CUBO, A Mathematical Journal Vol. 23, no. 02, pp. 333–341, August 2021 DOI: 10.4067/S0719-06462021000200333 On the conformally k-th Gauduchon condition and the conformally semi-Kähler condition on almost complex manifolds Masaya Kawamura 1 1 Department of General Education, National Institute of Technology, Kagawa College, 355, Chokushi-cho, Takamatsu, Kagawa, Japan 761-8058. kawamura-m@t.kagawa-nct.ac.jp ABSTRACT We introduce the k-th Gauduchon condition on almost com- plex manifolds. We show that if both the conformally k-th Gauduchon condition and the conformally semi-Kähler con- dition are satisfied, then it becomes conformally quasi- Kähler. RESUMEN Introducimos la k-ésima condición de Gauduchon en va- riedades casi complejas. Mostramos que si la k-ésima condición de Gauduchon conforme y la condición semi- Kähler conforme se satisfacen ambas, entonces la variedad es cuasi-Kähler conforme. Keywords and Phrases: Almost Hermitian manifold, k-th Gauduchon metric, semi-Kähler metric. 2020 AMS Mathematics Subject Classification: 32Q60, 53C15, 53C55. Accepted: 26 June, 2021 Received: 17 November, 2020 c©2021 M. Kawamura. This open access article is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License. http://cubo.ufro.cl/ http://dx.doi.org/10.4067/S0719-06462021000200333 https://orcid.org/0000-0003-1303-4237 334 Masaya Kawamura CUBO 23, 2 (2021) 1 Introduction S. Ivanov and G. Papadopoulous introduced the conditions on the Hermitian form such that ωl ∧ ∂∂̄ωk = 0 for 1 ≤ k + l ≤ n − 1, which is called the (l|k)-SKT condition. They have proven that every compact conformally balanced (l|k)-SKT manifold, k < n − 1, n > 2, is Kähler (cf. [5]). J. Fu, Z. Wang and D. Wu introduced and investigated the generalization of Gauduchon metrics, which is called k-th Gauduchon. The k-th Gauduchon condition is the case l = n − k − 1, 1 ≤ k ≤ n−1 of the (l|k)-SKT condition. By definition, (n−1)-th Gauduchon metrics are the usual Gauduchon metrics, astheno-Kähler metrics are examples of (n − 2)-th Gauduchon metrics, and pluriclosed metrics are in particular 1-st Gauduchon. They proved that there exists a non-Kähler 3-fold which can support a 1-Gauduchon metric and a balanced metric simultaneously (cf. [2]). Since K. Liu and X. Yang have shown that if a compact complex manifold is k-th Gauduchon for 1 ≤ k ≤ n − 2 and also balanced, then it must be Kähler, a 1-Gauduchon metric and a balanced metric on a non-Kähler 3-fold which Fu, Wang and Wu discovered must be different Hermitian metrics. Liu and Yang also have shown that the conformally Kählerianity is equivalent to that both the conformally k-th Gauduchon for 1 ≤ k ≤ n − 2, and the conformally balancedness are satisfied (cf. [7]). Our aim in this paper is to generalize the Liu-Yang’s equivalence [7, Corollary 1.17] to almost Hermitian geometry. Let (M2n,J) be an almost complex manifold with n ≥ 3 and let g be an almost Hermitian metric on M. Let {Zr} be an arbitrary local (1,0)-frame around a fixed point p ∈ M and let {ζr} be the associated coframe. Then the associated real (1,1)-form ω with respect to g takes the local expression ω = √ −1grk̄ζr ∧ ζk̄. We will also refer to ω as to an almost Hermitian metric. We introduce the definition of a Gauduchon metric and we define a k-th Gauduchon metric as follows. Definition 1.1. Let (M2n,J) be an almost complex manifold. A metric g is called a Gauduchon metric on M if g is an almost Hermitian metric whose associated real (1,1)-form ω = √ −1gij̄ζi∧ζj̄ satisfies d∗(Jd∗ω) = 0, where d∗ is the adjoint of d with respect to g, which is equivalent to d(Jd(ωn−1)) = 0, or ∂∂̄(ωn−1) = 0. When an almost Hermitian metric g is Gauduchon, the triple (M2n,J,g) will be called a Gauduchon manifold. For 1 ≤ k ≤ n − 1, an almost Hermitian metric ω is called k-th Gauduchon if it satisfies that ∂∂̄ωk ∧ ωn−k−1 = 0. Notice that the condition ∂∂̄ωk ∧ ωn−k−1 = 0 for 1 ≤ k ≤ n − 2 is not equivalent to d(Jd(ωk)) ∧ ωn−k−1 = 0 for 1 ≤ k ≤ n−2 since there exist A and Ā parts of the exterior differential operator d in the almost complex setting (Note that these conditions are equivalent in the case of k = n−1 as we confirmed in Definition 1.1 since then we have A(ωn−1) = Ā(ωn−1) = 0.). Hence the condition ∂∂̄ωk ∧ ωn−k−1 = 0 for 1 ≤ k ≤ n − 1 can be regarded as a natural extension of the Gauduchon condition on almost complex manifolds. We next introduce the definition of a semi-Kähler metric. CUBO 23, 2 (2021) On the conformally k-th Gauduchon condition and the conformally ... 335 Definition 1.2. Let (M2n,J) be an almost complex manifold. A metric g is called a semi-Kähler metric on M if g is an almost Hermitian metric whose associated real (1,1)-form ω = √ −1gij̄ζi∧ζj̄ satisfies d(ωn−1) = 0. When an almost Hermitian metric g is semi-Kähler, the triple (M2n,J,ω) will be called a semi-Kähler manifold. Recall that on an almost Hermitian manifold (M,J,g), a quasi-Kähler structure is an almost Hermitian structure whose real (1,1)-form ω satisfies (dω)(1,2) = ∂̄ω = 0, which is equivalent to the original definition of quasi-Kählerianity: DXJ(Y ) + DJXJ(JY ) = 0 for all vector fields X,Y (cf. [4]), where D is the Levi-Civita connection associated to g. It is important for us to study quasi-Kähler manifolds since they include the classes of almost Kähler manifolds and nearly Kähler manifolds. An almost Kähler or quasi-Kähler manifold with J integrable is a Kähler manifold. We define some conformally conditions. Definition 1.3. Let (M,J,ω) be an almost Hermitian manifold. We say ω is conformally k-th Gauduchon (resp. semi-Kähler, quasi-Kähler) if there exist a k-th Gauduchon (resp. semi-Kähler, quasi-Kähler) metric ω̃ and a smooth function F ∈ C∞(M,R) such that ω = eF ω̃. Our main result is as follows. Theorem 1.4. On a compact almost Hermitian manifold (M,J,ω), the following are equivalent: (1) (M,J,ω) is conformally quasi-Kähler. (2) (M,J,ω) is conformally k-th Gauduchon for 1 ≤ k ≤ n − 2, and conformally semi-Kähler. In particular, the following are also equivalent: (a) (M,J,ω) is quasi-Kähler. (b) (M,J,ω) is k-th Gauduchon for 1 ≤ k ≤ n − 2, and conformally semi-Kähler. This paper is organized as follows: in the second section, we recall some basic definitions and computations. In the last section, we will give a proof of the main result. Notice that we assume the Einstein convention omitting the symbol of sum over repeated indexes in all this paper. 2 Preliminaries 2.1 The Nijenhuis tensor of the almost complex structure Let M be a 2n-dimensional smooth differentiable manifold. An almost complex structure on M is an endomorphism J of TM, J ∈ Γ(End(TM)), satisfying J2 = −IdT M . The pair (M,J) is called 336 Masaya Kawamura CUBO 23, 2 (2021) an almost complex manifold. Let (M,J) be an almost complex manifold. We define a bilinear map on C∞(M) for X,Y ∈ Γ(TM) by 4N(X,Y ) := [JX,JY ] − J[JX,Y ] − J[X,JY ] − [X,Y ], (2.1) which is the Nijenhuis tensor of J. The Nijenhuis tensor N satisfies N(X,Y ) = −N(Y,X), N(JX,Y ) = −JN(X,Y ), N(X,JY ) = −JN(X,Y ), N(JX,JY ) = −N(X,Y ). For any (1,0)- vector fields W and V , N(V,W) = −[V,W ](0,1), N(V,W̄) = N(V̄ ,W) = 0 and N(V̄ ,W̄) = −[V̄ ,W̄ ](1,0) since we have 4N(V,W) = −2([V,W ] + √ −1J[V,W ]), 4N(V̄ ,W̄) = −2([V̄ ,W̄] − √ −1J[V̄ ,W̄ ]). An almost complex structure J is called integrable if N = 0 everywhere on M. Giving a complex structure on a differentiable manifold M is equivalent to giving an integrable almost complex structure on M. 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 2-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), ω(Y,X) = g(JY,X) = g(J2Y,JX) = −g(JX,Y ) = −ω(X,Y ) (2.2) and ω ∈ Γ( ∧2 T ∗M). We will also refer to the associated real fundamental (1,1)-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 (1,0)-frame {Zr} on (M,J) with an almost Hermitian metric g and let {ζr} be a local associated coframe with respect to {Zr}, i.e., ζi(Zj) = δij for i,j = 1, . . . ,n. Since g is almost Hermitian, its components satsfy gij = gīj̄ = 0 and gij̄ = gj̄i = ḡīj. We write T RM for the real tangent space of M. Then its complexified tangent space is given by T CM = T RM ⊗R C. By extending J C-linearly and g, ω, C-bilinearly to T CM, they are also defined on T CM and we observe that the complexified tangent space T CM can be decomposed as T CM = T 1,0M⊕T 0,1M, where T 1,0M, T 0,1M are the eigenspaces of J corresponding to eigenvalues √ −1 and − √ −1, respectively: T 1,0M = {X − √ −1JX ∣∣X ∈ TM}, T 0,1M = {X + √ −1JX ∣∣X ∈ TM}. (2.3) Let ΛrM = ⊕ p+q=r Λ p,qM for 0 ≤ r ≤ 2n denote the decomposition of complex differential r-forms into (p,q)-forms, where Λp,qM = Λp(Λ1,0M) ⊗ Λq(Λ0,1M), Λ1,0M = {α + √ −1Jα ∣∣α ∈ Λ1M}, Λ0,1M = {α − √ −1Jα ∣∣α ∈ Λ1M} (2.4) and Λ1M denotes the dual of TM. For any α ∈ Λ1M, we define Jα(X) = −α(JX) for X ∈ TM. Let (M2n,J,g) be an almost Hermitian manifold. An affine connection D on TM is called almost Hermitian connection if Dg = DJ = 0. For the almost Hermitian connection, we have the following Lemma (cf. [3, 9, 11]). CUBO 23, 2 (2021) On the conformally k-th Gauduchon condition and the conformally ... 337 Lemma 2.1. Let (M,J,g) be an almost Hermitian manifold with dimR M = 2n. Then for any given vector valued (1,1)-form Θ = (Θi)1≤i≤n, there exists a unique almost Hermitian connection D on (M,J,g) such that the (1,1)-part of the torsion is equal to the given Θ. If the (1,1)-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 ∇. Note that for any p-form ψ, there holds that dψ(X1, . . . ,Xp+1) = p+1∑ i=1 (−1)i+1Xi(ψ(X1, . . . ,X̂i, . . . ,Xp+1)) + ∑ i