we get Obviously, According to Definition 2, these mean that and are Levi-Civita Ricci-flat.
Conversely, we assume and are Levi-Civita Ricci-flat; according to Definition 2, we know that Since and are holomorphic, thus (52) and (53) are established. Then, taking (52), (53), (57), and (58) into (36), we obtain By Definition 2, (59) indicates that is Levi-Civita Ricci-flat.

Notation 13. Theorem 12 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 function and are holomorphic, then the DWP-Hermitian manifold is Levi-Civita Ricci-flat if and only if and are Levi-Civita Ricci-flat manifolds. Thus, we gave an effective way to construct Levi-Civita Ricci-flat DWP-Hermitian manifold.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grant Nos. 11761069 and 12061077).