On Kähler-like and G-Kähler-like almost Hermitian manifolds

Abstract We introduce Kähler-like, G-Kähler-like metrics on almost Hermitian manifolds. We prove that a compact Kähler-like and G-Kähler-like almost Hermitian manifold equipped with an almost balanced metric is Kähler. We also show that if a Kähler-like and G-Kähler-like almost Hermitian manifold satisfies B i¯j¯λBλji≥0 B_{\bar i\bar j}^\lambda B_{\lambda j}^i \ge 0 , then the metric is almost balanced and the almost complex structure is integrable, which means that the metric is balanced. We investigate a G-Kähler-like almost Hermitian manifold under some assumptions.


Introduction
The almost Hermitian geometry has been studied vigorously in last years such as in [14], [15], [16], [24] and [26]. We introduce the notion of a Kähler-like almost Hermitian metric, a G-Kähler-like almost Hermitian metric and an almost balanced metric and investigate compact almost Hermitian manifolds equipped with these three conditions. In the Hermitian case, Yang and Zheng examined the Hermitian curvature tensors of Hermitian metrics, as the curvature tensors satis es all the symmetry conditions of the curvature tensor of a Kähler metric in [23]. They called these metrics Kähler-like. When a manifold is compact, these metrics are more special than balanced metrics since they are always balanced, that is, d(ω n− ) = , where ω is the fundamental -form associated to a Hermitian metric and n is the complex dimension of the manifold. This fact has attracted attention in the reserch of non-Kähler Calabi-Yau manifolds. Their de nitions are as follows. Given a Hermitian manifold (M n , J, g), there are two canonical connections associated to g, the Chern connection ∇ and the Levi-Civita connection D. Denote R and R L the curvature tensor of these two connections respectively. Notice that in this whole paper, in the almost Hermitian case M n indicates that n = dim R M, in the Hermitian case M n means that n = dim C M.
De nition 1.1. (Kähler-like and G-Kähler-like [23]) A Hermitian metric g will be called Kähler-like, if R XȲ ZW = R ZȲ XW holds for any type ( , ) tangent vectors X, Y, Z and W. Similarly, if R L XYZW = R L XYZW = for any type ( , ) tangent vectors X, Y, Z and W, we will say that g is Gray-Kähler-like, or G-Kähler-like for short.
The G-Kähler-like condition was rstly introduced by Gray in [11]. Yang and Zheng showed that when R = R L , then g is Kähler in [23,Theorem 1.1], and they also showed that when the manifold is compact, either condition, the Kähler-likeness or the G-Kähler-likeness, would imply that the metric is balanced. In this sense, the Kähler-likeness and G-Kähler-likeness are more special classes of Hermitian metrics than satis es dω = . It is important for us to study quasi-Kähler manifolds since they include the classes of almost Kähler manifolds and nearly Kähler manifolds. Recall that an almost Hermitian manifold (M, J, g) is said to be nearly Kähler if (D X J)X = for any tangent vector eld X and DJ ≠ , where D is the Levi-Civita connection associated to g. An almost Kähler or quasi-Kähler manifold with J integrable is a Kähler manifold. We have shown the following result on compact Kähler-like almost Hermitian manifolds. Proposition 1.3. ([16, Theorem 1.1]) Let (M n , J, g) be a compact Kähler-like almost Hermitian manifold with n ≥ . If M n admits a positive ∂∂-closed (n − , n − )-form χ, then g is quasi-Kähler. In particular, if M n is compact, Kähler-like, and ∂∂(ω n− ) = , then g is quasi-Kähler. When n = , compactness implies that any Kähler-like metric is almost Kähler.
Note that in dimension , every quasi-Kähler manifold is almost Kähler. In general, there are known examples of quasi-Kähler manifolds which are not almost Kähler. The canonical example of a quasi-Kähler Chern-at manifold is the Iwasawa manifold M equipped with the almost Hermitian structure associated to the almost complex structure J de ned in [1]. The structure J is the unique invariant almost complex structure on M such that its associated fundamental form ω with respect to a canonical metric of M is quasi-Kähler and not-symplectic (cf. [7]).
Yang and Zheng raise the natural question of whether a Kähler-like and G-Kähler-like Hermitian manifold is Kähler or not. They gave the following partial answer to this question. We prove the generalized result of Proposition 1.4. Theorem 1.1. Let (M n , J, g) be a compact Kähler-like and G-Kähler-like almost balanced manifold with n ≥ . Then g is Kähler.
By applying Proposition 1.3 and Theorem 1.1, we get the following corollary. We show the following result.
Theorem 1.2. Let (M n , J, g) be a G-Kähler-like almost Hermitian manifold with n ≥ . Assume that ∇¯i w i + B λ ij B¯i λj ≥ . Then the metric g is almost balanced and the almost complex structure J is integrable (i.e. g is balanced).
From Lemma 4.2, we have ∇¯j w j = B λ ij B¯i λj on a Kähler-like almost Hermitian manifold. By combining this formula and Theorem 1.2, we obtain the following result. Corollary 1.2. Let (M n , J, g) be a Kähler-like and G-Kähler-like almost Hermitian manifold with n ≥ . Assume that B λ ij B¯i λj ≥ . Then the metric g is almost balanced and the almost complex structure J is integrable (i.e. g is balanced). This paper is organized as follows: in the second section, we recall some basic de nitions and computations. In the third section, we de ne Kähler-likeness and G-Kähler-likeness in another way. And we show some equivalences between the de nitions in De nition 1.2 and the ones in De nition 3.1. In the fourth section, by calculating the curvature tensor applying Proposition 1.2 and 1.4, we show Theorem 1.1 and 1.2. In the last section, we study the case when R ijkl = R L ijkl for all i, j, k, l = , . . . , n and B λ ij B¯i λj = , and then we give a proof of Theorem 3.1. Notice that we assume the Einstein convention omitting the symbol of sum over repeated indexes in all this paper.

Preliminaries . The Nijenhuis tensor of the almost complex structure
Let M be a n-dimensional smooth di erentiable manifold. An almost complex structure on M is an endomorphism J of TM, J ∈ Γ(End(TM)), satisfying J = −Id TM . The pair (M, J) is called an almost complex manifold. Let (M, J) be an almost complex manifold. We de ne a bilinear map on C ∞ (M) for X, Y ∈ Γ(TM) by  ( , ) , ). An almost complex structure J is called integrable if N = everywhere on M. Giving a complex structure on a di erentiable manifold M is equivalent to giving an integrable almost complex structure on M (cf. [18]). Let (M, J) be an almost complex manifold. A Riemannian metric g on M is called J-invariant if J is compatible with g, i.e., for any X, Y ∈ Γ(TM), g(X, Y) = g(JX, JY). In this case, the pair (J, g) is called an almost Hermitian structure. The fundamental -form ω associated to a J-invariant Riemannian metric g, i.e., an almost Hermitian metric, is determined by, for X, Y ∈ Γ(TM), ω(X, Y) = g(JX, Y). Indeed we have, for any X, Y ∈ Γ(TM), and ω ∈ Γ( T * M). We will also refer to the associated real fundamental ( , )-form ω as an almost Hermitian metric. The form ω is related to the volume form dVg by n!dVg = ω n . Let a local ( , )-frame {Zr} on (M, J) with an almost Hermitian metric g and let {ζ r } be a local associated coframe with respect to {Zr}, i.e., ζ i (Z j ) = δ i j for i, j = , . . . , n. Since g is almost Hermitian, its components satsfy g ij = g¯i¯j = and g ij = g¯j i =ḡ¯i j . With using these local frame {Zr} and coframe {ζ r }, we have and We write T R M for the real tangent space of M. Then its complexi ed tangent space is given by T C M = T R M ⊗ R C. By extending J C-linearly and g, ω, C-bilinearly to T C M, they are also de ned on T C M and we observe that the complexi ed tangent space T C M can be decomposed as (2.5) and Λ M denotes the dual of TM. Let (M n , J, g) be an almost Hermitian manifold. An a ne connection D on TM is called almost Hermitian connection if Dg = DJ = . For the almost Hermitian connection, we have the following Lemma (cf. [4], [9], [22], [26]). If the ( , )-part of the torsion of an almost Hermitian connection vanishes everywhere, then the connction is called the second canonical connection or the Chern connection. We will refer the connection as the Chern connection and denote it by ∇. (2.9) Notice that J is integrable if and only if the B¯r ij 's vanish.
Note that for any p-form ψ, there holds that for any vector elds X , . . . , X p+ on M (cf. [26]). We directly compute that According to the direct computation above, we may split the exterior di erential operator d : In terms of these components, the condition d = can be written as Notice that J is integrable if and only if A = , equivalently, if and only if∂ = .
For any real ( , From these computations above, we havē

. The torsion and the curvature on almost complex manifolds
Let a local ( , )-frame {Zr} on (M, J) with an almost Hermitian metric g and let {ζ r } be a local associated coframe with respect to {Zr}. Since the Chern connection ∇ preserves J, we are able to de ne the Christo el symbols: for i, j, r = , . . . , n, The torsion T = (T i ) of the Chern connection ∇ is de ned by Since the torsion T of the Chern connection ∇ has no ( , )-part; Here note that Bq jb , B q jb 's do not depend on g, which depend only on J since the mixed derivatives ∇ j Zb do not depend on g (cf. [22]).
The torsion T of ∇ has no ( , )-part and the only non-vanishing components are as follows: and on the other hand we have dζs(Z i , Z j ) = Ns ij , hence we obtain that Ts ij = Ns ij = −Bs ij . These computations tell us that T splits in , which tells us that ( , )-part of the Chern connection is uniquely determined by the Nijenhuis tensor (cf. [26]). Let (M, J, g) be an almost Hermitian manifold. Let {Zr} be a local ( , )-frame with respect to g and let {ζ r } be the associated coframe. Then the associated real ( , )-form ω with respect to g takes the local expression ω = √ − ζ r ∧ ζ¯r. With using these notations, we may compute where we used the skew-Hermitian property γ p r + γrp = , which can be obtained with using ∇g = (cf. [21]). Note that we obtain for any ( , )-form β, and so∂β = is equivalent to that ∇¯l βr = for any l, r = , . . . , n.
We also lower the index of torsion and denote it by T ijk = T s ij g sk . Note that T depends only on J and it can be regarded as the Nijenhuis tensor of J, that is, J is integrable if and only if T vanishes.
We denote by Ω the curvature of the Chern connection ∇. We can regard Ω as a section of Λ M ⊗ TM, Ω ∈ Γ(Λ M ⊗ TM) and Ω splits in Ω = H + R +Hwith The curvature form can be written by and we deduce that with using Γ p kp = Z k (log det g) − Bp kp , The curvature P is one of the Ricci-type curvatures of the Chern curvature. One has with an arbitrary ( , )frame {Zr} with respect to g, P ij = g kl Ω ijkl . We denote by S one of the Ricci-type curvatures of the Chern curvature, which is locally given by S ij = g kl Ω klij . In the Kähler case, S ij = P ij is the Ricci curvature. We introduce the rst Bianchi identity for the Chern curvature.
where the sum is taken over all cyclic permutations.
This identity induces the following formulae: .

Complexi ed Riemannian curvature tensor
We introduce the complexi ed Riemannian curvature tensor as in [17]. Let (M, g, D) be a n-dimensional Riemannian manifold with Levi-Civita connection D. The tangent bundle of M is denoted by T R M. The curvature tensor of (M, g, D) is de ned as Notice that the components of C-linear complexi ed curvature tensor have the same properties as the components of the real curvature tensor. The components of the complexi ed curvature tensor R ijkl satis es the following properties: R L ijkl = −R L jikl , R L ijkl = R L klij and the Bianchi identity:

Almost Hermitian Kähler-like and G-Kähler-like structure
We de ne the curvature with respect to the Levi-Civita connection D in the following way for tangent vectors X, Y , Z and W: An almost Hermitian manifold (M, J, g) satisfying that (cf. [12], [13] AHC-manifold (cf. [13]). Then AH ⊂ AH ⊂ AH , AH ⊂ AHC ⊂ AH , and AHC ∩ AH = AH . Note that if an AH -manifold is almost Kähler, then it is Kähler (cf. [12,Theorem 5.1]). Furthermore, it is known that an almost Kählerian or a nearly Kählerian AHC-manifold M n is Kählerian (cf. [13,Lemma 10.3]). It is well-known that a nearly Kähler manifold automatically satisfy the AH -condition (cf. [11]).
The curvature R L of the Levi-Civita connection D satis es the rst Bianchi identity: where the sum is taken over all cyclic permutations. The curvature Ω ( , ) = R of the Chern connection ∇ satis es (Cplx) R(x, y, z, w) = R(x, y, Jz, Jw) = R(Jx, Jy, z, w).
We de ne the Kähler-likeness in the way of [3, De nition 4] as follows.
De nition 3.1. Let M be an almost complex manifold endowed with an almost Hermitian structure. Let ∇ be a metric connection on it. We say that the curvature of ∇ is Kähler-like if it satis es (1Bnc) and (Cplx).
We see the following equivalences which are similar to the ones in [3, Remark 5].   We see that R satis es (1Bnc). Since we have assumed that R ijkl = R L ijkl for all i, j, k, l = , . . . , n and since we have that R kijl = , we obtain that where we used that R L kijl = since we have assumed that (M, J, g) is G-Kähler-like and also we used that R¯j kil = −R kjil . Therefore, R satis es (1Bnc). By applying Lemma 3.2, since the G-Kähler-likeness and that R ijkl = R L ijkl for all i, j, k, l = , . . . , n imply the Kähler-likeness, if the manifold is compact, we obtain the following corollary by applying Theorem 1.1. Corollary 3.1. Let (M n , J, g) be a compact G-Kähler-like almost balanced manifold with n ≥ . Assume that R ijkl = R L ijkl for all i, j, k, l = , . . . , n. Then g is Kähler.
By applying Corollary 1.2, we have the following result. Next, we introduce the following result for AH -manifolds.

De nition 3.2.
Let (M n , J) be an almost complex manifold. A metric g is called an almost Gauduchon metric on M if g is an almost Hermitian metric whose associated real ( , )-form ω = √ − g ij ζ i ∧ζ¯j satis es d * (Jd * ω) = , where d * is the adjoint of d with respect to g, which is equivalent to d(Jd(ω n− )) = . When an almost Hermitian metric g is almost Gauduchon, the triple (M n , J, g) will be called an almost Gauduchon manifold.
One has the following well-known result. We shall show that the torsion ( , )-form η satis es that Re(∂η) = if and only if g is almost Gauduchon in Lemma 4.4. By applying Theorem 1.2, then we obtain the following result. Corollary 3.3. Let (M n , J, g) be a G-Kähler-like almost Gauduchon manifold with n ≥ . Assume that B λ ij B¯i λj ≥ . Then the metric g is almost balanced and the almost complex structure J is integrable (i.e. g is balanced).
We consider the case of quasi-Kähler manifolds. The following lemma can be obtained easily to see the expression of (2.30).  We see that if the metric is pluriclosed and balanced at the same time, it must be Kähler as follows. If ω is pluriclosed and balanced, then ω is Kähler.
We de ne an almost pluriclosed metric on almost complex manifolds.
Then by combining Corollary 1.2 and Proposition 3.2, we obtain the following result.
Corollary 3.5. Let (M n , J, g) be a Kähler-like and G-Kähler-like almost Hermitian manifold with n ≥ . Assume that B λ ij B¯i λj ≥ and the metric g is almost pluriclosed. Then g is Kähler.
Obviously, when the metric g is Kähler, one has R = R L . Yang and Zheng have shown the following result in the Hermitian case. We study the case R L ijkl and R ijkl are equal in the almost Hermitian geometry. By applying Corollary 3.5 (or Theorem 1.1) and Proposition 1.3, we obtain the following.
Proof. By using a local g-unitary ( , )-frame {Zr}, we obtain the following computation: Therefore we obtain the desired formula.
Let (M n , J, g) be an almost Hermitian manifold and let ∇ be the Chern connection on M. Let {Zr} be a local ( , )-frame with respect to g around a xed point p ∈ M such that ∇Z i (p) = and g ij (p) = δ ij , and let {ζ r } be the associated coframe. The existence of such frames has been proven in [25]. In order to prove Theorem 1.1, we introduce the following lemma. By applying the rst Bianchi identity for the Chern curvature (2.40), we obtain the formula as follows. Recall the torsion ( , )-form η = T i ir ζ r = −wr ζ r , where wr = T i ri , as we de ned in the previous section. Notice that when a Hermitian manifold is Kähler-like, we obtain that∂η = as we see below. From the equation (2.31), we have∂η = ∇¯l wr ζ r ∧ ζ¯l. Hence∂η = is equivalent to that ∇¯l wr = for any l, r = , . . . , n. By applying (4.3), we can obtain the same result as in the rst statement of Proposition 4.1, since we have ∇¯l wr = Bq ir B iql = Nq ir N iql = for any r, l = , . . . , n on a Kähler-like Hermitian manifold. Proof. As we see in section 2, we have ∂ω = √ − T ijk ζ i ∧ ζ j ∧ ζk. Then a direct calculation shows that where we used that η = −w i ζ i = −(n − ) ∂ω ∧ ω n− ω n− . Similarly, we obtain that Recall that the metric g is said to be almost balanced if ω n− is closed. Since we have Aω n− =Āω n− = , we get dω n− = ∂ω n− +∂ω n− . Therefore, these identities (4.4), (4.5) show that g is almost balanced if and only if η = .
From Lemma 4.3, we obtain the following result by applying Proposition 1.2. Let (M n , J, g) be an almost Hermitian manifold. We say that g is almost Gauduchon if the associated real ( , )-form ω with respect to g satis es d * (Jd * ω) = , where d * is the adjoint of d with respect to g, which is equivalent to d(Jd(ω n− )) = . We have the following equivalence. where we have used that Aη ∧ ω n− =Āη ∧ ω n− = and ∂η ∧ ω n− = .
Proof of Theorem 1.1. Suppose that (M n , J, g) be a compact Kähler-like and G-Kähler-like almost balanced manifold with n ≥ . Let D be the Levi-Civita connection and R L denotes its curvature tensor with respect to g. Recall the de nition of curvature operator: for type ( , ) tangent vectors X, Y, Z, The curvature tensor is de ned for type ( , ) tangent vectors X, Y, Z and W as for any type ( , ) tangent vectors X, Y, Z.
For an arbitrary chosen xed point p ∈ M, let {Zr} be a local ( , )-frame at p such that ∇Zr(p) = and g ij (p) = δ ij . By applying (4.11), we compute at p, Similarly, we have at p, Since we have 20) we then get at p, Hence, we have at p, where we used the de nitions (4.9), (4.10), the formula (4.1), and that ∇ [Zk ,Z¯l] Z i (p) = , and Rk¯l ij = .
By taking k = i, l = j and sum them over, we obtain at p, where we used that Tk ij = −Bk ij . Since we have assumed that g is almost balanced, which is equivalent to that Bq ik B iql = for all k, l = , . . . , n on a compact Kähler-like almost Hermitian manifold from Proposition 1.2, then especially we have that B λ ij B¯i λj = . Also we have wr = for all r = , . . . , n since it is equivalent to the almost balancedness from Lemma 4.3. Combining these, we get R L ijīj = |T | . In addition, since we have assumed that M is G-Kähler-like as well, we then obtain at p, which implies that we have that Tk ij (p) = for all i, j and k. Since the point p was chosen arbitrary, we obtain that Tk ij = for all i, j and k on whole M, which is equivalent to that J is integrable. Then we may apply Proposition 1.4, and we conclude that g is Kähler.
Next, we give a proof of Theorem 1.2.
Proof of Theorem 1.2. As in the proof of Theorem 1.1, for an arbitrary chosen xed point p ∈ M, we choose a local ( , )-frame {Zr} at p such that ∇Zr(p) = and g ij (p) = δ ij . As in the proof of Theorem 1.1, we use the formula (4.23), and since we have assumed the G-Kähler-likeness, we have at p, Since the xed point p was chosen arbitrary, the equation holds on whole M, since we have assumed that ∇¯i w i + B λ ij B¯i λj ≥ . Then we get that w = and T = , which tell us the desired result since g is almost balanced if and only if w i = for all i = , . . . , n, and also since J is integrable if and only if T vanishes. Now, we prove Corollary 3.3.

Proof of Corollary 3.3.
For an arbitrary chosen xed point p ∈ M, we choose a local ( , )-frame {Zr} at p such that ∇Zr(p) = and g ij (p) = δ ij . From Lemma 4.4, we have that the torsion ( , )-form η satis es Re(∂η) = is equivalent to that the metric g is almost Gauduchon. Note that Re(∂η) = implies that Re(∇¯l w k ) = for all k, l = , . . . , n, especially we then have Re(∇¯i w i ) = . As in the proof of Theorem 1.1 and 1.2, we again use the formula (4.23), and since we have assumed the G-Kähler-likeness, we have at p,  J, g) be a G-Kähler-like quasi-Kähler manifold satisfying B λ ij B¯i λj ≥ . For an arbitrary chosen xed point p ∈ M, we choose a local ( , )-frame {Zr} at p such that ∇Zr(p) = and g ij (p) = δ ij . From Lemma 3.4, we have that T k ij = for all i, j, k = , . . . , n, especially we then have w i = for all i = , . . . , n. As we see in the proof of Theorem 1.1 and 1.2, since we have assumed the G-Kähler-likeness, we have at p, = R L ijīj = B λ ij B¯i λj + |T | . holds on whole M, since we have assumed that B λ ij B¯i λj ≥ . Then we get that T = , which tells us that J is integrable. Therefore, g is Kähler.

Proof of Theorem 3.1
Let (M n , J, g) be a compact almost Hermitian manifold. Let ∇ be the Chern connection withrespect to g on (M n , J, g). Let {Zr} be a local unitary ( , )-frame with respect to g around a xed point p ∈ M. Note that unitary frames always exist locally since we can take any frame and apply the Gram-Schmidt process. Then with respect to a local g-unitary frame, we have g ij = δ ij , Z k (g ij ) = for any i, j, k = , . . . , n, and the hence we obtain R ijkr = R jīrk by using a local unitary frame with respect to g. In order to prove Theorem 3.1, we prepare the following lemma. where we used that R kīik = R iīkk . Since we have assumed that R ijkl = R L ijkl for all i, j, k, l = , . . . , n, we obtain = − |T | − (B¯i kλ B λ ik + Bλ ik B ikλ ) + |Bλ ik + B¯i kλ − Bk λi | , Here notice that J is integrable if and only if T vanishes. Therefore we conclude that, by applying Corollary 1.2, under the assumptions in Theorem 3.1, we have that J is integrable and |T | = at the same time. Since T = is equivalent to that (M, J, g) is quasi-Kähler (Lemma 3.4), and since the quasi-Kählerity indicates the Kählerity when J is integrable, we obtain the desired result.