Second Chern-Einstein metrics on four-dimensional almost-Hermitian manifolds

Giuseppe Barbaro; Mehdi Lejmi

doi:10.1515/coma-2022-0150

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Second Chern-Einstein metrics on four-dimensional almost-Hermitian manifolds

Giuseppe Barbaro and Mehdi Lejmi

Published/Copyright: July 14, 2023

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From the journal Complex Manifolds Volume 10 Issue 1

Abstract

We study four-dimensional second Chern-Einstein almost-Hermitian manifolds. In the compact case, we observe that under a certain hypothesis, the Riemannian dual of the Lee form is a Killing vector field. We use that observation to describe four-dimensional compact second Chern-Einstein locally conformally symplectic manifolds, and we give some examples of such manifolds. Finally, we study the second Chern-Einstein problem on unimodular almost-abelian Lie algebras, classifying those that admit a left-invariant second Chern-Einstein metric with a parallel non-zero Lee form.

Keywords: almost-Hermitian Metrics; Chern-Einstein metrics; Hermitian connections; Weyl connection

MSC 2010: 53C55 (primary); 53B35 (secondary)

1 Introduction

An almost-Hermitian manifold ( M , g , J ) is a manifold M equipped with a Riemannian metric g and a g -orthogonal almost-complex structure J . The almost-Hermitian structure ( g , J ) induces the fundamental 2-form F ( ⋅ , ⋅ ) = g ( J ⋅ , ⋅ ) . The Lee form θ associated to the almost-Hermitian structure ( g , J ) is defined as follows:

d F n − 1 = 1 n − 1 θ ∧ F n − 1 ,

where d is the exterior derivative and 2 n is the real dimension of the manifold. The almost-Hermitian metric g (with a unit total volume) is called Gauduchon if δ g θ = 0 , where δ g is the adjoint of d with respect to g , and almost-Kähler if d F = 0 . On the other hand, if the almost-complex structure J is integrable, then the pair ( g , J ) is a Hermitian structure, and it is Kähler if it is almost-Kähler. A four-dimensional almost-Hermitian manifold ( M , g , J ) is locally conformally symplectic (LCS) if d θ = 0 , while in higher dimension, the LCS condition becomes d F = θ ∧ F [60,62] (for a general introduction to the subject, see [14]). In the integrable case, LCS is locally conformally Kähler (LCK).

In the Hermitian and almost-Hermitian geometry, there are some natural connections other than D g the Levi-Civita one. Among these we have the Chern connection ∇ [32,41,42] defined as the unique connection preserving the almost-Hermitian structure and having J -anti-invariant torsion; the Bismut connection ∇ B [32], which is, in the integrable case, the unique connection preserving the Hermitian structure and having skew-symmetric torsion; and the canonical Weyl connection D W defined as the unique torsion-free connection such that D W g = θ ⊗ g . Many different Einstein-type equations can be associated to any of these connections. See, for example, [20,21,31,36,47,49–53] for the Einstein-Weyl problem and [56] for the Bismut-Einstein problem in the Hermitian case.

In this note, we focus on the second Chern-Einstein problem, which is stated as follows (for more details, see [7,28,29,33]). We define the second Chern-Ricci form r as follows:

r = R ∇ ( F ) ,

that is, the image of the fundamental form by the Chern curvature R ∇ .

Definition

Given an almost-Hermitian manifold ( M , g , J ) of real dimension 2 n , the metric g is said to be second Chern-Einstein if

r = tr F r n F .

In the Hermitian case, second Chern-Einstein metrics were studied in refs. [7,8,22,28,33,43,54,57]. The condition of being second Chern-Einstein is conformally invariant (even in the non-integrable case [40]). D W is invariant under conformal change of the metric, which led to explore the relation between Einstein-Weyl metrics and second Chern-Einstein metrics. It turns out that in real dimension 4, in the Hermitian case, the Einstein-Weyl condition and the second Chern-Einstein condition (as well as the Bismut-Einstein condition) are equivalent [33, Theorem 1], and so the only compact Hermitian non-Kähler 4-manifold admitting a second Chern-Einstein metric is the Hopf surface [33]. Even if this equivalence is no longer true for almost-Hermitian structures, some crucial relations persist, see, for example Corollary 3 and Proposition 24.

In this note, we investigate the existence of second Chern-Einstein metrics on four-dimensional almost-Hermitian manifolds. In Section 3, we collect some general results about the geometry of these manifolds. In particular, we observe the following (for the analog in the Einstein-Weyl case see [58])

Proposition

[Proposition 6] Let ( M , g ˜ , J ) be a compact four-dimensional almost-Hermitian manifold such that g ˜ is a second Chern-Einstein metric. Suppose that the Gauduchon metric g in the conformal class [ g ˜ ] satisfies ( D g θ ) sym , J , − = 0 , where D g is the Levi-Civita connection of g and θ is the Lee form of ( g , J ) . Then, the g-Riemannian dual of θ is a Killing vector field of g. Here, ( ⋅ ) sym , J , − denotes the g-symmetric J-anti-invariant part.

Taking twice the trace of the curvature of D W , we obtain the conformal scalar curvature s W , which enters the pictures through our two main results, Theorems 15 and 19. In details, we have the following

Theorem

[Theorem 15] Suppose that ( M , g ˜ , J ) is a four-dimensional compact locally conformally symplectic manifold and g ˜ is a second Chern-Einstein metric. Suppose that the Gauduchon metric g in the conformal class [ g ˜ ] satisfies ( D g θ ) sym , J , − = 0 , where θ is the Lee form of ( g , J ) . Then, either

( M , g , J ) is a second Chern-Einstein almost-Kähler manifold or
θ is D g -parallel and the conformal scalar curvature s W is non-positive. Moreover, s W is identically zero if and only if J is integrable, and so ( M , J ) is a Hopf surface as described in [33, Theorem 2]. Furthermore, if s W is nowhere zero, then χ = σ = 0 , where χ and σ are the Euler class and signature of M, respectively.

In Section 4, we give examples of compact four-dimensional second Chern-Einstein almost-Hermitian manifolds. Some examples are locally conformally symplectic, and some satisfy the condition ( D g θ ) s y m , J , − = 0 . We also remark that in those examples, the second Chern-Einstein metrics may have positive or zero Chern scalar curvature. It is observed that in the integrable case (in higher dimension), second Chern-Einstein Hermitian non-Kähler metrics with negative Chern scalar curvature are still missing [8].

Finally, in Section 5, we study the Bismut-Einstein and the second Chern-Einstein problems on four-dimensional almost-abelian Lie groups. By using this class of manifolds, we show the existence in the almost-Hermitian case of Bismut-Einstein metrics with d J d F = 0 , while we recall that in the Hermitian non-Kähler case, examples of this kind are still missing. We also give a classification of four-dimensional unimodular almost-abelian Lie algebras equipped with a left-invariant almost-Hermitian non-Hermitian second Chern-Einstein metric such that θ is D g -parallel and non-zero.

Theorem

[Theorem 32] Let g be a four-dimensional unimodular almost-abelian Lie algebra equipped with an almost-Hermitian non-Hermitian structure ( g , J ) such that the Lee form θ is D g -parallel and non-zero. Suppose that ( g , J ) is a solution to the second Chern-Einstein problem. Then g is isomorphic to one of the following Lie algebras

A 3 , 6 ⊕ A 1 : [ e 1 , e 3 ] = − e 2 , [ e 2 , e 3 ] = e 1 .
A 3 , 4 ⊕ A 1 : [ e 1 , e 3 ] = e 1 , [ e 2 , e 3 ] = − e 2 .

The simply connected Lie groups associated to these Lie algebras admit compact quotients by discrete subgroups.

Here, we use the same notation of Lie algebras as [48] (for a classification of locally conformally symplectic Lie algebras, see [4,5]).

2 Preliminaries

In all the following, ( M , g , J ) will be an almost-Hermitian manifold of real dimension 4. We denote by D g the Levi-Civita connection associated to the Riemannian metric g and by R ∇ , R W , R g , and R B the curvature tensors of ∇ , D W , D g , and ∇ B , respectively, where we use the following convention R X , Y ∇ = ∇ [ X , Y ] − [ ∇ X , ∇ Y ] , etc. Moreover, given a 2-tensor ψ , we denote by ψ J , + its J -invariant part, ψ J , − its J -anti-invariant part, ψ sym its g -symmetric part, and ψ anti-sym its g -anti-symmetric part. Moreover, a 2-form ϕ can be decomposed into a g -orthogonal sum ϕ = ϕ + + ϕ − , where ϕ + is self-dual, i.e. ∗ g ϕ + = ϕ + and ϕ − is anti-self-dual ∗ g ϕ − = − ϕ − under the action of the Riemannian Hodge operator ∗ g .

It follows from [32] that the Chern connection ∇ is related to the Levi-Civita connection D g by

(1) ∇ X Y = D X g Y − 1 2 θ ( J X ) J Y − 1 2 θ ( Y ) X + 1 2 g ( X , Y ) θ ♯ + g ( X , N ( ⋅ , Y ) ♯ ) ,

where ♯ is the isomorphism induced by the metric g between 1-forms and vector fields, and N is the Nijenhuis tensor defined by

4 N ( X , Y ) = [ J X , J Y ] − [ X , Y ] − J [ J X , Y ] − J [ X , J Y ] .

Furthermore, the canonical Weyl connection D W is related to the Levi-Civita connection D g by

(2) D X W Y = D X g Y − 1 2 θ ( X ) Y − 1 2 θ ( Y ) X + 1 2 g ( X , Y ) θ ♯ ,

We also remark that (for more details, see [37])

(3) g ( ( D Z W J ) X , J Y ) = − 2 g ( N ( X , Y ) , Z ) ,

and that, on any four-dimensional almost-Hermitian manifold, Gauduchon proved [32, Proposition 1] that

(4) g ( N ( X , Y ) , Z ) + g ( N ( Y , Z ) , X ) + g ( N ( Z , X ) , Y ) = 0 ,

for any vector fields X , Y , Z .

The almost-complex structure J is integrable if and only if N vanishes [45]. Hence, D W preserves J if and only if J is integrable.

We will now list some of the most relevant curvatures that can be obtained tracing the curvature tensors R ∇ , R W , R g , R B together with their relations. To define them, we will consider a J -adapted g -orthonormal frame of the tangent bundle { e 1 , e 2 = J e 1 , e 3 , e 4 = J e 3 } .

The first Chern-Ricci form ρ ∇ (called also the Hermitian Ricci form) of the Chern connection ∇ is defined as follows:

ρ ∇ ( X , Y ) = 1 2 ∑ i = 1 4 g ( R X , Y ∇ e i , J e i ) ,

similarly, the Bismut-Ricci form Ric B is defined by

Ric B ( X , Y ) = 1 2 ∑ i = 1 4 g ( R X , Y B e i , J e i ) .

These 2-forms ρ ∇ and Ric B are closed, and they are representatives of the first Chern class 2 π c 1 ( T M , J ) in de Rham cohomology. Indeed, they differ by the exact factor d J θ , i.e.

(5) Ric B = ρ ∇ + d J θ ,

see [32, equation (2.7.6)] (see also [1,13] and the references therein). We also remark that these forms are not necessarily J -invariant.

We denote by r the second Chern-Ricci form of ∇ defined by

r ( X , Y ) = 1 2 ∑ i = 1 4 g ( R e i , J e i ∇ X , Y ) ,

or equivalently,

r = R ∇ ( F ) .

r is a J -invariant 2-form but not closed in general.

Similarly, R W ( F ) is given by the following formula:

R W ( F ) = 1 2 ∑ i = 1 4 g ( R e i , J e i W X , Y ) .

The tensor R W ( F ) is a 2-form and it is not J -invariant in general. However, when J is integrable, D W preserves J and so R W ( F ) = ( R W ( F ) ) J , + is J -invariant. The Weyl Ricci tensor Ric W is defined in [33] as follows:

Ric W ( X , Y ) = ∑ i = 1 4 g ( R e i , X W e i , Y ) .

Note that the tensor Ric W is symmetric (this is only true in dimension 4). On the other hand, the tensor Ric ˜ W , defined as (see, e.g. [51])

Ric ˜ W ( X , Y ) = ∑ i = 1 4 g ( R X , e i W Y , e i ) ,

is not symmetric in general. Its anti-symmetric part is d θ , while its symmetric part is Ric W , i.e. [2],

Ric ˜ W = Ric W + d θ .

We define the Riemannian Ricci tensor Ric g as

Ric g ( X , Y ) = ∑ i = 1 4 g ( R e i , X g e i , Y ) ,

and the ⋆ -Ricci tensor ρ ⋆ as

ρ ⋆ ( X , Y ) = R g ( F ) ( X , J Y ) = 1 2 ∑ i = 1 4 g ( R e i , J e i g X , J Y ) .

From the definition, it follows that ρ ⋆ ( X , Y ) = ρ ⋆ ( J Y , J X ) so ρ ⋆ is symmetric if and only it is J -invariant.

We define the Hermitian scalar curvature s H as

s H = ∑ i = 1 4 r ( e i , J e i ) ,

the Riemannian scalar curvature s g as s g = ∑ i = 1 4 Ric g ( e i , e i ) , the conformal scalar curvature s W as s W = ∑ i = 1 4 Ric W ( e i , e i ) , and the ⋆ -scalar curvature s ⋆ as s ⋆ = ∑ i = 1 4 ρ ⋆ ( e i , e i ) .

The conformal scalar curvature s W is then related to the Riemannian scalar curvature s g by [2]

(6) s W = s g − 3 δ g θ − 3 2 ‖ θ ‖ 2 ,

where ‖ θ ‖ 2 = g ( θ , θ ) . Furthermore, for any almost-Hermitian manifold of dimension 4, we have that [59]

(7) ( ρ ⋆ ) sym − ( Ric g ) J , + = s ⋆ − s g 4 g .

On the other hand, s ⋆ is related to the Riemannian scalar curvature s g by (see [38,55], while in the integrable case, see [61])

(8) s ⋆ − s g = − 2 δ g θ − ‖ θ ‖ 2 + 2 ‖ N ‖ 2 .

3 Second Chern-Einstein metrics

Let ( M , g , J ) be an almost-Hermitian manifold of real dimension 4. We recall that with our notations, the metric g is said to be second Chern-Einstein if

r = s H 4 F ,

(we note that s H is not necessarily constant here).

3.1 Relation between second Chern-Ricci and Weyl-Ricci tensors

Under a conformal change g ˜ = e 2 f g , the conformal variation of the second Chern-Ricci form r of ( g , J ) is given by ([40], in the Hermitian case, see [30])

(9) r ˜ = r + ( Δ g f + g ( θ , d f ) ) F ,

where r ˜ is the second Ricci form of ( g ˜ , J ) and Δ g is the Riemannian Laplacian of g . The condition of being second Chern-Einstein, in the almost-Hermitian setting, is then conformally invariant. Hence, we investigate the relation between the curvature R ∇ of the Chern connection ∇ and the curvature R W of the Weyl connection D W .

Proposition 1

The curvatures R ∇ and R W are related by

R X , Y ∇ Z = R X , Y W Z − 1 2 ( d J θ ) ( X , Y ) J Z − 1 2 ( d θ ) ( X , Y ) Z − 1 2 ( D X W ( D Y W J ) − D Y W ( D X W J ) − D [ X , Y ] W J ) J Z + 1 4 ( ( D X W J ) ( D Y W J ) − ( D Y W J ) ( D X W J ) ) Z .

Proof

From (2) and (1), we obtain that

∇ X Y = D X W Y + 1 2 θ ( X ) Y − 1 2 θ ( J X ) J Y + g ( X , N ( ⋅ , Y ) ) .

From (3), we obtain that

∇ X Y = D X W Y + 1 2 θ ( X ) Y − 1 2 θ ( J X ) J Y + 1 2 ( D X W J ) J Y .

Then, we compute the curvature.□

When J is integrable, the relation reduces to [33]

R ∇ = R W − 1 2 ( d J θ ) ⊗ J − 1 2 ( d θ ) ⊗ I d .

We remark that the part ( R X , Y W ) J , − of R X , Y W that anti-commutes with J is given precisely by (see, e.g. [37, equation (2.15)])

(10) ( R X , Y W ) J , − = 1 2 ( D X W ( D Y W J ) − D Y W ( D X W J ) − D [ X , Y ] W J ) .

We obtain then the following

Corollary 2

Let ( M , g , J ) be an almost-Hermitian four-dimensional manifold. Then,

R X , Y ∇ Z = ( R X , Y W ) J , + Z − 1 2 ( d J θ ) ( X , Y ) J Z − 1 2 ( d θ ) ( X , Y ) Z + 1 4 ( ( D X W J ) ( D Y W J ) − ( D Y W J ) ( D X W J ) ) Z ,

where ( R X , Y W ) J , + is the part of R X , Y W that commutes with J.

It follows from Corollary 2 that we can relate the second Chern-Ricci curvature with R W ( F ) extending the relation in the integrable case [33, Theorem 1]

Corollary 3

Let ( M , g , J ) be an almost-Hermitian four-dimensional manifold. Then,

r = ( R W ( F ) ) J , + + 1 2 ( δ g θ + ‖ θ ‖ 2 ) F − 1 4 ‖ N ‖ 2 F .

In particular, g is second Chern-Einstein if and only if ( R W ( F ) ) J , + is a multiple of F.

Proof

We first have that g ( d J θ , F ) = − δ g θ − ‖ θ ‖ 2 , and g ( d θ , F ) = 0 . Moreover, using (3), we see that

∑ i = 1 4 g ( ( D e i W J ) ( D J e i W J ) − ( D J e i W J ) ( D e i W J ) Z , V ) = − 2 ∑ i = 1 4 g ( ( D e i W J ) ( D e i W J ) Z , J V ) , = 2 ∑ i = 1 4 g ( ( D e i W J ) Z , ( D e i W J ) J V ) , = 4 ∑ i = 1 4 g ( N ( Z , ( D e i W J ) J V ) , J e i ) , = 4 ∑ i , j = 1 4 g ( N ( Z , e j ) , J e i ) g ( ( D e i W J ) J V , e j ) , = 8 ∑ i , j = 1 4 g ( N ( Z , e j ) , J e i ) g ( N ( J V , e j ) , J e i ) , = − 8 ∑ i , j = 1 4 g ( N ( J Z , e j ) , e i ) g ( N ( V , e j ) , e i ) .

Now, we use the crucial fact that we are in dimension 4 with ‖ N ‖ 2 = 8 ‖ N ( e 1 , e 3 ) ‖ 2

∑ i , j = 1 4 g ( N ( J Z , e j ) , e i ) g ( N ( V , e j ) , e i ) = ∑ i , j , k , l = 1 4 g ( J Z , e k ) g ( V , e l ) g ( N ( e k , e j ) , e i ) g ( N ( e l , e j ) , e i ) , = ∑ j , k , l = 1 4 g ( J Z , e k ) g ( V , e l ) g ( N ( e k , e j ) , N ( e l , e j ) ) , = ∑ j , k = 1 4 g ( J Z , e k ) g ( V , e k ) g ( N ( e k , e j ) , N ( e k , e j ) ) , = 1 4 ‖ N ‖ 2 g ( J Z , V ) .

The result then follows.□

The canonical Weyl connection D W is said to be Einstein-Weyl if Ric W (or equivalently the symmetric part of Ric ˜ W ) is proportional to the metric g . When J is integrable, the metric g is second Chern-Einstein if and only if D W is Einstein-Weyl [33, Theorem 1]. This is due to the fact that Ric W ( J X , Y ) = R W ( F ) ( X , Y ) because R X , Y W commutes with J .

From (2), we also obtain

Corollary 4

Let ( M , g , J ) be an almost-Hermitian four-dimensional manifold. Then,

R W ( F ) ⋅ , J ⋅ J , + = ( ρ ⋆ ) sym + ( D g θ ) sym , J , + − 1 4 ‖ θ ‖ 2 g + 1 2 ( θ ⊗ θ ) J , + .

Then, combining Corollary 3 with Corollary 4 and using (7) and (8), we deduce the following.

Corollary 5

Let ( M , g , J ) be an almost-Hermitian four-dimensional manifold. Then,

r ⋅ , J ⋅ = ( Ric g ) J , + + 1 4 ‖ N ‖ 2 g + ( D g θ ) sym , J , + + 1 2 ( θ ⊗ θ ) J , + .

3.2 The conditions ( D g θ ) sym , J , − = 0 and ℒ θ ♯ g = 0

In [29], Gauduchon proved that every conformal class of an almost-Hermitian metric has a unique representative g (up to constant), which satisfies δ g θ = 0 . We call such metric (with a unit total volume) a Gauduchon metric. With this assumption, by using Corollory 5, we can prove the following (when the metric is Einstein-Weyl, see [58] and also [31])

Proposition 6

Let ( M , g ˜ , J ) be a compact four-dimensional almost-Hermitian manifold and g ˜ is a second Chern-Einstein metric. Suppose that the Gauduchon metric g in the conformal class [ g ˜ ] satisfies ( D g θ ) sym , J , − = 0 , where θ is the Lee form of g. Then, the g-Riemannian dual of θ is a Killing vector field of g.

Proof

The condition r = s H 4 F is conformally invariant. Then, from Corollory 5, we have for the Gauduchon metric g

s H 4 g = ( Ric g ) J , + + 1 4 ‖ N ‖ 2 g + ( D g θ ) sym , J , + + 1 2 ( θ ⊗ θ ) 1 , 1 .

We recall that δ g θ = − g ( D g θ , g ) . Taking the inner product with ( D g θ ) sym = ( D g θ ) sym , J , + and integrating, we have

− ∫ M s H 4 δ g θ v g = ∫ M g ( ( Ric g ) J , + , ( D g θ ) sym ) − 1 4 ‖ N ‖ 2 δ g θ + ‖ ( D g θ ) sym ‖ 2 + 1 2 g ( ( θ ⊗ θ ) J , + , ( D g θ ) sym ) v g ,

where v g is the volume form. Since the metric is Gauduchon, i.e. δ g θ = 0 , we obtain

∫ M ‖ ( D g θ ) sym ‖ 2 v g = − ∫ M g ( ( Ric g ) J , + , ( D g θ ) sym ) + 1 2 g ( ( θ ⊗ θ ) J , + , ( D g θ ) sym ) v g , = − ∫ M g ( Ric g , D g θ ) + 1 2 g ( θ ⊗ θ , D g θ ) v g , = − ∫ M g ( δ g Ric g , θ ) + 1 2 g ( δ g ( θ ⊗ θ ) , θ ) v g , = − ∫ M g − 1 2 d s g , θ + 1 2 ( δ g θ ‖ θ g ‖ 2 − g ( D θ g θ , θ ) ) v g , = − ∫ M − 1 2 s g δ g θ + 1 2 − 1 2 g ( d ‖ θ ‖ g 2 , θ ) v g , = 1 4 ∫ M ‖ θ ‖ g 2 δ g θ v g = 0 ,

where we used the contracted Bianchi identity δ g Ric g = − 1 2 d s g . The lemma follows.□

We have seen that in the integrable case second Chern-Einstein is equivalent to Weyl-Einstein (also in greater dimension if we ask the LCK condition). Then, second Chern-Einstein together with LCK implies that D g θ = 0 and that both θ ♯ , J θ ♯ are Killing real holomorphic vector fields, see [31,49]. However, in the almost-Hermitian case, the condition ( D g θ ) sym , J , − = 0 is necessary, see the examples in Section 4.

Moreover, we remark that ( D g θ ) sym , J , − = 0 is equivalent to ( ℒ θ ♯ J ) sym = 0 , where ℒ is the Lie derivative, so the flow of the vector field θ ♯ does not necessarily preserve J , i.e. θ ♯ is not necessarily a real holomorphic vector field. We also remark that when M is compact, the condition that J θ ♯ is a Killing vector field implies that ( M , g , J ) is LCS, i.e. d θ = 0 . Indeed, applying the Lie derivative ℒ J θ ♯ to the relation F = g ( J ⋅ , ⋅ ) , we obtain

d θ = − 2 ( D g J θ ) J ⋅ , ⋅ sym − g ( ℒ J θ ♯ J ⋅ , ⋅ ) .

Hence, if ( D g J θ ) sym = 0 , then d θ is J -anti-invariant. Thus, d θ is a self-dual d -exact 2-form on a compact manifold so d θ = 0 . However, the converse is not true in general: if d θ = 0 , then only the J -invariant part of ( D g J θ ) sym vanishes, i.e. ( D g J θ ) sym , J , + = 0 so J θ ♯ is not necessarily a Killing vector field.

3.3 Constant scalar curvatures

Let g be a second Chern-Einstein metric. If the Hermitian scalar curvature s H is a constant (respectively a non-constant function), then g is called a strong (respectively weak) second Chern-Einstein metric [7]. Since the conformal change of the second Chern-Ricci form (9) is the same as in the integrable case, we can generalize [7, Theorem B] to the almost-Hermitian setting [6,19,29,34,39].

Definition 7

Let g be the Gauduchon metric in the conformal class [ g ˜ ] . Then, the fundamental constant [9,12,27] is

C ( M , [ g ˜ ] , J ) = ∫ M s H F n n ! ,

where s H is the Hermitian scalar curvature of the almost-Hermitian structure ( g , J ) inducing the fundamental form F .

Theorem 8

[7, Theorem B][40, Corollary 5.10] Let ( M , g ˜ , J ) be a compact four-dimensional almost-Hermitian manifold and suppose that g ˜ is a weak second Chern-Einstein metric. Then, there is a representative in [ g ˜ ] such that its Hermitian scalar curvature has the same sign as C ( M , [ g ˜ ] , J ) . Moreover, if C ( M , [ g ˜ ] , J ) ≤ 0 , then there is a strong second Chern-Einstein representative in [ g ˜ ] .

Moreover, we can prove that s W is constant under some conditions (in the Einstein-Weyl case, see [51, Proposition 2.1]). First, thanks to Corollary 5 and Proposition 6, we obtain the following

Lemma 9

Suppose that ( M , g ˜ , J ) is a compact four-dimensional almost-Hermitian manifold, where g ˜ is a second Chern-Einstein metric. Suppose that the Gauduchon metric g in the conformal class [ g ˜ ] satisfies ( D g θ ) sym , J , − = 0 , where θ is the Lee form of g. Then

(11) ( Ric g ) J , + = s H 4 g − 1 4 ‖ N ‖ 2 g − 1 2 ( θ ⊗ θ ) J , + .

Then we can prove that

Proposition 10

Let ( M , g ˜ , J ) be a compact four-dimensional almost-Hermitian manifold and g ˜ is a second Chern-Einstein metric. Suppose that the Gauduchon metric g in the conformal class [ g ˜ ] satisfies ( D g θ ) sym , J , − = 0 , where θ is the Lee form of g. Moreover, suppose that Ric g is J-invariant and J θ ♯ is a Killing vector field. Then the conformal scalar curvature s W of g is constant.

Proof

We apply the codifferential δ g to (11), and we use the contracted Bianchi identity δ g Ric g = − 1 2 d s g . We obtain

− 1 2 d s g = − 1 4 d s H + 1 4 d ( ‖ N ‖ 2 ) − 1 4 δ g ( θ ⊗ θ ) − 1 4 δ g ( J θ ⊗ J θ ) , = − 1 4 d s H + 1 4 d ( ‖ N ‖ 2 ) − 1 4 ( δ g θ ⊗ θ − D θ ♯ g θ ) − 1 4 ( δ g J θ ⊗ θ − D J θ ♯ g J θ ) , = − 1 4 d s H + 1 4 d ( ‖ N ‖ 2 ) + 1 4 D θ ♯ g θ + 1 4 D J θ ♯ g J θ .

Hence

d s g = 1 2 d s H − 1 2 d ( ‖ N ‖ 2 ) − 1 2 D θ ♯ g θ − 1 2 D J θ ♯ g J θ .

On the other hand, from the trace of (11), we have s g = s H − ‖ N ‖ 2 − 1 2 ‖ θ ‖ 2 . Thus,

d ( s H − ‖ N ‖ 2 ) = d ( ‖ θ ‖ 2 ) − D θ ♯ g θ − D J θ ♯ g J θ .

By applying again the codifferential δ g , we have

(12) Δ g ( s H − ‖ N ‖ 2 ) = Δ g ( ‖ θ ‖ 2 ) − δ g D θ ♯ g θ − δ g D J θ ♯ g J θ ,

where Δ g is the Laplacian. From Proposition 6, θ ♯ is Killing. Thus, by using Cartan formula and because ℒ θ ♯ θ = 0 , we obtain

(13) δ g D θ ♯ g θ = 1 2 δ g ( d θ ( θ ♯ , ⋅ ) ) = − 1 2 δ g d ( ‖ θ ‖ 2 ) = − 1 2 Δ g ( ‖ θ ‖ 2 ) .

Similarly, because J θ ♯ is a Killing vector field, we have δ g D J θ ♯ g J θ = − 1 2 Δ g ( ‖ θ ‖ 2 ) . From (12), we obtain

Δ g ( s H − ‖ N ‖ 2 − 2 ‖ θ ‖ 2 ) = 0 .

Since M is compact and from (6), we obtain that s H − ‖ N ‖ 2 − 2 ‖ θ ‖ 2 = s g − 3 2 ‖ θ ‖ 2 = s W is a constant.□

We can also deduce from (2), Proposition 6 and Lemma 9 that

Corollary 11

Ric W = Ric g − 1 2 ( ‖ θ ‖ 2 g − θ ⊗ θ ) , = s W 4 g + ( Ric g ) J , − + 1 2 ( θ ⊗ θ ) J , − .

In particular, the metric g is Einstein-Weyl if and only if

(14) ( Ric g ) J , − = − 1 2 ( θ ⊗ θ ) J , − .

3.4 The main theorems

By combining the equations (7) and (8), we obtain

(15) ( ρ ⋆ ) sym − ( Ric g ) J , + = − 1 4 ( 2 δ g θ + ‖ θ ‖ 2 − 2 ‖ N ‖ 2 ) g .

We have already computed ( Ric g ) J , + in (11); now we want to compute ρ ⋆ ( θ ♯ , θ ♯ ) so that we can evaluate (15) on ( θ ♯ , θ ♯ ) and obtain a condition on the almost-Hermitian structure when g is second Chern-Einstein. We will assume that the Riemannian dual of θ is a Killing vector field.

Lemma 12

Suppose that ( M , g , J ) is a four-dimensional compact almost-Hermitian manifold. Suppose that the Riemannian dual θ ♯ of the Lee form θ is a Killing vector field. Then,

ρ ⋆ ( θ ♯ , X ) = − 1 2 ( d θ ) J , − ( θ ♯ , X ) + g ( d θ , N X ) ,

for any vector field X, where N X ( Y , Z ) = g ( N ( Y , Z ) , X ) .

Proof

Let α be a 1-form. Then

( δ g ( D g α ) J , + − δ g ( D g α ) J , − ) ( X ) = − ∑ i = 1 4 ( D e i g ( ( D g α ) J , + − ( D g α ) J , − ) ) ( e i , X ) , = ∑ i = 1 4 − D e i g ( D J e i g α ( J X ) ) + D g α ( J D e i g e i , J X ) + D g α ( J e i , J D e i g X ) , = ∑ i = 1 4 − D e i g ( D J e i g α ( J X ) ) + D g α ( D e i g ( J e i ) , J X ) + D g α ( J e i , D e i g ( J X ) ) − D g α ( ( D e i g J ) e i , J X ) − D g α ( J e i , ( D e i g J ) X ) , = ∑ i = 1 4 − ( D e i g ( D J e i g α ) ) ( J X ) + D g α ( D e i g ( J e i ) , J X ) − D g α ( ( D e i g J ) e i , J X ) − D g α ( J e i , ( D e i g J ) X ) , = ∑ i = 1 4 1 2 g ( R e i , J e i g α ♯ , J X ) − D g α ( J e i , ( D e i g J ) X ) − D g α ( J θ ♯ , J X ) , = ρ ⋆ ( α ♯ , X ) − D g α ( J θ ♯ , J X ) − ∑ i = 1 4 D g α ( J e i , ( D e i g J ) X ) .

On the other hand, by using that g ( d θ , F ) = 0 and δ g = − ∗ g d ∗ g , we have

δ g ( D g θ ) J , + − δ g ( D g θ ) J , − = 1 2 δ g ( d θ ) J , + − 1 2 δ g ( d θ ) J , − , = 1 2 δ g ( d θ ) − − 1 2 δ g ( d θ ) + , = − 1 2 ∗ g d ∗ g ( d θ ) − + 1 2 ∗ g d ∗ g ( d θ ) + , = 1 2 ∗ g d ( d θ ) − + 1 2 ∗ g d ( d θ ) + , = 1 2 ∗ g d ( ( d θ ) − + d ( d θ ) + ) = 1 2 ∗ g d ( d θ ) = 0 .

We deduce then

(16) ρ ⋆ ( θ ♯ , X ) = D g θ ( J θ ♯ , J X ) + ∑ i = 1 4 D g θ ( J e i , ( D e i g J ) X ) .

Now, we would like to compute the second term on the right-hand side of (16). We first recall that

( D X g J ) Y = 1 2 g ( X , Y ) J θ ♯ + 1 2 θ ( J Y ) X + 1 2 g ( J X , Y ) θ ♯ − 1 2 θ ( Y ) J X + 2 ( g ( N ( Y , ⋅ ) , J X ) ) ♯ ,

which can be easily deduced from (2) and (3). Hence,

(17) ∑ i = 1 4 D g θ ( J e i , ( D e i g J ) X ) = ∑ i = 1 4 D J e i g θ 1 2 g ( e i , X ) J θ ♯ + D J e i g θ 1 2 θ ( J X ) e i + D J e i g θ 1 2 g ( J e i , X ) θ ♯ − D J e i g θ 1 2 θ ( X ) J e i + 2 g ( N ( X , D J e i g θ ♯ ) , J e i ) , = 1 2 g ( D J X g θ , J θ ) + 1 2 g ( D X g θ , θ ) + 2 ∑ i = 1 4 g ( N ( X , D J e i g θ ♯ ) , J e i ) ,

where we use the fact that g ( d θ , F ) = δ g θ = 0 . Moreover, by using (4), we compute the third term in (17)

∑ i = 1 4 g ( N ( X , D J e i g θ ♯ ) , J e i ) = ∑ i , j = 1 4 g ( D J e i θ ♯ , e j ) g ( N ( X , e j ) J e i ) , = ∑ i , j = 1 4 − g ( D J e i g θ ♯ , e j ) g ( N ( e j , J e i ) , X ) − ( D J e i g θ ♯ , e j ) g ( N ( J e i , X ) , e j ) , = ∑ i , j = 1 4 − g ( D J e i g θ ♯ , e j ) g ( N ( e j , J e i ) , X ) − ( D e j g θ ♯ , J e i ) g ( N ( X , J e i ) , e j ) .

Thus,

∑ i = 1 4 g ( N ( X , D J e i g θ ♯ ) , J e i ) = − 1 2 ∑ i , j = 1 4 g ( D J e i g θ ♯ , e j ) g ( N ( e j , J e i ) , X ) , = 1 2 ∑ i , j = 1 4 g ( D e i g θ ♯ , e j ) g ( N ( e i , e j ) , X ) , = 1 2 g ( d θ , N X ) .

From (17), we deduce that

∑ i = 1 4 D g θ ( J e i , ( D e i g J ) X ) = 1 2 g ( D J X g θ , J θ ) + 1 2 g ( D X g θ , θ ) + g ( d θ , N X ) .

Finally, from (16) and (17), we conclude that

ρ ⋆ ( θ ♯ , X ) = D g θ ( J θ ♯ , J X ) + 1 2 g ( D J X g θ , J θ ) + 1 2 g ( D X g θ , θ ) + g ( d θ , N X ) , = 1 2 D g θ ( J θ ♯ , J X ) + 1 2 g ( D X g θ , θ ) + g ( d θ , N X ) , = − 1 2 ( d θ ) J , − ( θ ♯ , X ) + g ( d θ , N X ) .

The lemma follows.□

As consequences of Lemma 12, we obtain

Corollary 13

Suppose that ( M , g , J ) is a four-dimensional compact locally conformally symplectic manifold. Suppose that the Riemannian dual θ ♯ of the Lee form θ is a Killing vector field. Then,

ρ ⋆ ( θ ♯ , X ) = 0 ,

for any vector field X.

We also extend [33, Lemma 2] to the almost-Hermitian setting.

Corollary 14

Suppose that ( M , g , J ) is a compact four-dimensional almost-Hermitian manifold. Suppose that the Riemannian dual θ ♯ of the Lee form θ is a Killing vector field. Then,

ρ ⋆ ( θ ♯ , θ ♯ ) = g ( d θ , N θ ♯ ) ,

where N θ ♯ ( X , Y ) = g ( N ( X , Y ) , θ ♯ ) .

Notice that when J is integrable ρ ⋆ ( θ ♯ , θ ♯ ) = 0 . As a matter of fact, this condition simplifies in a natural way the problem, thus we will now prove our main theorems with the assumption that d θ = 0 (Theorem 15) or N θ ♯ = 0 (Theorem 19).

Theorem 15

Suppose that ( M , g ˜ , J ) is a four-dimensional compact locally conformally symplectic manifold such that g ˜ is a second Chern-Einstein metric. Suppose that the Gauduchon metric g in the conformal class [ g ˜ ] satisfies ( D g θ ) sym , J , − = 0 , where θ is the Lee form of g. Then, either

( M , g , J ) is a second Chern-Einstein almost-Kähler manifold, or
θ is D g -parallel and the conformal scalar curvature s W is non-positive and the ⋆ -scalar curvature s ⋆ is a positive constant. Moreover, s W is identically zero if and only if J is integrable and so ( M , J ) is a Hopf surface as described in [33, Theorem 2]. Furthermore, if s W is nowhere zero, then χ = σ = 0 , where χ and σ are the Euler class and signature of M respectively.

Proof

From Proposition 6, θ ♯ is a Killing vector field. Since d θ = 0 , it follows that θ is D g -parallel, and so θ ♯ has a constant length. By combining Corollaries 14 and 11 and (15), we obtain

(18) ( s H + ‖ N ‖ 2 − 2 ‖ θ ‖ 2 ) ‖ θ ‖ 2 = 0 .

Hence, either θ = 0 or s H = 2 ‖ θ ‖ 2 − ‖ N ‖ 2 . Now, if s H = 2 ‖ θ ‖ 2 − ‖ N ‖ 2 , then s g = 3 2 ‖ θ ‖ 2 − 2 ‖ N ‖ 2 . Hence, from (6), we obtain that

s W = − 2 ‖ N ‖ 2 .

Thus, s W ≡ 0 if and only if J is integrable. Moreover, if s W is nowhere zero, then 5 χ + 6 σ = 0 using [10, Lemma 3]. The existence of a non-trivial Killing vector field of constant length implies χ = 0 by Hopf theorem [35], hence, χ = σ = 0 . Furthermore, by using (8), we have that s ⋆ = s g − ‖ θ ‖ 2 + 2 ‖ N ‖ 2 = 3 2 ‖ θ ‖ 2 − 2 ‖ N ‖ 2 − ‖ θ ‖ 2 + 2 ‖ N ‖ 2 = 1 2 ‖ θ ‖ 2 , and so s ⋆ is constant.□

Remark 16

We recall that there are many restrictions to the existence of a non-zero Killing vector field of constant length on a Riemannian manifold, see, e.g. [15,46]

Remark 17

When J is integrable, if θ is D g -parallel (the metric g is called Vaisman), then both θ ♯ and J θ ♯ are Killing and real holomorphic vector fields [23].

Gauduchon proved in [31] (in dimension 3, see also [53]) that if the conformal scalar curvature of a compact Einstein-Weyl manifold is non-positive but non-identically zero, then the Gauduchon metric is an almost-Kähler Riemannian Einstein metric (i.e. Ric g is proportional to g ). We can then deduce the following.

Corollary 18

[31, Theorem 3][37, Corollary 4.2] Let ( M , g ˜ , J ) be a four-dimensional compact locally conformally symplectic manifold. Suppose that the Gauduchon metric g ∈ [ g ˜ ] is a second Chern-Einstein and an Einstein-Weyl metric. Then ( M , g , J ) is either an almost-Kähler Riemannian Einstein manifold or a Hopf surface as described in [33].

Proof

It follows from Theorem 15 that either g is almost-Kähler or s W is non-positive. The corollary follows from the fact that if the metric g is an almost-Kähler Einstein-Weyl metric, then g is a Riemannian Einstein metric [31].□

The vanishing of N θ ♯ means that θ ♯ is g -orthogonal to Span ( N ) , which is the distribution spanned by all the vector fields N ( X , Y ) . Moreover, if N θ ♯ = 0 , then N J θ ♯ = 0 . In fact, in real dimension 4, at each point, the dimension of Span ( N ) is equal to 0 or 2 [18] (see also [44] for more details). A similar proof to Theorem 15 gives the following.

Theorem 19

Suppose that ( M , g ˜ , J ) is a four-dimensional compact almost-Hermitian manifold such that g ˜ is a second Chern-Einstein metric. Suppose that the Gauduchon metric g in the conformal class [ g ˜ ] satisfies ( D g θ ) sym , J , − = 0 , N θ ♯ = 0 , and θ ♯ is of constant length, where θ is the Lee form of g. Then, either

( M , g , J ) is a second Chern-Einstein almost Kähler manifold or
θ ♯ is a non-zero Killing vector field of constant length and the conformal scalar curvature s W is non-positive and the ⋆ -scalar curvature s ⋆ is a positive constant. Moreover, s W is identically zero if and only if J is integrable and so ( M , J ) is a Hopf surface as described in [33, Theorem 2]. Furthermore, if s W is nowhere zero, then χ = σ = 0 , where χ and σ are the Euler class and signature of M, respectively.

Now, we would like to investigate the condition N θ ♯ = 0 and see if it can be implied by the J -invariance of different Ricci forms.

Lemma 20

Let ( M , J , g ) be a four-dimensional almost-Hermitian manifold. Then

ρ ∇ = R W ( F ) − d J θ − ( d θ ) J ⋅ , ⋅ J , − − 1 2 ∑ i = 1 4 ( N ( e i , e j ) ) ♭ ∧ ( J N ( e i , e j ) ) ♭ ,

where ♭ is the isomorphism between vector fields and 1-forms induced by g. In particular,

( ρ ∇ ) J , − = ( R W ( F ) ) J , − − ( d J θ ) J , − − ( d θ ) J ⋅ , ⋅ J , − .

Proof

From Proposition 1, we have that

2 ρ ∇ ( X , Y ) = ∑ i = 1 4 g ( R X , Y ∇ e i , J e i ) , = ∑ i = 1 4 g ( R X , Y W e i , J e i ) − 1 2 ( d J θ ) ( X , Y ) g ( J e i , J e i ) − 1 2 ( d θ ) ( X , Y ) g ( e i , J e i ) − 1 2 g ( ( D X W ( D Y W J ) − D Y W ( D X W J ) − D [ X , Y ] W J ) J e i , J e i ) + 1 4 g ( ( ( D X W J ) ( D Y W J ) − ( D Y W J ) ( D X W J ) ) e i , J e i ) , = ∑ i = 1 4 g ( R X , Y W e i , J e i ) − 2 ( d J θ ) ( X , Y ) + 1 4 ∑ i = 1 4 g ( ( ( D X W J ) ( D Y W J ) − ( D Y W J ) ( D X W J ) ) e i , J e i ) .

Because D W is torsion free, and using the relation,

g ( R X , Y W Z , W ) + g ( R X , Y W W , Z ) = ( d θ ) ( X , Y ) g ( Z , W ) ,

we can easily deduce the following relation for any vector fields X , Y , Z , W

2 g ( R X , Y W Z , W ) = 2 g ( R Z , W W X , Y ) − d θ ( Z , W ) g ( X , Y ) + d θ ( X , Y ) g ( Z , W ) − d θ ( X , W ) g ( Y , Z ) − d θ ( Y , Z ) g ( X , W ) + d θ ( Y , W ) g ( X , Z ) + d θ ( X , Z ) g ( Y , W ) .

In particular,

∑ i = 1 4 g ( R e i , J e i W X , Y ) = ∑ i = 1 4 g ( R X , Y W e i , J e i ) + d θ ( J X , Y ) + d θ ( X , J Y ) .

Hence,

(19) 2 ρ ∇ ( X , Y ) = 2 R W ( F ) ( X , Y ) − d θ ( J X , Y ) − d θ ( X , J Y ) − 2 ( d J θ ) ( X , Y ) + 1 4 ∑ i = 1 4 g ( ( ( D X W J ) ( D Y W J ) − ( D Y W J ) ( D X W J ) ) e i , J e i ) .

Moreover, by using (3), we have

∑ i = 1 4 g ( ( D X W J ) ( D Y W J ) e i , J e i ) = − 2 ∑ i = 1 4 g ( N ( ( D Y W J ) e i , e i ) , X ) , = − 2 ∑ i , j = 1 4 g ( N ( e j , e i ) , X ) g ( ( D Y W J ) e i , e j ) , = − 4 ∑ i , j = 1 4 g ( N ( e i , e j ) , X ) g ( J N ( e i , e j ) , Y ) .

Hence

∑ i = 1 4 g ( ( ( D X W J ) ( D Y W J ) − ( D Y W J ) ( D X W J ) ) e i , J e i ) = − 4 ∑ i , j = 1 4 g ( N ( e i , e j ) , X ) g ( J N ( e i , e j ) , Y ) + 4 ∑ i , j = 1 4 g ( N ( e i , e j ) , Y ) g ( J N ( e i , e j ) , X ) , = − 4 ∑ i , j = 1 4 ( ( N ( e i , e j ) ) ♭ ∧ ( J N ( e i , e j ) ) ♭ ) ( X , Y ) .

From (19), we deduce that

□ ρ ∇ ( X , Y ) = R W ( F ) ( X , Y ) − 1 2 ( d θ ( J X , Y ) + d θ ( X , J Y ) ) − ( d J θ ) ( X , Y ) − 1 2 ∑ i , j = 1 4 ( N ( e i , e j ) ) ♭ ∧ ( J N ( e i , e j ) ) ♭ ( X , Y ) .

Remark 21

We can compute the J -anti-invariant part of R W ( F ) , and it is given by

( R W ( F ) ) J , − = − ∑ i = 1 4 g ( ( D e i g N ) ( ⋅ , ⋅ ) , e i ) + 3 2 N θ ♯ ,

where N θ ♯ = g ( N ( ⋅ , ⋅ ) , θ ♯ ) .

Corollary 22

Let ( M , g , J ) be a four-dimensional almost-Hermitian manifold. Suppose that ρ ∇ and R W ( F ) are J-invariant. Then N θ ♯ = 0 .

Proof

From Lemma 20, we obtain that

(20) ( d J θ ) J , − = − ( d θ ) J ⋅ , ⋅ J , − .

By applying the Lie derivative ℒ θ ♯ to the relation F = g ( J ⋅ , ⋅ ) and using the Cartan formula, we obtain

d J θ = − ‖ θ ‖ 2 F + θ ∧ J θ + 2 ( D g θ ) J ⋅ , ⋅ sym + g ( ℒ θ ♯ J ⋅ , ⋅ ) .

In particular,

(21) ( d J θ ) J , − = ( ℒ θ ♯ J ) anti-sym .

Similarly, we have

d θ = − 2 ( D g J θ ) J ⋅ , ⋅ sym − g ( ℒ J θ ♯ J ⋅ , ⋅ ) .

In particular,

(22) ( d θ ) J ⋅ , ⋅ J , − = g ( J ( ℒ J θ ♯ J ) anti-sym ⋅ , ⋅ ) .

By combining (20), (21), and (22) we deduce that

(23) ( ℒ J θ ♯ J ) anti-sym = J ( ℒ θ ♯ J ) anti-sym .

Now, for any almost-Hermitian manifold, we have

ℒ J θ ♯ J − J ( ℒ θ ♯ J ) = 4 N ( θ ♯ , ⋅ ) .

On the other hand, from (4), we see that ( g ( ( N ( θ ♯ , ⋅ ) , ⋅ ) ) anti-sym ) = − 2 N θ ♯ , so that

g ( ( ℒ J θ ♯ J ) anti-sym − J ( ℒ θ ♯ J ) anti-sym ⋅ , ⋅ ) = − 2 N θ ♯ .

From (23), we deduce that N θ ♯ = 0 .□

Corollary 23

Suppose that ( M , g ˜ , J ) is a compact four-dimensional almost-Hermitian manifold and g ˜ is a second Chern-Einstein metric. Suppose that the Gauduchon metric g in the conformal class [ g ˜ ] satisfies ( D g θ ) sym , J , − = 0 and that ρ ∇ and R W ( F ) are J-invariant. Then conclusions of Theorem 19 hold.

Proof

ρ ∇ and R W ( F ) being J -invariant implies that N θ ♯ = 0 by Corollary 22. We apply then Theorem 19.□

3.5 Relation between second Chern-Ricci and Bismut-Ricci tensors

Finally, we relate the second Chern-Ricci tensor with the Bismut-Ricci tensor. This will give us the possibility to compute it on almost-abelian Lie groups in Section 5. Thanks to Corollary 3 and Lemma 20, we see that both the first Chern-Ricci form ρ ∇ and the second Chern-Ricci form r can be expressed in terms of R W ( F ) . Hence, we obtain the following

Proposition 24

Let ( M , g , J ) be a four-dimensional almost-Hermitian manifold. Then,

r = ( ρ ∇ + d J θ ) J , + + 1 4 ( 2 δ g θ + 2 ‖ θ ‖ 2 − ‖ N ‖ 2 ) F + 1 2 ∑ i , j 4 ( N ( e i , e j ) ) ♭ ∧ ( J N ( e i , e j ) ) ♭ .

In particular, thanks to (5),

r = ( Ric B ) J , + + 1 4 ( 2 δ g θ + 2 ‖ θ ‖ 2 − ‖ N ‖ 2 ) F + 1 2 ∑ i , j 4 ( N ( e i , e j ) ) ♭ ∧ ( J N ( e i , e j ) ) ♭ .

From this proposition, we see that in the integrable case, the second Chern-Einstein problem agrees with the Bismut-Einstein problem

( Ric B ) J , + = λ F , for some function λ .

As a matter of fact, on a four-dimensional Hermitian manifold,

R W ( F ) = ( Ric B ) J , + .

This also follows from Lemma 20 and (5).

Remark 25

One can check that the crucial property of four-dimensional manifolds that leads to these relations between the second Chern-Ricci form, the first Bismut-Ricci form, and R W ( F ) is d F = θ ∧ F . As a consequence, in higher dimension 2 n , if we assume the Hermitian structure to be locally conformally Kähler, we obtain similar relations [1]. For example, computations analogous to the one we did here can show that on a LCK manifold

R W ( F ) = Ric 1 1 − n ,

where Ric ( t ) is the first Ricci form of the Gauduchon connection ∇ t introduced in [32], see Section 5 for the definition.

4 Four-dimensional compact examples with almost-Hermitian second Chern-Einstein metrics

Here, we use the same notation of Lie algebras as [48].

4.1 Conformally locally symplectic Lie algebras associated to compact solvmanifolds

The Lie algebra A 3 , 6 ⊕ A 1 : the structure of the Lie algebra is
[ e 1 , e 3 ] = − e 2 , [ e 2 , e 3 ] = e 1 ,
where { e 1 , e 2 , e 3 , e 4 } is a basis of A 3 , 6 ⊕ A 1 . The associated simply connected group to the Lie algebra A 3 , 6 ⊕ A 1 admits lattices (see, e.g. [4,5,17], in the notation of [4] A 3 , 6 ⊕ A 1 corresponds to r 3 , 0 ′ × R ). We consider the almost-complex structure
J e 1 = e 3 , J e 2 = e 4 .
The almost-complex structure J is non-integrable because N ( e 1 , e 2 ) = 1 4 e 3 .

We consider the following J -compatible metric g
g = ( 5 − 1 ) ( e 1 ⊗ e 1 + e 3 ⊗ e 3 ) + e 2 ⊗ e 2 + e 4 ⊗ e 4 ,
where { e 1 , e 2 , e 3 , e 4 } is the dual basis.

The pair ( g , J ) induces the fundamental form
F = ( 5 − 1 ) e 13 + e 24 ,
where e 13 = e 1 ∧ e 3 etc. Remark that the form d F = e 134 . Moreover, the Lee form is given by
θ = 1 ( 5 − 1 ) e 4 .
Hence, d θ = 0 . Moreover, ( D g θ ) sym , J , − = 0 . On the other hand, the second Chern-Ricci form is given by
r = 1 4 e 13 + 1 4 ( 5 − 1 ) e 24 ,
so the metric g is a second Chern-Einstein metric with a positive Hermitian scalar curvature s H = 1 5 − 1 . Thus, θ is D g -parallel. We also remark that N θ ♯ = 0 and the first Chern-Ricci form ρ ∇ = 1 2 e 13 is J-invariant.
The Lie algebra A 4 , 1 : the structure of the Lie algebra is
[ e 2 , e 4 ] = e 1 , [ e 3 , e 4 ] = e 2 ,
where { e 1 , e 2 , e 3 , e 4 } is a basis of A 4 , 1 . The associated simply connected group to the Lie algebra A 4 , 1 admits lattices (see for example [4,5,17], in the notation of [4] A 4 , 1 corresponds to n 4 ). We consider the almost-complex structure
J e 1 = e 3 , J e 2 = e 4 .
The almost-complex structure J is non-integrable because N ( e 1 , e 2 ) = 1 4 e 2 .

We consider the following J -compatible metric g
g = 1 2 ( e 1 ⊗ e 1 + e 3 ⊗ e 3 ) + e 2 ⊗ e 2 + e 4 ⊗ e 4 ,
where { e 1 , e 2 , e 3 , e 4 } is the dual basis.

The pair ( g , J ) induces the fundamental form
F = 1 2 e 13 + e 24 .
Remark that the form d F = 1 2 e 234 . Moreover, the Lee form is given by
θ = − 1 2 e 3 .
Hence, d θ = 0 . However, in this example, ( D g θ ) sym , J , − does not vanish so θ ♯ is not a Killing vector field. Explicitly, ( D g θ ) sym , J , − = 1 2 ( e 2 ⊗ e 4 + e 4 ⊗ e 2 ) . On the other hand, the second Chern-Ricci form r vanishes, so the metric g is a second Chern-Einstein metric with vanishing Hermitian scalar curvature. We also remark that N θ ♯ = 0 and the first Chern-Ricci form vanishes.
The Lie algebra A 4 , 8 : the structure of the Lie algebra is
[ e 2 , e 3 ] = e 1 , [ e 2 , e 4 ] = e 2 , [ e 3 , e 4 ] = − e 3 ,
where { e 1 , e 2 , e 3 , e 4 } is a basis of A 4 , 8 . The associated simply connected group to the Lie algebra A 4 , 8 admits lattices (see, e.g. [5,17], in the notation of [5] A 4 , 8 corresponds to d 4 ) We consider the almost-complex structure
J e 1 = e 4 , J e 2 = e 3 .
The almost-complex structure J is non-integrable because N ( e 1 , e 2 ) = 1 2 e 3 . We consider the following J -compatible metric g
g = ∑ i = 1 4 e i ⊗ e i ,
where { e 1 , e 2 , e 3 , e 4 } is the dual basis.

The pair ( g , J ) induces the fundamental form
F = e 14 + e 23 .
Remark that the form d F = − e 234 . Moreover, the Lee form is given by
θ = − e 4 .
Hence, d θ = 0 . However, in this example, ( D g θ ) sym , J , − does not vanish so θ ♯ is not a Killing vector field. Explicitly, ( D g θ ) sym , J , − = e 3 ⊗ e 3 − e 2 ⊗ e 2 . On the other hand, the second Chern-Ricci form r vanishes, so the metric g is a second Chern-Einstein metric with vanishing Hermitian scalar curvature. We also remark that N θ ♯ = 0 and the first Chern-Ricci form vanishes.

4.2 Non-conformally locally symplectic Lie algebra associated to compact solvmanifolds

The Lie algebra A 4 , 10 : the structure of the Lie algebra is

[ e 2 , e 3 ] = e 1 , [ e 2 , e 4 ] = − e 3 , [ e 3 , e 4 ] = e 2 ,

where { e 1 , e 2 , e 3 , e 4 } is a basis of A 4 , 10 . The associated simply connected group to the Lie algebra A 4 , 8 admits lattices (see, e.g. [5,17], in the notation of [5] A 4 , 10 corresponds to d 4 , 0 ′ ). We consider the almost-complex structure

J e 1 = e 3 , J e 2 = e 4 .

The almost-complex structure J is non-integrable because N ( e 1 , e 2 ) = 1 4 e 2 + 1 4 e 3 .

We consider the following J -compatible metric g

g = 1 + 17 8 ( e 1 ⊗ e 1 + e 3 ⊗ e 3 ) + ( e 2 ⊗ e 2 + e 4 ⊗ e 4 ) ,

where { e 1 , e 2 , e 3 , e 4 } is the dual basis.

The pair ( g , J ) induces the fundamental form

F = 1 + 17 8 e 13 + e 24 .

Remark that the form d F = − 1 + 17 8 e 124 . Moreover, the Lee form is given by

θ = − 1 + 17 8 e 1 .

Hence d θ ≠ 0 . In this example, ( D g θ ) sym , J , − vanishes. On the other hand, the second Chern-Ricci form is

r = 1 + 17 32 e 13 + 1 4 e 24

so the metric g is a second Chern-Einstein metric with positive Hermitian scalar curvature s H = 1 . Hence, θ ♯ is a Killing vector field but not D g -parallel. We also remark that in this example N θ ♯ ≠ 0 , and the first Chern-Ricci form ρ ∇ = 1 2 e 24 − 1 2 e 34 is not J -invariant.

5 Special metrics on almost-abelian Lie algebras

An almost-abelian Lie group G is a Lie group whose Lie algebra g has a codimension-one abelian ideal n ⊂ g . Given an almost-Hermitian left-invariant structure ( g , J ) on a 2 n -dimensional almost-abelian Lie group G , define n 1 ≔ n ∩ J n and J 1 ≔ J ∣ n 1 . Then we can choose an orthonormal basis { e 1 , … , e 2 n } for g such that

n = span R ⟨ e 1 , … , e 2 n − 1 ⟩ and J e i = e 2 n − i + 1 for i = 1 , … , n .

Hence, the fundamental form F ( ⋅ , ⋅ ) ≔ g ( J ⋅ , ⋅ ) associated to the almost-Hermitian structure ( J , g ) is

F = e 1 ∧ e 2 n + e 2 ∧ e 2 n − 1 + ⋯ + e n ∧ e n + 1 ,

given in terms of the dual left-invariant frame { e 1 , … , e 2 n } .

The algebra structure of g is completely described by the adjoint map

ad e 2 n : g → g x ↦ [ e 2 n , x ] .

The matrix associated to this endomorphism is

(24) ad e 2 n ∣ n = a b v A , a ∈ R , b , v ∈ n 1 , A ∈ gl ( n 1 ) .

The data ( a , b , v , A ) completely characterizes the almost-Hermitian structure ( g , J ) . For example, the integrability of J can be expressed in terms of ( a , b , v , A ) asking that b = 0 and A ∈ gl ( n 1 , J 1 ) , where gl ( n 1 , J 1 ) denotes endomorphisms of n 1 commuting with J 1 , see [11, Lemma 4.1].

On an almost-abelian almost-Hermitian Lie group ( G , [ ⋅ , ⋅ ] ( a , b , v , A ) , J , g ) , the Lee form is given by

(25) θ = J δ g F = ( J v ) ♭ − ( tr A ) e 2 n ,

with respect to the adapted unitary basis { e 1 , … , e 2 n } , see, e.g. [24].

Given an almost-Hermitian manifold ( M , g , J ) , Gauduchon introduced in [32] a one-parameter family ∇ ( t ) of canonical Hermitian connections, i.e. ∇ ( t ) g = ∇ ( t ) J = 0 . In this family, ∇ ( 1 ) = ∇ corresponds to the Chern connection, while ∇ ( − 1 ) = ∇ B corresponds to the Bismut connection. Any canonical connection ∇ ( t ) induces the associated first Ricci form

Ric ( t ) = 1 2 ∑ i = 1 2 n g ( R X , Y ∇ t e i , J e i ) ,

where R ∇ ( t ) denotes the curvature tensor of ∇ ( t ) .

The first Ricci forms of the canonical connections on a Lie group ( G , g ) equipped with an almost-Hermitian structure ( g , J ) were computed in [63]. In particular, for any parameter t ∈ R , these are given by

Ric ( t ) ( X , Y ) = − 1 2 { tr ( ad [ X , Y ] ∘ J ) − t tr ad J [ X , Y ] + ( t − 1 ) g ( F , d [ X , Y ] ♭ ) } .

Then, a direct computation leads to the following lemma.

Lemma 26

Let ( G , [ ⋅ , ⋅ ] ( a , b , v , A ) , g , J ) be an almost-abelian almost-Hermitian Lie group, endowed with an adapted unitary basis { e 1 , … , e 2 n } , determining the algebraic data ( a , b , v , A ) by (24). Then, the first Ricci form associated to the canonical connection ∇ t is

Ric ( t ) = − 1 2 { ( 2 a 2 + t a tr A + ( 1 − t ) ∣ v ∣ 2 + b ⋅ v ) e 1 + ( ( 2 a + t tr A ) b + A t b + ( 1 − t ) A t v ) ♭ } ∧ e 2 n .

In particular,

(26) Ric B = − 1 2 { ( 2 a 2 − a tr A + 2 ∣ v ∣ 2 + b ⋅ v ) e 1 ∧ e 2 n + ( ( 2 a − tr A ) b + A t b + 2 A t v ) ♭ ∧ e 2 n } .

Before studying the second Chern-Einstein problem, we examine the Bismut-Einstein problem on almost-Hermitian almost-abelian Lie groups of any dimension. We analyze the Einstein condition together with the request that d J d F = 0 . Indeed, in Hermitian geometry, the Einstein problem for the Bismut connection is stated as follows.

Definition 27

[25,26] Let ( M , g , J ) be a Hermitian manifold with g a pluriclosed metric (meaning that d J d F = 0 ). Then g is said to be a Bismut-Hermitian-Einstein metric if ( Ric B ) J , + = λ F for some function λ .

These metrics were first studied in [56] as fixed points of a parabolic flow of metrics, the pluriclosed flow. We shall remark that there is a lack of examples of such metrics: the only known are the Kähler-Einstein and the Bismut-flat metrics, the latter meaning that the whole Bismut curvature tensor vanishes ( R B = 0 ). In the following, we prove that on Hermitian almost-abelian Lie groups the Bismut-Hermitian-Einstein metrics are Kähler. However, as soon as we drop the integrability assumption, we are able to find non-almost-Kähler metrics that satisfy the Bismut-Einstein equation and such that d J d F = 0 .

5.1 Bismut-Einstein problem

We study here the Bismut-Einstein problem, ( Ric B ) J , + = λ F for λ ∈ R , on an almost-Hermitian almost-abelian Lie group ( G , [ ⋅ , ⋅ ] ( a , b , v , A ) , g , J ) . Thanks to equation (26), we see that to obtain a solution to the Einstein problem λ must vanish. Thus, we obtain the conditions

(27) 2 a 2 − a tr A + 2 ∣ v ∣ 2 + b ⋅ v = 0 , ( 2 a − tr A ) b + A t b + 2 A t v = 0 ,

Moreover, from a direct computation of d J d F , we obtain the following lemma.

Lemma 28

Let ( G , [ ⋅ , ⋅ ] ( a , b , v , A ) , g , J ) be an almost-Hermitian almost-abelian Lie group, endowed with an adapted unitary basis { e 1 , … , e 2 n } , determining the algebraic data ( a , b , v , A ) by (24). Then, the metric satisfies d J d F = 0 (here J d F = − d F ( J ⋅ , J ⋅ , J ⋅ ) ) if and only if for any x , y , z ∈ n 1

⟨ b , x ⟩ ( ⟨ A J y , z ⟩ − ⟨ A J z , y ⟩ ) − ⟨ b , y ⟩ ( ⟨ A J x , z ⟩ − ⟨ A J z , x ⟩ ) + ⟨ b , z ⟩ ( ⟨ A J x , y ⟩ − ⟨ A J y , x ⟩ ) = 0 , a ( ⟨ A J y , z ⟩ − ⟨ A J z , y ⟩ ) + ⟨ A J A y , z ⟩ − ⟨ A J z , A y ⟩ + ⟨ A J y , A z ⟩ − ⟨ A J A z , y ⟩ = 0 .

Remark 29

In the Hermitian case, i.e. when b = 0 and A commutes with J , the condition d J d F = 0 (the metric g is then called SKT [16]) is equivalent to

(28) a A + A 2 + A t A ∈ so ( n 1 ) ,

as showed in [11, Lemma 4.4].

Thus, in the Hermitian case, taking the trace in (28), we see that a tr A ≤ 0 with equality if and only if A ∈ so ( n 1 ) . This, together with the first equation in (27) (with b = 0 ), implies that v = 0 and A ∈ so ( n 1 ) , and hence, also tr A = 0 and then θ = 0 equation (25). We have the following:

Proposition 30

On a Hermitian almost-Abelian Lie group ( G , [ ⋅ , ⋅ ] ( a , b , v , A ) , g , J ) , the Bismut-Hermitian-Einstein metrics are Kähler-Einstein.

On the other hand, if we take A ∈ so ( n 1 ) commuting with J , but b ≠ 0 ≠ v , we obtain an almost-Hermitian almost-abelian Lie group ( G , [ ⋅ , ⋅ ] ( a , b , v , A ) , g , J ) , which satisfies d J d F = 0 but is not almost-Kähler (i.e. θ ≠ 0 ). Then, any solution of the equations

(29) 2 a 2 + 2 ∣ v ∣ 2 + b ⋅ v = 0 2 a b + A t b + 2 A t v = 0

gives a first Bismut-Einstein metric on G . We focus on the four-dimensional unimodular case ( a = − t r A ), which corresponds to a = 0 .

Theorem 31

Let g be a four-dimensional unimodular almost-abelian Lie algebra equipped with an almost-Hermitian structure ( g , J ) such that A ∈ so ( n 1 ) and commutes with J, in particular, it satisfies d J d F = 0 . Suppose that ( g , J ) is a solution to the Bismut-Einstein problem. Then g is isomorphic to one of the following Lie algebras

A 3 , 6 ⊕ A 1 : [ e 1 , e 3 ] = − e 2 , [ e 2 , e 3 ] = e 1 . The solution is not almost-Kähler.
A 3 , 1 ⊕ A 1 : [ e 1 , e 2 ] = e 3 . The solution is almost-Kähler.

The simply connected Lie groups associated to these Lie algebras admit compact quotients by discrete subgroups.

Proof

The isomorphism classes of almost-abelian Lie algebras can be described using Jordan forms of ad e 4 ∣ n up to scaling (see [4, Lemma 2.1] and [3]). Denote by A j i the ( i , j )th element of A . Equations (27) become

(30) 2 ∣ v ∣ 2 + b ⋅ v = 0 , A 2 1 ( b 1 + 2 v 1 ) = 0 , A 2 1 ( b 2 + 2 v 2 ) = 0 .

We have then two cases:

A 2 1 = 0 . Then,
- either b ⋅ v = − 2 ∣ v ∣ 2 < 0 : the canonical Jordan form of ad e 4 ∣ n up to scaling is 0 0 0 0 0 1 0 − 1 0 , which corresponds to A 3 , 6 ⊕ A 1 . v ≠ 0 so d F ≠ 0 .
- or b ⋅ v = v = 0 : the canonical Jordan form of ad e 4 ∣ n up to scaling is 0 0 0 0 0 1 0 0 0 , which corresponds to A 3 , 1 ⊕ A 1 . v = tr A = 0 so d F = 0 .
A 2 1 ≠ 0 . Then, b = − 2 v ≠ 0 and the canonical Jordan form of ad e 4 ∣ n up to scaling is 0 0 0 0 0 1 0 − 1 0 , which corresponds to A 3 , 6 ⊕ A 1 . v ≠ 0 so d F ≠ 0 .□

5.2 Second Chern-Einstein problem

Here we study the second Chern-Einstein problem on an almost-Hermitian almost-abelian Lie group ( G , [ ⋅ , ⋅ ] ( a , b , v , A ) , g , J ) of real dimension 4. We have seen that in the integrable case it reduces to the Bismut-Einstein problem (Proposition 24), studied in the previous section; while in the non-integrable case, a factor depending on the Nijenhuis tensor pops up:

(31) 1 2 ∑ i , j 4 ( N J ( e i , e j ) ) ♭ ∧ ( J N J ( e i , e j ) ) ♭ .

Choose an adapted unitary basis { e 1 , e 2 , e 3 , e 4 } for g , determining the algebraic data ( a , b , v , A ) . Then (31) can be written as follows

(32) 1 2 ∑ i , j 4 ( N J ( e i , e j ) ) ♭ ∧ ( J N J ( e i , e j ) ) ♭ = ∣ b ∣ 2 e 1 ∧ e 4 + ( ( A 1 2 + A 2 1 ) 2 + ( A 1 1 − A 2 2 ) 2 ) e 2 ∧ e 3 + ( b 2 ( A 1 1 − A 2 2 ) − b 1 ( A 2 1 + A 1 2 ) ) e 1 ∧ e 2 + ( b 2 ( A 1 2 + A 2 1 ) + b 1 ( A 1 1 − A 2 2 ) ) e 1 ∧ e 3 + ( b 1 ( A 1 1 − A 2 2 ) + b 2 ( A 2 1 + A 1 2 ) ) e 2 ∧ e 4 + ( b 1 ( A 1 2 + A 2 1 ) − b 2 ( A 1 1 − A 2 2 ) ) e 3 ∧ e 4 ,

where b = ( b 1 , b 2 ) and A j i is the ( i , j ) th element of A .

Theorem 32

Let g be a four-dimensional unimodular almost-abelian Lie algebra equipped with an almost-Hermitian non-Hermitian structure ( g , J ) such that the Lee form θ is D g -parallel and non-zero. Suppose that ( g , J ) is a solution to the second Chern-Einstein problem. Then g is isomorphic to one of the following Lie algebras

A 3 , 6 ⊕ A 1 : [ e 1 , e 3 ] = − e 2 , [ e 2 , e 3 ] = e 1 .
A 3 , 4 ⊕ A 1 : [ e 1 , e 3 ] = e 1 , [ e 2 , e 3 ] = − e 2 .

The simply connected Lie groups associated to these Lie algebras admit compact quotients by discrete subgroups.

Remark 33

A classification of four-dimensional almost-abelian Lie algebras with locally conformally symplectic structure is given in [4] (see also [5]). In the notation of [4], the aforementioned Lie algebras correspond, respectively, to r 3 , 0 ′ × R and r 3 , − 1 × R .

Proof

We recall that the isomorphism classes of almost-abelian Lie algebras can be described using Jordan forms of ad e 4 ∣ n up to scaling (see [4, Lemma 2.1] and [3]).

The Lie algebra g is unimodular, i.e. tr A = − a . Thanks to equations (26) and Proposition 24, we obtain

2 ∣ b ∣ 2 − 3 a 2 − b ⋅ v − 2 ∣ v ∣ 2 = 2 ( A 1 1 − A 2 2 ) 2 + 2 ( A 2 1 + A 1 2 ) 2 , 3 a b 1 + A 1 1 b 1 + A 1 2 b 2 + 2 A 1 1 v 1 + 2 A 1 2 v 2 = 4 b 1 ( A 1 1 − A 2 2 ) + 4 b 2 ( A 2 1 + A 1 2 ) , 3 a b 2 + A 2 1 b 1 + A 2 2 b 2 + 2 A 2 1 v 1 + 2 A 2 2 v 2 = 4 b 1 ( A 2 1 + A 1 2 ) − 4 b 2 ( A 1 1 − A 2 2 ) .

The Lee form is given by

θ = v 1 e 3 − v 2 e 2 + a e 4 .

Then, the condition D g θ = 0 implies the following:

a = 0 , b 1 v 2 = b 2 v 1 , v 1 ( A 2 1 − A 1 2 ) = 0 , v 2 ( A 2 1 − A 1 2 ) = 0 , v 1 ( A 2 1 + A 1 2 ) = 2 A 1 1 v 2 , v 2 ( A 2 1 + A 1 2 ) = 2 A 2 2 v 1 .

Suppose that v 1 ≠ 0 . Then the aforementioned equations imply that A 1 1 = A 2 2 = A 2 1 = A 1 2 = 0 and

2 b 1 2 − b 1 v 1 − 2 v 1 2 = 0 , 2 b 2 2 − b 2 v 2 − 2 v 2 2 = 0 .

We remark that b ⋅ v ≠ 0 . We have then two cases:

b ⋅ v > 0 : the canonical Jordan form of ad e 4 ∣ n up to scaling is 0 0 0 0 1 0 0 0 − 1 , which corresponds to A 3 , 4 ⊕ A 1 .
b ⋅ v < 0 : the canonical Jordan form of ad e 4 ∣ n up to scaling is 0 0 0 0 0 1 0 − 1 0 , which corresponds to A 3 , 6 ⊕ A 1 .

Now, if we suppose that v 1 = 0 , then θ = v 2 e 2 ≠ 0 implies that v 2 ≠ 0 . We can then deduce that b 1 = A 1 1 = A 2 2 = A 2 1 = A 1 2 = 0 . Because J is non-integrable, we have b 2 ≠ 0 . We conclude that 2 b 2 2 − b 2 v 2 − 2 v 2 2 = 0 , with b 2 v 2 ≠ 0 . We then obtain the same canonical Jordan forms as mentioned earlier.□

Acknowledgement

The authors are thankful to Daniele Angella for his valuable comments, and also to the anonymous Referee for their suggestions which helpd the exposition of the paper. The first author would also like to thank the CUNY Graduate Center of New York, Mehdi Lejmi, and Scott Wilson for their warm hospitality.

Funding information: The first author is supported by GNSAGA of INdAM and by project PRIN2017 “Real and Complex Manifolds: Topology, Geometry and holomorphic dynamics” (code 2017JZ2SW5). The second author is supported by the Simons Foundation Grant #636075.
Conflict of interest: The authors declare that they have no conflict of interest.

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Received: 2023-03-09

Accepted: 2023-06-15

Published Online: 2023-07-14

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/coma-2022-0150

Keywords for this article

almost-Hermitian Metrics; Chern-Einstein metrics; Hermitian connections; Weyl connection

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