Locally Conformally Flat Doubly Twisted Product Complex Finsler Manifolds

Xiao, Wei; He, Yong; Li, Shuwen; Ni, Qihui

doi:https://doi.org/10.1155/2022/5129687

Journal of Mathematics

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Research Article | Open Access

Volume 2022 | Article ID 5129687 | https://doi.org/10.1155/2022/5129687

Locally Conformally Flat Doubly Twisted Product Complex Finsler Manifolds

Wei Xiao,¹Yong He,¹Shuwen Li,¹and Qihui Ni¹

Academic Editor: Bibhas Ranjan Majhi

Received27 Mar 2022

Revised28 Apr 2022

Accepted10 May 2022

Published10 Jun 2022

Abstract

Let be a doubly twisted product manifold of two strongly pseudoconvex complex Finsler manifolds and . 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.

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.

Let be a complex manifold of complex dimension . We denote as the holomorphic tangent bundle of and as the complement of zero section in . We denote as the local holomorphic coordinates on and as the induced local holomorphic coordinates on the holomorphic tangent bundle .

Definition 1. (see [33]). A strongly pseudoconvex complex Finsler metric on a complex manifold is a continuous function satisfying(i) is smooth on .(ii), for all .(iii), for all and .(iv)The Levi matrix (or complex Hessian matrix), is positive definite on .

In this study, we denote the inverse matrix of such that . We also use the notion in [33], that is, the derivatives of with respect to the -coordinates and -coordinates are separated by semicolon; for instance,

In the following, we denote as the partial derivative with respect to the local coordinates on M and as the partial derivative with respect to the fiber coordinates .

The Chern–Finsler connection was first constructed in [34] and systemically studied in [33]. Let be the Chern–Finsler connection associated to a strongly pseudoconvex complex Finsler metric . The Chern–Finsler complex nonlinear connection associated to is given by

The connection 1-forms of are given bywhereand

The complex Rund connection associated to a strongly pseudoconvex complex Finsler metric was first introduced in [35] and were systemically studied in [36, 37]. Let be the complex Rund connection; then, the connection 1-forms of are given bywhere are defined by equation (5). It is clearly from equation (4) that are just the horizontal part of .

Definition 2. (see [37, 38]). A complex Finsler manifold is said to be modeled on a complex Minkowski space if the horizontal connection coefficients of the Chern–Finsler connection or complex Rund connection coefficients depend only on the coordinates of the base manifold , i.e., .

Definition 3. (see [36]). Let be a complex Finsler metric on . If there exits an open cover such that, on each , the function is a function of the fiber coordinate only, then the complex Finsler metric will be a complex locally Minkowski metric.

The necessary and sufficient condition of a complex Finsler metric on to be a complex locally Minkowski metric is that it is modeled on a complex Minkowski metric and the complex Rund connection coefficients on is holomorphic. That is, the horizontal coefficients of the Chern–Finsler connection or complex Rund connection coefficients satisfy the following conditions:

Definition 4. (see [38]). Let and be two strongly pseudoconvex complex Finsler metric on complex manifold . A conformal change of is the change for a smooth function on .

Definition 5. (see [38]). A complex Finsler manifold is said to be locally conformally flat if it is locally conformal to a locally Minkowski space; that is, there exhibits an open covering with a family of locally defined smooth functions such that the metric is a complex locally Minkowski metric on .

3. Doubly Twisted Product of Complex Finsler Manifolds

Let and be two strongly pseudoconvex complex Finsler manifolds with and ; then, is a strongly pseudoconvex complex Finsler manifold with .

Let and be the holomorphic tangent bundles of and , respectively. Let and be natural projection maps; then, and be the holomorphic tangent maps induced by and , respectively. Note that and for every with and .

Definition 6. (see [21]). Let and be two strongly pseudoconvex complex Finsler manifolds and be smooth functions. The doubly twisted product (abbreviated as DTP) complex Finsler manifold of and is the product complex manifold endowed with the complex Finsler metric given byfor and . The functions and are called twisted functions. The DTP-complex Finsler manifold of and is denoted by .

In the case of or , the corresponding DTP-complex Finsler manifold is called twisted product complex Finsler manifold. If , becomes the product complex Finsler manifold. If neither nor is a constant, then we call a nontrivial (proper) DTP-complex Finsler manifold of and .

Notation: lowercase Greek indices such as will run from 1 to , whereas lowercase Latin indices such as will run from 1 to , and lowercase Latin indices with a prime such as will run from to . Quantities associated to and are denoted with upper indices 1 and 2, respectively, such as .

Denote , andso equation (9) is equal to

Proposition 1. (see [21]). Let be a doubly twisted product manifold of complex Finsler manifolds and . Then, the fundamental tensor matrix of is given bywith its inverse matrix given by

Lemma 1. (see [21]). Let be a doubly twisted product manifold of two strongly pseudoconvex complex Finsler manifolds and . Then, the Chern–Finsler complex nonlinear connection coefficients associated to are given by

Proposition 2. (see [21]). Let be a doubly twisted product manifold of two strongly pseudoconvex complex Finsler manifolds and ; the horizontal coefficients of the Chern–Finsler connection associated to are given by

4. Locally Conformally Flat Doubly Twisted Product Complex Finsler Manifolds

Let be a complex Finsler metric on . A conformal transformation of is a change , where is a smooth real function. We denote by and use the symbol “” to mark the geometric objects associated to the complex Finsler metric , e.g., are the horizontal coefficients of Chern–Finsler connection associate to .

Let and be locally conformally flat complex Finsler manifolds. In this section, we shall give the necessary and sufficient conditions for a doubly twisted product manifold of and to be locally conformally flat.

Proposition 3. Let be a doubly twisted product manifold of two strongly pseudoconvex complex Finsler manifolds and , and is conformal transformation of . Then, the coefficients of Chern–Finsler connection associated to are given bywhere is a smooth real function.

Proof. Using equation (3), we obtainAccording to Lemma 1, we haveSimilarly, we can get other equations of Proposition 3.

Proposition 4. Let be a doubly twisted product manifold of two strongly pseudoconvex complex Finsler manifolds and , and is conformal transformation of . Then, the horizontal coefficients of the Chern–Finsler connection associated to are given bywhere is a smooth real function.

Proof. According to equation (5) and Proposition 2, by straightforward computation, we obtainSimilarly, we can obtain other equations of Proposition 4.

Theorem 1. Let be a doubly twisted product manifold of two strongly pseudoconvex complex Finsler manifolds and ; then, is locally conformally flat if and only if there exists an open covering and a family of smooth functions such that

Proof. According to Definition 5, is locally conformally flat if and only if there exists an open covering and a family of smooth functions on such that is a locally Minkowski metric on . Notice that the complex Finsler metric 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 is holomorphic. Thus, we obtainSubmit the equations of Proposition 4 into equations (22) and (23); after a straightforward computation, we haveNoticing that , so if and only if . Similarly, we know that if and only if .
Thus, we can obtain equation (21).

Corollary 1. Let be a twisted product manifold of two strongly pseudoconvex complex Finsler manifolds and ; then, is locally conformally flat if and only if is locally conformally flat, and there exists an open covering and a family of smooth functions such that

Proof. Since be a twisted product complex Finsler manifold, soBy plunging equation (26) into equation (21), we obtainFrom equation (29) and noticing , we obtain that is a locally conformally flat manifold.
Submit equations (31) into equations (28) and (30), respectively; we can obtain equation (25).

Theorem 2. Let be a doubly twisted product manifold of two locally conformally flat complex Finsler manifolds and ; then, is locally conformally flat if and only if and are pluriharmonic functions on and , respectively, and there exists an open covering and a family of smooth functions such that

Proof. According to Theorem 1, is locally conformally flat if and only if equation (21) holds on .
Since and are locally conformally flat complex Finsler manifolds, thus,Plunging equations (33) and (34) into equation (21), equation (21) can be simplified toEquations (35) and (36) mean that and are pluriharmonic functions on and , respectively. Therefore, is locally conformally flat if and only if and are pluriharmonic functions on and , respectively, and equation (32) holds on .

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 and are pluriharmonic functions on and , 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.

Data Availability

Previously reported data were used to support this study and are available at 10.4208/jms.v55n2.22.04.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by National Natural Science Foundation of China (Grant nos. 11761069 and 12061077).

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Copyright © 2022 Wei Xiao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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