On a k-th Gauduchon almost Hermitian manifold

In [3], J. Fu, Z. Wang and D. Wu introduced the k-th Gauduchonmetric as a generalized version of the Gauduchonmetric. From the de nition, we see that (n−1)-th Gauduchonmetrics are the usual Gauduchonmetrics, and astheno-Kähler metrics (cf. [8]) are examples of (n − 2)-th Gauduchon metrics, pluriclosed metrics are 1-st Gauduchonmetrics. A. Fino and L. Ugarte have shown that for each k = 1, . . . , [ n 2 ]−1, a Hermitianmetric is k-th Gauduchon if and only if it is (n − k − 1)-th Gauduchon on a complex nilmanifold (cf. [2, Lemma 4.7]). A. Latorre and L. Ugarte have investigated the k-th Gauduchon condition on homogeneous compact complex manifolds (cf. [12]). In [1], H. Chen, L. Chen and X. Nie investigated the k-th Gauduchon metric and characterized it on a compact Hermitian manifold. They also obtained a direct corollary that if the manifold is k-th Gauduchon, then one has s ≥ ŝ, where s is the Chern scalar curvature and ŝ is the Riemannian type scalar curvature.We extend thier characterization result to almost Hermitian geometry.We have already de ned the k-th Gauduchon metric on almost Hermitian manifolds and have obtained some results in [10, 11]. Let (M2n , J) be an almost complex manifold of real dimension 2n with n ≥ 3 and let g be an almost Hermitian metric on M. Let {Zr} be an arbitrary local (1, 0)-frame around a xed point p ∈ M and let {ζ r} be the associated coframe. Then the associated real (1, 1)-form ω with respect to g takes the local expression ω = √ −1grk̄ζ r ∧ ζ k̄. We will also refer to ω as to an almost Hermitian metric in the present paper. We de ne a Gauduchon metric and a k-th Gauduchon metric on an almost Hermitian manifold in the following.


Introduction
In [3], J. Fu, Z. Wang and D. Wu introduced the k-th Gauduchon metric as a generalized version of the Gauduchon metric. From the de nition, we see that (n − )-th Gauduchon metrics are the usual Gauduchon metrics, and astheno-Kähler metrics (cf. [8]) are examples of (n − )-th Gauduchon  A. Latorre and L. Ugarte have investigated the k-th Gauduchon condition on homogeneous compact complex manifolds (cf. [12]). In [1], H. Chen, L. Chen and X. Nie investigated the k-th Gauduchon metric and characterized it on a compact Hermitian manifold. They also obtained a direct corollary that if the manifold is k-th Gauduchon, then one has s ≥ŝ, where s is the Chern scalar curvature andŝ is the Riemannian type scalar curvature. We extend thier characterization result to almost Hermitian geometry. We have already de ned the k-th Gauduchon metric on almost Hermitian manifolds and have obtained some results in [10,11].
Let (M n , J) be an almost complex manifold of real dimension n with n ≥ and let g be an almost Hermitian metric on M. Let {Zr} be an arbitrary local ( , )-frame around a xed point p ∈ M and let {ζ r } be the associated coframe. Then the associated real ( , )-form ω with respect to g takes the local expression ω = √ − g rk ζ r ∧ ζk. We will also refer to ω as to an almost Hermitian metric in the present paper. We de ne a Gauduchon metric and a k-th Gauduchon metric on an almost Hermitian manifold in the following.
Note that in the case of n = , a Gauduchon metric and a k-th Gauduchon metric are both almost pluriclosed (see De nition 1.3).
One has the following well-known result.
Proposition 1.1. (cf. [4]) Let (M n , J, ω) be a compact almost Hermitian manifold with n ≥ . Then there exists a smooth function u, unique up to addition of a constant, such that the conformal almost Hermitian metric e u ω is Gauduchon.
We characterize the k-th Gauduchon condition on a compact almost Hermitian manifold as follows.
Theorem 1.1. Let (M n , J, ω) be compact almost Hermitian manifold with n ≥ and let k be an integer such that ≤ k ≤ n − . Then the following are equivalent.
where s is the Chern scalar curvature andŝ is a Riemannian type scalar curvature of the metric ω with respect to the Chern connection (see (2.10)), and B¯r ij 's are the structure coe cients of Lie bracket (see (2.7), (2.8) for more detail). Note that B¯r ij B ī rj means that we sum over repeated indices i, j and r with respect to the metric ω.
Corollary 1.1. Let (M n , J, ω) be a compact k-th Gauduchon, semi-Kähler manifold with n ≥ and let k be an integer such that ≤ k ≤ n − . Then ω is quasi-Kähler.
We de ne an almost pluriclosed metric on almost complex manifolds. Note that in the case of complex manifolds, an almost pluriclosed metric is called a pluriclosed metric.
De nition 1.3. Let (M, J) be an almost complex manifold. An almost Hermitian metric ω is called almost pluriclosed if the metric ω satis es that ∂∂ω = . When an almost Hermitian metric ω is almost pluriclosed, the triple (M, J, ω) is called an almost pluriclosed manifold.
We introduce the following well-known result in Hermitian geometry. This paper is organized as follows: in the second section, we recall some basic de nitions and computations in almost Hermitian geometry. In the third section, we give a proof of main result and by applying it, we show some corollaries. Notice that we assume the Einstein convention omitting the symbol of sum over repeated indices in all this paper.

Preliminaries . The bundle of real k-forms and the interior product
Let M be a real n-dimensional smooth di erentiable manifold and let h be a Riemannian metric on M. In a local coordinate (x , x , . . . , x n ) on M, we write h = h ij dx i dx j . Denote (h ij ) the inverse matrix of (h ij ), ≤ i, j ≤ n. Then the metric h induces an inner product ·, · on the cotangent bundle T * M by dx i , dx j = h ij . Let Λ k T * M be the bundle of real k-forms for ≤ k ≤ n. The inner product induced by h on Λ k T * M is given by We de ne the interior product ι X φ ∈ Λ k− T * M for vector elds X, X , . . . ,X k− on M and φ ∈ Λ k T * M by Note that we have De neX := h(X, ·) ∈ T * M, then we obtain that for φ ∈ Λ k+ T * M and ψ ∈ Λ k T * M (cf. [1, (2.3)]), ι X φ, ψ = φ,X ∧ ψ . (2.5) . The Nijenhuis tensor of the almost complex structure The complexi ed tangent vector bundle is given by T C M = TM ⊗ R C for the real tangent vector bundle TM. By extending J C-linearly and g, C-bilinearly to T C M, they are also de ned on T C M and we observe that the complexi ed tangent vector bundle T C M can be decomposed as and Λ M denotes the dual of T C M. Let {Zr} be a local ( , )-frame with respect to 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. We write g ij := g(Z i , Z¯j). The fundamental ( , )-form ω associated to g is locally given by ω = √ − g ij ζ i ∧ζ¯j. Since g is almost Hermitian, its components satsfy g ij = g¯i¯j = and g ij = g¯j i =ḡ¯i j . By using these local frame {Zr} and coframe {ζ r }, we have Let (M n , J, g) be an almost Hermitian manifold. An a ne connection D on T C M is called almost Hermitian connection if Dg = DJ = . For the almost Hermitian connection, we have the following Lemma (cf. [5], [15], [18]). 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 ∇.

. The torsion and the curvature on almost complex manifolds
Since the Chern connection ∇ preserves J, we have where Γ r ij = g rs Z i (g js ) − g rs g jl B¯l is . We can obtain that since the ( , )-part of the torsion of the Chern connection vanishes everywhere. Note that the mixed derivatives ∇ i Z¯j do not depend on g (cf. [15]).
which has no ( , )-part and the only non-vanishing components are as follows: We 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 ⊗ Λ , M, Ω ∈ Γ(Λ M ⊗ Λ , M) and Ω splits in The curvature form can be expressed by Ω i j = dγ i j + γ i s ∧ γ s j . In terms of Zr's, we have We can write Ω = (Ω i j ) = Ω ( , ) + Ω ( , ) + Ω ( , ) = H + R +H, with We deduce that P ij = g kl R ijkl , S ij = g kl R klij R ij = g kl H ijkl , R¯i¯j = g kl H¯i¯j kl .
We de ne the Chern scalar curvature s and a Riemannian type scalar curvatureŝ of the metric ω with respect to the Chern connection: s := g ij g kl R ijkl = g ij P ij = g ij S ij ,ŝ := g il g kj R ijkl .
where the sum is taken over all cyclic permutations.
This identity induces the following formula: where used that R ijkl = R¯i¯j kl = .
. The exterior di erential operator Note that for any p-form ψ, there holds that for any vector elds X , . . . , X p+ on M (cf. [18]). We directly compute that For any ( , )-form β, we have that ∂β = ∂ k β¯j ζ k ∧ ζ¯j, and from (2.10) and (2.13), which gives that where ω is the associated real ( , )-form with respect to the almost Hermitian metric g. 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
We de ne the total inner product by where dVg is the volume form de ned by dVg := ω n n! .
We say that the torsion ( , )-form η is holomorphic when it satis es that∂η = .
We can obtain the following lemma in almost Hermitian geometry as well. Proof. As in [1], we compute that which gives us that *τ = − √ − τ ∧ ω n− (n− )! . Then, by applying Lemma 2.3 and (3.2), we obtain that where we used that *ω = ω n− (n− )! ,∂ω n− = −η∧ω n− , τ∧η, ω = τ, where s is the Chern scalar curvature andŝ is a Riemannian type scalar curvature of the metric ω with respect to the Chern connection in (2.11). Notice that B¯r ij B ī rj means that we sum over repeated indices i, j and r with respect to the metric ω.
On the other hand, we have from (2.12), where we used that T s si = −T s is = −w i and B¯r is = −B¯r si . By combining (3.5) with (3.6), and by summing over indices i, j with respect to the metric ω, we obtain that s −ŝ = ∂∂ * ω, ω + B¯r si B s rī . (3.7) By applying Lemma 3.4 for τ = η, we obtain where we used that η = √ − ∂ * ω. By plugging (3.8) into (3.7), we obtain the desired formula (3.4).
Lemma 3.7. Let (M n , J, ω) be a compact almost Hermitian manifold with n ≥ and let k be an integer such that ≤ k ≤ n − . Then one has that *( Lemma 3.7 implies the equivalence in Theorem 1.1.
Proof of Corollary 1.1. By combining Lemma 3.3 with Proposition 3.1, we conclude that if a compact almost Hermitian manifold is semi-Kähler, since then we have that η = , we obtain that s −ŝ = −∂ * η + |η| + B¯r ij B ī rj = B¯r ij B ī rj . (3.14) If additionally, it is also k-th Gauduchon for an integer k such that ≤ k ≤ n − , from the equivalence in Theorem 1.1, we obtain that since we have that ∂ * ω = √ − η = , By combining (3.14) with (3.15) for ≤ k ≤ n − , we must have that ∂ω = , which tells us that the metric ω is quasi-Kähler.
By combining Corollary 1.1 with Corollary 3.1, we obtain the following result.
Corollary 3.2. Let (M n , J, ω) be a compact k-th Gauduchon almost Hermitian manifold with n ≥ and let k be an integer such that ≤ k ≤ n − . If the torsion ( , )-form η is holomorphic on M, then ω is quasi-Kähler.
By combining Corollary 1.2 with Corollary 3.1, we obtain the following result. The following equivalence can be obtained from Theorem 1.1 and Lemma 3.5, 3.6.