1. Introduction
Warped product and twisted product are important methods used to construct new classes of geometric spaces, and these models are widely applied in theoretical physics. In 1969, warped product was firstly introduced by O’Neill and Bishop to construct Riemannian manifolds with negative sectional curvature [
1]. In 2001, Kozma, Peter and Varga [
2] extended the warped product to real Finsler manifolds. Asanov [
3,
4] obtained some models of relativity theory by studying the warped product Finsler metric. In 2018, the notion of warped product was extended to Hermitian geometry by the work of He and Zhang [
5], and they obtained the necessary and sufficient conditions for the compact nontrivial doubly warped product (abbreviated as DWP) Hermitian manifold to have constant holomorphic sectional curvature.
The notion of twisted product, as a generalization of warped product, was first introduced by Chen [
6]. In 1993, Ponge and Reckziegel [
7] extended twisted product to pseudo-Riemannian manifolds. Then, Fernández-López showed that a mixed Ricci-flat twisted product semi-Riemannian manifold can be expressed as a warped product semi-Riemannian manifold [
8]. In 2017, Kazan and Sahin [
9] deeply investigated the twisted product and multiply twisted product semi-Riemannian manifolds, which further promoted the development of twisted product in Riemannian geometry. Kozma, Peter and Shimada [
10] extended the twisted product to real Finsler manifolds and studied some geometric properties relating to Cartan connection, geodesic and completeness. Recently, Xiao and He [
11] extended the twisted product to complex Finsler manifolds and gave the formulae of holomorphic curvature and Ricci scalar curvature of the doubly twisted product (abbreviated as DTP) complex Finsler manifold. In light of the above results, we shall extend the twisted product to Hermitian manifold, and attempt to derive the Chern curvature, Chern Ricci curvature, Chern Ricci scalar curvature and holomorphic sectional curvature of the twisted product Hermitian manifold. In addition, we intend to find the necessary and sufficient conditions for the compact Hermitian manifold to have constant holomorphic sectional curvature.
One of the most important problems in geometry is to characterize Chern flat or Chern Ricci-flat manifolds. In 1967, Tani [
12] firstly gave the definition of Ricci-flat space in Riemannian geometry. Later, Bando and Kobayashi [
13] constructed Ricci-flat metrics on Einstein-Kähler manifolds. Liu and Yang [
14] obtained the sufficient and necessary conditions for the Hopf manifold to be Levi-Civita Ricci-flat. Recently, Ni and He [
15] gave the necessary and sufficient conditions for DWP-Hermitian manifold to be Levi-Civita Ricci-flat. In 2012, Di Scala [
16] showed that quasi-Kähler Chern flat almost Hermitian structures on compact manifolds correspond to complex parallelizable Hermitian structures satisfying the second Gray identity. Wu and Zheng [
17] proved that the compact Hermitian manifold with complex dimension 3, having vanishing real bisectional curvature, must be Chern flat. Based on the above mentioned studies, we are interested in the condition under which the twisted product Hermitian manifold is Chern flat or Chern Ricci-flat.
The structure of this paper is as follows. In
Section 2, we briefly recall some basic concepts of Hermitian geometry and related symbolic conventions. In
Section 3, we shall extend the concept of twisted product to Hermitian geometry, and derive the Chern connection coefficients of a twisted product Hermitian manifold. In
Section 4, we shall give the formulae of Chern curvature, Chern Ricci curvature and Chern Ricci scalar curvature of the twisted product Hermitian manifold. In
Section 5, we focus on investigating the twisted product Hermitian manifold with constant holomorphic sectional curvature. In
Section 6, under the condition that the logarithm of the twisted function is pluriharmonic, we shall show that the twisted product Hermitian manifold is Chern flat or Chern Ricci-flat if and only if
and
are Chern flat or Chern Ricci-flat, respectively.
2. Preliminary
In this section, we briefly introduce the definitions and notations which we need in this paper.
Let
be a n-dimensional Hermitian manifold with complex structure
J and Hermitian metric
G. Let
denote the complexified tangent bundle of
M, which can be decomposed as
where
and
are eigenspaces of
J corresponding to eigenvalues
and
, respectively.
Let
denote the local holomorphic coordinates on
M, then vector fields
and
form the basis of
and
, respectively, where
. On the Hermitian holomorphic tangent bundle
, the coefficients of Chern connection ∇ are [
18]
and their complex conjugate.
Definition 1 ([
18])
. Let ∇
be the Chern connection, its Chern curvature tensor K on the Hermitian manifold is defined bywhere Definition 2 ([
14])
. The first and the second Chern Ricci curvature on the Hermitian manifold are defined byrespectively, where Definition 3 ([
14])
. The Chern Ricci scalar curvature on the Hermitian manifold is defined by For research purposes, we introduce the following two definitions.
Definition 4 ([
19])
. Let D be open in . A function is said to be pluriharmonic if it satisfies the differential equations Definition 5 ([
20])
. The complex Laplace operatoris a second-order elliptic partial differential operator with smooth coefficients. Clearly, if f is a pluriharmonic function, then .
3. Twisted Product Hermitian Manifold
Let and be two Hermitian manifolds with and , respectively, then is a Hermitian manifold with .
We denote and , so . Let be the natural projection maps, then .
Let and be the holomorphic tangent bundle of and , respectively. Denote and , then . Let be the holomorphic tangent maps induced by and , then , , where z is called the base coordinates (or points) on M and v is called the fiber coordinates (or tangent directions).
For the reader’s convenience, the lowercase Greek indices like ,⋯ run from 1 to m + n, the lowercase Latin indices like run from 1 to m, while the lowercase Latin indices with a prime like run from m + 1 to m + n. Quantities associated with and are denoted with upper indices 1 and 2, respectively; for example, , are Chern connection coefficients of and , respectively. In the following, we use the Einstein summation convention.
Definition 6. Let and be two Hermitian manifolds. Let be a positive smooth function. The twisted product Hermitian manifold is the product manifold endowed with the Hermitian metric :for and . The function f is called the twisted function and G is called the twisted product Hermitian metric for simplicity. In particular, if f only depends on , then is a warped product Hermitian manifold. If f only depends on , then is the product Hermitian manifold.
Denote
Then, the fundamental tensor matrix
of
G has the following forms
its inverse matrix
is also given by
Proposition 1. Let be a twisted product Hermitian manifold. Then, the Chern connection coefficients associated with G are given by Proof. By putting
in (1), we have
Plunging (13) and (14) into (17), we can obtain
Similarly, the other equalities of Proposition 1 can be deduced. □
4. Curvatures of Twisted Product Hermitian Manifold
In this section, we shall derive the Chern curvature, Chern Ricci curvature and Chern Ricci scalar curvature of the twisted product Hermitian manifold.
Proposition 2. Let be a twisted product Hermitian manifold. Then, the coefficients of Chern curvature tensor are given by Proof. By putting
in (4), we have
Substituting the second equality of (15) into (22), and using (4), we have
Similarly, we can obtain other equalities of Proposition 2. □
Proposition 3. Let be a twisted product Hermitian manifold. Then, Proof. By putting
in (3), we have
Plunging (13) and the second equality of (18) into (30), a trivial caculation yields
Similarly, we can obtain other equalities of Proposition 3. □
Proposition 4. Let be a twisted product Hermitian manifold. Then, the coefficients of the first and the second Chern Ricci curvature tensor are given byand Proof. Letting
in (6), we have
Substituting (14), (24) and (27) into (35), and noticing that (10), we can obtain
Similarly, we can obtain other equalities of Proposition 4. □
Theorem 1. Let be a twisted product Hermitian manifold. Then, the Chern Ricci scalar curvature of G along a nonzero vector is given by Proof. According to (8), we have
Substituting (14), (31) and (32) into (37), after a straightfoward computation, we see that
Thus, we complete the proof. □
According to Definitions 4 and 5, we can obtain the following.
Corollary 1. Let be a twisted product Hermitian manifold. Suppose is a pluriharmonic function, then
5. Holomorphic Sectional Curvature of Twisted Product Hermitian Manifold
In this section, we would like to derive the holomorphic sectional curvature of the twisted product Hermitian manifold, and give the necessary and sufficient conditions for the compact twisted product Hermitian manifold to have constant holomorphic sectional curvature.
Definition 7 ([
21])
. Let be a Hermitian manifold. Then, the holomorphic sectional curvature of G along a nonzero vector is defined by Theorem 2. Let be a twisted product Hermitian manifold. Then, the holomorphic sectional curvature of G along a nonzero vector is given by Proof. According to (28), (29) and (38), we have
Using (27) and noting that
, we have
Similarly, we can obtain
Plunging (41)–(45) into (40), we can obtain (39). □
According to Definition 4, we can easily obtain
Corollary 2. Let be a twisted product Hermitian manifold. Suppose is a pluriharmonic function, then Theorem 3 ([
21])
. Let be a compact Hermitian manifold. Then, M has constant holomorphic sectional curvature κ if and only if, at every point of M,where Proposition 5. Let be a twisted product Hermitian manifold. Then, Proof. By putting
in (47), we have
By using (27), we obtain
Similar calculations give the rest of the equalities of Proposition 5. □
Theorem 4. Let be a compact twisted product Hermitian manifold. Then, G has constant holomorphic sectional curvature κ if and only if and the following equalities hold Proof. According to Theorem 3, (13) and (53),
has constant holomorphic sectional curvature if and only if
Substituting (13) and (48)–(52) into (56a)–(56e), and noticing that
, (56a)–(56e) are thus equivalent to the following equalities
The above equalities are equivalent to
In fact, contracting (57b) with
and
successively, and noticing that
, we can obtain (58b). Contracting (57c) and (57d) with
, respectively, we can obtain (58c) and (58d). Contracting (57e) with
and
successively, and noticing that
, we can obtain (58e).
Proof of the necessity.
Let us suppose that
, combining (58a) and (58b), we have
since
,
depend only on
, which says that
f only depends on
. These are contradicted by the fact that
is a twisted product Hermitian manifold. Thus,
Plunging (60) into (58a), (58b) and (58e), we can check that (58a)–(58e) can be simplified as (55a)–(55e).
Next, we prove the sufficiency.
Suppose that and (55a)–(55e) hold; this immediately confirms that (57a)–(57e) hold, i.e., has constant holomorphic sectional curvature . Thus, we complete the proof. □
6. Chern Flat and Chern Ricci-Flat Twisted Product Hermitian Manifolds
Let and be two Chern flat or Chern Ricci-flat Hermitian manifolds, respectively. We would like to know under what conditions the twisted product Hermitian manifold is Chern flat or Chern Ricci-flat.
Definition 8 ([
22])
. A Hermitian manifold is called Chern flat ifwhere K is the Chern curvature tensor. Definition 9 ([
22])
. A Hermitian manifold is called Chern Ricci-flat ifwhere is the first Chern Ricci curvature tensor. Theorem 5. Let be a twisted product Hermitian manifold. Suppose is pluriharmonic, then is Chern flat if and only if and are Chern flat.
Proof. Since
is pluriharmonic, then
According to Definition 8 and (2),
is Chern flat if and only if
Using Proposition 3 and (61)–(64), and noticing that
, (65) is equivalent to following equalities
which means that
and
are Chern flat. □
Theorem 6. Let be a twisted product Hermitian manifold. Suppose is pluriharmonic, then is Chern Ricci-flat if and only if and are Chern Ricci-flat.
Proof. Suppose that
is pluriharmonic, then
By Definition 9 and (5),
is Chern Ricci flat if and only if
Using (31), (32) and (67), (68) is equivalent to the following equalities
Which means that
and
are Chern Ricci flat. □
7. Conclusions
In this paper, we extended the twisted product to Hermitian manifold. Based on this, we confirmed that the compact twisted product Hermitian manifold has constant holomorphic sectional curvature if and only if and a system of differential equations holds. Under the condition that the logarithm of the twisted function is pluriharmonic, we obtained the necessary and sufficient conditions for the twisted product Hermitian manifold to be Chern flat or Chern Ricci-flat, respectively, so then we gave an effective way to construct Chern flat or Chern Ricci-flat Hermitian manifolds.