Some Remarks on Existence of a Complex Structure on the Compact Six Sphere

Abstract

The existence or nonexistence of a complex structure on a differential manifold is a central problem in differential geometry. In particular, this problem on ${\mathrm{S}}^6$ was a long-standing unsolved problem, and differential geometry is an important tool. Recently, G. Clemente found a necessary and sufficient condition for almost-complex structures on a general differential manifold to be complex structures by using a covariant exterior derivative in three articles. However, in two of them, G. Clemente used a stronger condition instead of the published one. From there, G. Clemente proved the nonexistence of the complex structure on ${\mathrm{S}}^6$. We study the related differential operators and give some examples of nilmanifolds. And we prove that the earlier condition is too strong for an almost complex structure to be integrable. In another word, we clarify the situation of this problem.

1. Introduction

A fundamental problem in the theory of complex manifolds is the characterization of the orientable manifolds of even dimension that have a complex structure. Among the even-dimensional spheres, only the 2-sphere ${\mathrm{S}}^2$ and the 6-sphere ${\mathrm{S}}^6$ have almost-complex structures. ${\mathrm{S}}^2$ is a complex manifold. There exists many known almost-complex structures on ${\mathrm{S}}^6$; however, these known almost complex structures are not complex structures.

For some examples related to this topic, see [1,2,3,4,5,6,7,8,9], and also [10,11]. The problem of the existence or non-existence of a complex structure on a differential manifold is a central problem in differential geometry. In particular, this problem on ${\mathrm{S}}^6$ is a long-standing unsolved problem that was followed with interest by many great masters of mathematics, such as S. S. Chern [12], Hopf, K. Kodaira [13], S. T. Yau [14] and so on [1]. This paper [1] gives a short historical introduction to the problem.

Here, we would like to mention that the aim of [3,4] was to calculate the algebraic dimension of a complex ${\mathrm{S}}^6$, if there is one. In the first paper, there was a flaw, so they published a second paper with exactly the same title, the same authors and the same journal twenty years later. One of the possible goals of their second effort was to disprove a claim by Professor Etesi, who suggested that there is a conjugate orbit in $G_2$ that is diffeomorphic to ${\mathrm{S}}^6$ and is a complex submanifold. This was also the purpose of [5]. That is, in [5], we prove that the orbit cannot be a complex submanifold. We use the method of Tits’ fibration for the homogeneous complex structure on $G_2$. However, in a further version, Professor Etesi still conjectures that any complex structure on any simply connected compact Lie group has an algebraic dimension of zero. One notice that the compact Lie group $G_2$ in [5,6] has an algebraic dimension of 6, since the second Betti number $b_2\left(G_2\right)=0$. The reason for this is that the fiber of the Tits’ fibration (Cf. [5,6]) has a complex dimension of 1. Then, due to the algebraic dimension of $G_2$ being ≥ the complex dimension of the base, which is a complex six-dimension projective rational homogeneous manifold, the algebraic dimension must be six since $G_2$ cannot be Kähler.

In [6], we provde a different perspective on [5], which also disproves Etesi’s claim. In particular, our proof shows that if the conjugate orbit S in [5,6,10,15] is a complex submanifold of $G_2$, then the algebraic dimension of S is three. In [5], we can only determine that, if the conjugate orbit S is a complex submanifold, then the algebraic dimension of S is $\ge 2$. We hope that this article might also serve as a partial survey of some recent progress in this direction and provide a different perspective view to the research in [1,16].

Let M be an even dimensional ($n\ge 2$) compact almost-complex manifold. The space of almost-complex structures on M can be described as follows:

$$AC\left(M\right):=\{A\in {\mathrm{\Omega }}^1(M,TM)|:A\circ A=-\mathrm{id}\},$$

where $TM$ is the tangent bundle of M. The manifold M is complex if it supports $A\in AC\left(M\right)$, such that the Nijenhuis tensor of A,

$$N_A(\zeta ,\eta )=[A\left(\zeta \right),A\left(\eta \right)]-A([A\left(\zeta \right),\eta ]+[\zeta ,A\left(\eta \right)])-[\zeta ,\eta ],$$

vanishes for all vector fields $\zeta ,\eta \in \mathfrak{X}\left(M\right)$. In this case, A is called an integrable almost-complex structure or a complex structure.

Let ∇ be a symmetric connection on TM, and ${\mathrm{d}}^{\nabla }$ be the associated covariant exterior differential, which is given as the map ${\mathrm{d}}^{\nabla }:{\mathrm{\Omega }}^k(M,TM)\to {\mathrm{\Omega }}^{k+1}(M,TM)$. If $\alpha \in {\mathrm{\Omega }}^k(M,TM)$ and ${\zeta }_0$, ${\zeta }_1$, ⋯, ${\zeta }_k$ are $k+1$ vector fields,

$$\left({\mathrm{d}}^{\nabla }\alpha \right)({\zeta }_0,{\zeta }_1,\cdots ,{\zeta }_k)=\sum _{i=0}^k{(-1)}^i\left({\nabla }_{{\zeta }_i}\alpha \right)({\zeta }_0,\cdots ,{\widehat{\zeta }}_i,\cdots ,{\zeta }_k).$$

Recently, G. Clemente found a proposition [17] that says $A\in AC\left(M\right)$ is integrable if and only if ${\mathrm{d}}^{\nabla }A(A(·),A(·))-{\mathrm{d}}^{\nabla }A=0$. See also [18]. Here, the identity means that if $\xi$ and $\eta$ are two vector fields, then $\left({\mathrm{d}}^{\nabla }A\right)(A\xi ,A\eta )=\left({\mathrm{d}}^{\nabla }A\right)(\xi ,\eta )$. This holds especially whenever $d^{\nabla }A=0$. We notice that the Kähler condition is equivalent to $\nabla A=0$ (Cf. [19] page 70), which also implies that ${\mathrm{d}}^{\nabla }A=0$.

But in an early version [17,20] proposed in the summer of 2021, there was another proposition stating that $A\in AC\left(M\right)$ is integrable if and only if ${\mathrm{d}}^{\nabla }A=0$. And the identity ${\mathrm{d}}^{\nabla }\circ {\mathrm{d}}^{\nabla }=0$ was used in the proof of the nonexistence of any complex structure on ${\mathrm{S}}^6$. It was Professor Zejun Hu who told the first author about her preprint and asked for a comment on the correctness of her argument. This work was then performed in the fall of 2021. One might call a complex structure J satisfying ${\mathrm{d}}^{\nabla }J=0$ a weakly Kähler complex structure, and a tangent value 1-form A with ${\mathrm{d}}^{\nabla }\circ {\mathrm{d}}^{\nabla }A=0$ a semi-weakly Kähler almost complex structure. What she actually proved was that there is no weakly Kähler complex structure on ${\mathrm{S}}^6$.

In this article, we will give a representation of ${\mathrm{d}}^{\nabla }$ and ${\mathrm{d}}^{\nabla }\circ {\mathrm{d}}^{\nabla }$ in local coordinates, and compute ${\mathrm{d}}^{\nabla }A$ and ${\mathrm{d}}^{\nabla }\circ {\mathrm{d}}^{\nabla }A$ for some complex structures. Using the examples in [9], we obtain many non-weakly Kähler complex structures on nil-manifolds. This implies that the integrability is weaker than weakly Kähler property. Actually, our main theorems can be stated as follows.

Main Theorem 1. There exists a complex structure that is not semi-weakly Kähler, with some given Riemannian metrics on some nilmanifolds.

Main Theorem 2. There is no semi-weakly Kähler complex structure on ${\mathrm{S}}^6$ with the standard metric.

Corollary 1. 

There exists a complex structure that is not weakly Kähler, with some given Riemannian metrics on some nilmanifolds.

Corollary 2. 

There is no weakly Kähler complex structure on ${\mathrm{S}}^6$ with the standard metric.

It is known that a weakly Kähler $J\in AC\left(M\right)$ is integrable but a semi-weakly Kähler $J\in AC\left(M\right)$ might not be integrable. Now, we raise the following question, which is still unsolved:

Question 1. 

Are weakly Kähler and semi-weakly Kähler almost complex structures equivalent?

If the answer is no, we might ask the following question:

Question 2. 

Is there any semi-weakly Kähler almost-complex structure that is not integrable?

In fact, to our limited knowledge, we do not have any example of nil-manifold with a weakly Kähler structure that is not Kähler. Please see Section 4.

Therefore, one has: Kähler ⊂ weakly Kähler ⊂ integrable; and weakly Kähler ⊂ semi-weakly Kähler.

G. Clemente’s early suggestion was that weakly Kähler = integrable. This is not true according to Main Theorem 1.

However, Main Theorem 2 is still true.

Nil-manifolds and solv-manifolds are very important classes of differentiable manifolds. For examples, see [9,21]. This article also serves as a partial exposition of these manifolds, and therefore of the calculation in [21,22]. This result was announced in [16]. We also use this opportunity to thank Professor Zejun Hu for showing us [17], which led to this work.

2. Properties of the Covariant Exterior Derivative

For convenience, we will show a few properties of ${\mathrm{d}}^{\nabla }$ here, and some of the proof is mainly attributed to G. Clemente [17].

Lemma 1. 

If $\alpha \in {\mathrm{\Omega }}^k(M,TM)$ and ${\zeta }_0$, ${\zeta }_1$, ⋯, ${\zeta }_k$ are $k+1$ vector fields, then

$$\begin{array}{cc}\hfill \left(d^{\nabla }\alpha \right)({\zeta }_0,{\zeta }_1,\cdots ,{\zeta }_k)& =\sum _{i=0}^k{(-1)}^i{\nabla }_{{\zeta }_i}\left(\alpha ({\zeta }_0,\cdots ,{\widehat{\zeta }}_i,\cdots ,{\zeta }_k)\right)\hfill \\ \hfill & +\sum _{i<j}{(-1)}^{i+j}\alpha ([{\zeta }_i,{\zeta }_j],{\zeta }_0,\cdots ,{\widehat{\zeta }}_i,\cdots ,{\widehat{\zeta }}_j,\cdots ,{\zeta }_k).\hfill \end{array}$$

In particular, if $A\in {\mathrm{\Omega }}^1(M,TM)$ and $\xi ,\eta$ are 2 vector fields, then

$$\left(d^{\nabla }A\right)(\xi ,\eta )={\nabla }_{\xi }\left(A\left(\eta \right)\right)-{\nabla }_{\eta }\left(A\left(\xi \right)\right)-A\left([\xi ,\eta ]\right).$$

Proof. 

According to the properties of the connection, we also obtain

$$\begin{array}{cc}\hfill \left({\mathrm{d}}^{\nabla }\alpha \right)({\zeta }_0,{\zeta }_1,\cdots ,{\zeta }_k)& =\sum _{i=0}^k{(-1)}^i\left({\nabla }_{{\zeta }_i}\alpha \right)({\zeta }_0,\cdots ,{\widehat{\zeta }}_i,\cdots ,{\zeta }_k)\hfill \\ \hfill & =\sum _{i=0}^k{(-1)}^i{\zeta }_i\left(\alpha ({\zeta }_0,\cdots ,{\widehat{\zeta }}_i,\cdots ,{\zeta }_k)\right)\hfill \\ \hfill & -\sum _{i=0}^k{(-1)}^i\sum _{j=0,j\ne i}^k\alpha ({\zeta }_0,\cdots ,{\nabla }_{{\zeta }_j}{\zeta }_i,\cdots ,{\widehat{\zeta }}_j,\cdots ,{\zeta }_k)\hfill \\ \hfill & =\sum _{i=0}^k{(-1)}^i{\zeta }_i\left(\alpha ({\zeta }_0,\cdots ,{\widehat{\zeta }}_i,\cdots ,{\zeta }_k)\right)\hfill \\ \hfill & -\sum _{i=0}^k{(-1)}^{i+j}\sum _{j=0,j\ne i}^k\alpha ({\nabla }_{{\zeta }_j}{\zeta }_i,{\zeta }_0,\cdots ,{\widehat{\zeta }}_i,\cdots ,{\widehat{\zeta }}_j,\cdots ,{\zeta }_k)\hfill \\ \hfill & =\sum _{i=0}^k{(-1)}^i{\zeta }_i\left(\alpha ({\zeta }_0,\cdots ,{\widehat{\zeta }}_i,\cdots ,{\zeta }_k)\right)\hfill \\ \hfill & -\sum _{i<j}{(-1)}^{i+j}\alpha ({\nabla }_{{\zeta }_j}{\zeta }_i-{\nabla }_{{\zeta }_i}{\zeta }_j,\cdots ,{\nabla }_{{\zeta }_j}{\zeta }_i,\cdots ,{\widehat{\zeta }}_j,\cdots ,{\zeta }_k)\hfill \\ \hfill & =\sum _{i=0}^k{(-1)}^i{\nabla }_{{\zeta }_i}\left(\alpha ({\zeta }_0,\cdots ,{\widehat{\zeta }}_i,\cdots ,{\zeta }_k)\right)\hfill \\ \hfill & +\sum _{i<j}{(-1)}^{i+j}\alpha ([{\zeta }_i,{\zeta }_j],{\zeta }_0,\cdots ,{\widehat{\zeta }}_i,\cdots ,{\widehat{\zeta }}_j,\cdots ,{\zeta }_k).\hfill \end{array}$$

□

For our purpose, we need the local representations of ${\mathrm{d}}^{\nabla }$ and ${\mathrm{d}}^{\nabla }\circ {\mathrm{d}}^{\nabla }$. Take the basis $\left\{{\tilde{e}}_i\right\}$ of $\mathfrak{X}\left(M\right)$ (the vector fields on M) and its dual basis $\left\{e^i\right\}$. The coefficients of connection ${\mathrm{\Gamma }}_{ij}^k$ are defined as

$${\nabla }_{{\tilde{e}}_i}{\tilde{e}}_j={\mathrm{\Gamma }}_{ij}^k{\tilde{e}}_k.$$

For $A\in {\mathrm{\Omega }}^1(M,TM)$ and $\xi \in \mathfrak{X}\left(M\right)$, which have local representations,

$$A=u_{ij}{\tilde{e}}_i\otimes e^j,\xi ={\xi }^i{\tilde{e}}_i,$$

we have

$$A\xi =u_{ij}{\tilde{e}}_i\otimes e^j\left({\xi }^k{\tilde{e}}_k\right)=u_{ij}{\xi }^j{\tilde{e}}_i,$$

which means that A acts on $\xi$ in the same way that matrix $\mathit{A}=\left(u_{ij}\right)$, which is called the associate matrix of A, acts on the vector $\mathit{\xi }={({\xi }^1,{\xi }^2,\cdots ,{\xi }^n)}^T$. In this sense, we say $A\in {\mathrm{\Omega }}^1(M,TM)$ is degenerated or non-degenerated if its associate matrix $\mathit{A}=\left(u_{ij}\right)$ is degenerated or non-degenerated. One can easily check the following:

(1)

$A\xi =0$ is some nonzero section $\xi \in \mathfrak{X}\left(M\right)$ if and only if A is degenerated.

(2)

${\mathit{A}}^2=-I_n$ and A is non-degenerated if $A\in AC\left(M\right)$, where $I_n$ is the unit matrix of order n.

Lemma 2 

(See [17]). $A\in AC\left(M\right)$ is integrable if and only if the following holds:

$$\left(d^{\nabla }A\right)(A\xi ,A\eta )=\left(d^{\nabla }A\right)(\xi ,\eta ),$$

where ξ and η are two arbitrary vector fields.

Proof. 

Since $A\in AC\left(M\right)$, A is non-degenerated, and it is sufficient to show that

$$N_A(\xi ,\eta )=A\circ \left[\left({\mathrm{d}}^{\nabla }A\right)(A\xi ,A\eta )-\left({\mathrm{d}}^{\nabla }A\right)(\xi ,\eta )\right].$$

According to Lemma 2 and $A\circ A=-\mathrm{id}$, we can obtain

$$\begin{array}{c}{\displaystyle A\circ \left[\left({\mathrm{d}}^{\nabla }A\right)(A\xi ,A\eta )-\left({\mathrm{d}}^{\nabla }A\right)(\xi ,\eta )\right]}\hfill \\ =A\left({\nabla }_{A\left(\xi \right)}\left(A\left(A\left(\eta \right)\right)\right)-{\nabla }_{A\left(\eta \right)}\left(A\left(A\left(\xi \right)\right)\right)-A\left(\left[A\right(\xi ),A(\eta \left)\right]\right)\right)\hfill \\ -A\left({\nabla }_{\xi }\left(A\left(\eta \right)\right)-{\nabla }_{\eta }\left(A\left(\xi \right)\right)-A\left([\xi ,\eta ]\right)\right)\hfill \\ =A\left(-{\nabla }_{A\left(\xi \right)}\eta +{\nabla }_{A\left(\eta \right)}\xi -{\nabla }_{\xi }\left(A\left(\eta \right)\right)+{\nabla }_{\eta }\left(A\left(\xi \right)\right)\right)\hfill \\ +\left[A\right(\xi ),A(\eta \left)\right]-[\xi ,\eta ]\hfill \\ =-A\left[A\right(\xi ),\eta ]-A[\xi ,A(\eta \left)\right]+\left[A\right(\xi ),A(\eta \left)\right]-[\xi ,\eta ].\hfill \end{array}$$

Considering that ∇ is a symmetric connection, or equally a torsion-free connection, the second to last equality holds. □

Lemma 3. 

If $A\in {\mathrm{\Omega }}^1(M,TM)$ and $\xi ,\eta ,\zeta \in \mathfrak{X}\left(M\right)$ have local representations

$$A=u_{ij}{\tilde{e}}_i\otimes e^j,\xi ={\xi }^i{\tilde{e}}_i,\eta ={\eta }^j{\tilde{e}}_j,and\zeta ={\zeta }^k{\tilde{e}}_k,$$

then

$$\left(d^{\nabla }A\right)(\xi ,\eta )={\xi }^i\left[{\tilde{e}}_i\left(u_{kj}\right)-{\tilde{e}}_j\left(u_{ki}\right)+u_{lj}{\mathrm{\Gamma }}_{il}^k-u_{li}{\mathrm{\Gamma }}_{jl}^k+u_{ks}{\mathrm{\Gamma }}_{ji}^s-u_{ks}{\mathrm{\Gamma }}_{ij}^s\right]{\eta }^j{\tilde{e}}_k,$$

and

$$\begin{array}{cc}\left(d^{\nabla }\left(d^{\nabla }A\right)\right)(\xi ,\eta ,\zeta )& ={\xi }^i{\eta }^j{\zeta }^k{u}_{tk}({\tilde{e}}_i\left({\mathrm{\Gamma }}_{jt}^l\right)-{\tilde{e}}_j\left({\mathrm{\Gamma }}_{it}^l\right)+{\mathrm{\Gamma }}_{jt}^s{\mathrm{\Gamma }}_{is}^l-{\mathrm{\Gamma }}_{it}^s{\mathrm{\Gamma }}_{js}^l-{\mathrm{\Gamma }}_{ij}^s{\mathrm{\Gamma }}_{st}^l+{\mathrm{\Gamma }}_{ji}^s{\mathrm{\Gamma }}_s^{t}l\hfill \\ & {\tilde{e}}_j\left({\mathrm{\Gamma }}_{kt}^l\right)-{\tilde{e}}_k\left({\mathrm{\Gamma }}_{jt}^l\right)+{\mathrm{\Gamma }}_{kt}^s{\mathrm{\Gamma }}_{js}^l-{\mathrm{\Gamma }}_{jt}^s{\mathrm{\Gamma }}_{ks}^l-{\mathrm{\Gamma }}_{jk}^s{\mathrm{\Gamma }}_{st}^l+{\mathrm{\Gamma }}_{kj}^s{\mathrm{\Gamma }}_s^{t}l\hfill \\ & {\tilde{e}}_k\left({\mathrm{\Gamma }}_{it}^l\right)-{\tilde{e}}_i\left({\mathrm{\Gamma }}_{kt}^l\right)+{\mathrm{\Gamma }}_{it}^s{\mathrm{\Gamma }}_{ks}^l-{\mathrm{\Gamma }}_{kt}^s{\mathrm{\Gamma }}_{is}^l-{\mathrm{\Gamma }}_{ki}^s{\mathrm{\Gamma }}_{st}^l+{\mathrm{\Gamma }}_{ik}^s{\mathrm{\Gamma }}_{st}^l){\tilde{e}}_l.\hfill \end{array}$$

Proof. 

Firstly, through direct calculation, we obtain

$$\begin{array}{cc}\hfill {\nabla }_{\xi }\left(A\eta \right)& ={\nabla }_{{\xi }^i{\tilde{e}}_i}\left(u_{kj}{\eta }^j{\tilde{e}}_k\right)={\xi }^i\left({\tilde{e}}_i\left(u_{kj}{\eta }^j\right){\tilde{e}}_k+u_{kj}{\eta }^j{\mathrm{\Gamma }}_{ik}^l{\tilde{e}}_l\right)\hfill \\ \hfill & ={\xi }^i\left({\tilde{e}}_i\left(u_{kj}\right){\eta }^j+u_{kj}{\tilde{e}}_i\left({\eta }^j\right)+u_{lj}{\eta }^j{\mathrm{\Gamma }}_{il}^k\right){\tilde{e}}_k,\hfill \\ \hfill {\nabla }_{\eta }\left(A\xi \right)& ={\eta }^j\left({\tilde{e}}_j\left(u_{ki}\right){\xi }^i+u_{ki}{\tilde{e}}_j\left({\xi }^i\right)+u_{li}{\xi }^i{\mathrm{\Gamma }}_{jl}^k\right){\tilde{e}}_k,\hfill \\ \hfill {\nabla }_{\xi }\eta & ={\nabla }_{{\xi }^i{\tilde{e}}_i}\left({\eta }^j{\tilde{e}}_j\right)={\xi }^i\left({\tilde{e}}_i\left({\eta }^k\right)+{\eta }^j{\mathrm{\Gamma }}_{ij}^k\right){\tilde{e}}_k,\hfill \\ \hfill A\left({\nabla }_{\xi }\eta \right)& =\left(u_{st}{\tilde{e}}_s\otimes e^t\right)\left({\xi }^i\left({\tilde{e}}_i\left({\eta }^k\right)+{\eta }^j{\mathrm{\Gamma }}_{ij}^k\right){\tilde{e}}_k\right)\hfill \\ \hfill & =u_{sk}{\xi }^i\left({\tilde{e}}_i\left({\eta }^k\right)+{\eta }^j{\mathrm{\Gamma }}_{ij}^k\right){\tilde{e}}_s=u_{ks}{\xi }^i\left({\tilde{e}}_i\left({\eta }^s\right)+{\eta }^j{\mathrm{\Gamma }}_{ij}^s\right){\tilde{e}}_k,\hfill \\ \hfill A\left({\nabla }_{\eta }\xi \right)& =u_{ks}{\eta }^j\left({\tilde{e}}_j\left({\xi }^s\right)+{\xi }^i{\mathrm{\Gamma }}_{ji}^s\right){\tilde{e}}_k,\hfill \end{array}$$

and hence

$$\left({\mathrm{d}}^{\nabla }A\right)(\xi ,\eta )={\xi }^i\left[{\tilde{e}}_i\left(u_{kj}\right)-{\tilde{e}}_j\left(u_{ki}\right)+u_{lj}{\mathrm{\Gamma }}_{il}^k-u_{li}{\mathrm{\Gamma }}_{jl}^k+u_{ks}{\mathrm{\Gamma }}_{ji}^s-u_{ks}{\mathrm{\Gamma }}_{ij}^s\right]{\eta }^j{\tilde{e}}_k.$$

For the second equality, let $R^{\nabla }$ be the curvature of ∇, i.e.,

$$R^{\nabla }(\xi ,\eta )\zeta ={\nabla }_{\xi }{\nabla }_{\eta }\zeta -{\nabla }_{\eta }{\nabla }_{\xi }\zeta -{\nabla }_{[\xi ,\eta ]}\zeta .$$

According the notation of curvature, we obtain

$$\begin{array}{cc}\hfill & \left({\mathrm{d}}^{\nabla }\left({\mathrm{d}}^{\nabla }A\right)\right)(\xi ,\eta ,\zeta )\hfill \\ \hfill =& \left({\nabla }_{\xi }\left({\mathrm{d}}^{\nabla }A\right)\right)(\eta ,\zeta )-\left({\nabla }_{\eta }\left({\mathrm{d}}^{\nabla }A\right)\right)(\xi ,\zeta )+\left({\nabla }_{\zeta }\left({\mathrm{d}}^{\nabla }A\right)\right)(\xi ,\eta )\hfill \\ \hfill =& {\nabla }_{\xi }(\left({\mathrm{d}}^{\nabla }A\right)(\eta ,\zeta ))-\left({\mathrm{d}}^{\nabla }A\right)({\nabla }_{\xi }\eta ,\zeta )-\left({\mathrm{d}}^{\nabla }A\right)(\eta ,{\nabla }_{\xi }\zeta )\hfill \\ \hfill & -{\nabla }_{\eta }(\left({\mathrm{d}}^{\nabla }A\right)(\xi ,\zeta ))+\left({\mathrm{d}}^{\nabla }A\right)({\nabla }_{\eta }\xi ,\zeta )+\left({\mathrm{d}}^{\nabla }A\right)(\xi ,{\nabla }_{\eta }\zeta )\hfill \\ \hfill & +{\nabla }_{\zeta }(\left({\mathrm{d}}^{\nabla }A\right)(\xi ,\eta ))-\left({\mathrm{d}}^{\nabla }A\right)({\nabla }_{\zeta }\xi ,\eta )-\left({\mathrm{d}}^{\nabla }A\right)(\xi ,{\nabla }_{\zeta }\eta )\hfill \\ \hfill =& {\nabla }_{\xi }\left(\left({\mathrm{d}}^{\nabla }A\right)(\eta ,\zeta ))-{\nabla }_{\eta }\right(\left({\mathrm{d}}^{\nabla }A\right)(\xi ,\zeta ))+{\nabla }_{\zeta }(\left({\mathrm{d}}^{\nabla }A\right)(\xi ,\eta ))\hfill \\ \hfill & -\left({\mathrm{d}}^{\nabla }A\right)({\nabla }_{\xi }\eta -{\nabla }_{\eta }\xi ,\zeta )-\left({\mathrm{d}}^{\nabla }A\right)({\nabla }_{\zeta }\xi -{\nabla }_{\xi }\zeta ,\eta )-\left({\mathrm{d}}^{\nabla }A\right)({\nabla }_{\eta }\zeta -{\nabla }_{\zeta }\eta ,\xi )\hfill \\ \hfill =& {\nabla }_{\xi }\left(\left({\mathrm{d}}^{\nabla }A\right)(\eta ,\zeta ))-{\nabla }_{\eta }\right(\left({\mathrm{d}}^{\nabla }A\right)(\xi ,\zeta ))+{\nabla }_{\zeta }(\left({\mathrm{d}}^{\nabla }A\right)(\xi ,\eta ))\hfill \\ \hfill & -\left({\mathrm{d}}^{\nabla }A\right)([\xi ,\eta ],\zeta )+\left({\mathrm{d}}^{\nabla }A\right)([\xi ,\zeta ],\eta )-\left({\mathrm{d}}^{\nabla }A\right)([\eta ,\zeta ],\xi )\hfill \\ \hfill =& {\nabla }_{\xi }\left({\nabla }_{\eta }\left(A\left(\zeta \right)\right)-{\nabla }_{\zeta }\left(A\left(\eta \right)\right)-A\left([\eta ,\zeta ]\right)\right)-{\nabla }_{\eta }\left({\nabla }_{\xi }\left(A\left(\zeta \right)\right)-{\nabla }_{\zeta }\left(A\left(\xi \right)\right)-A\left([\eta ,\zeta ]\right)\right)\hfill \\ \hfill & +{\nabla }_{\zeta }\left({\nabla }_{\xi }\left(A\left(\eta \right)\right)-{\nabla }_{\eta }\left(A\left(\xi \right)\right)-A\left([\xi ,\eta ]\right)\right)-\left({\nabla }_{[\xi ,\eta ]}\left(A\left(\zeta \right)\right)-{\nabla }_{\zeta }\left(A\left([\xi ,\eta ]\right)\right)-A\left([[\xi ,\eta ],\zeta ]\right)\right)\hfill \\ \hfill & +\left({\nabla }_{[\xi ,\zeta ]}\left(A\left(\eta \right)\right)-{\nabla }_{\eta }\left(A\left([\xi ,\zeta ]\right)\right)-A\left([[\xi ,\zeta ],\eta ]\right)\right)\hfill \\ \hfill & -\left({\nabla }_{[\eta ,\zeta ]}\left(A\left(\xi \right)\right)-{\nabla }_{\xi }\left(A\left([\eta ,\zeta ]\right)\right)-A\left([[\eta ,\zeta ],\xi ]\right)\right)\hfill \\ \hfill =& R^{\nabla }(\xi ,\eta )\left(A\left(\zeta \right)\right)+R^{\nabla }(\eta ,\zeta )\left(A\left(\xi \right)\right)-R^{\nabla }(\xi ,\zeta )\left(A\left(\eta \right)\right)+A([[\xi ,\eta ],\zeta ]-[[\xi ,\zeta ],\eta ]+[[\eta ,\zeta ],\xi ])\hfill \\ \hfill =& R^{\nabla }(\xi ,\eta )\left(A\left(\zeta \right)\right)+R^{\nabla }(\eta ,\zeta )\left(A\left(\xi \right)\right)+R^{\nabla }(\zeta ,\xi )\left(A\left(\eta \right)\right).\hfill \end{array}$$

Here, we use the Jacobi identity of the bracket product. According to the tensor properties of $R^{\nabla }$,

$$\begin{array}{cc}\hfill {\nabla }_{{\tilde{e}}_i}{\nabla }_{{\tilde{e}}_j}{\tilde{e}}_k& ={\nabla }_{{\tilde{e}}_i}\left({\mathrm{\Gamma }}_{jk}^l{\tilde{e}}_l\right)={\tilde{e}}_i\left({\mathrm{\Gamma }}_{jk}^l\right){\tilde{e}}_l+{\mathrm{\Gamma }}_{jk}^l{\nabla }_{{\tilde{e}}_i}{\tilde{e}}_l={\tilde{e}}_i\left({\mathrm{\Gamma }}_{jk}^l\right){\tilde{e}}_l+{\mathrm{\Gamma }}_{jk}^l{\mathrm{\Gamma }}_{il}^s{\tilde{e}}_s\hfill \\ \hfill & =\left({\tilde{e}}_i\left({\mathrm{\Gamma }}_{jk}^l\right)+{\mathrm{\Gamma }}_{jk}^s{\mathrm{\Gamma }}_{is}^l\right){\tilde{e}}_l,\hfill \\ \hfill {\nabla }_{{\tilde{e}}_j}{\nabla }_{{\tilde{e}}_i}{\tilde{e}}_k& =\left({\tilde{e}}_j\left({\mathrm{\Gamma }}_{ik}^l\right)+{\mathrm{\Gamma }}_{ik}^s{\mathrm{\Gamma }}_{js}^l\right){\tilde{e}}_l,\hfill \\ \hfill ]{\tilde{e}}_i,{\tilde{e}}_j]& ={\nabla }_{{\tilde{e}}_i}{\tilde{e}}_j-{\nabla }_{{\tilde{e}}_j}{\tilde{e}}_i={\mathrm{\Gamma }}_{ij}^l{\tilde{e}}_l-{\mathrm{\Gamma }}_{ji}^l{\tilde{e}}_l=({\mathrm{\Gamma }}_{ij}^l-{\mathrm{\Gamma }}_{ji}^l){\tilde{e}}_l,\hfill \\ \hfill {\nabla }_{[{\tilde{e}}_i,{\tilde{e}}_j]}{\tilde{e}}_k& ={\nabla }_{({\mathrm{\Gamma }}_{ij}^l-{\mathrm{\Gamma }}_{ji}^l){\tilde{e}}_l}{\tilde{e}}_k=({\mathrm{\Gamma }}_{ij}^l-{\mathrm{\Gamma }}_{ji}^l){\mathrm{\Gamma }}_{lk}^s{\tilde{e}}_s=({\mathrm{\Gamma }}_{ij}^s-{\mathrm{\Gamma }}_{ji}^s){\mathrm{\Gamma }}_{sk}^l{\tilde{e}}_l,\hfill \\ \hfill R^{\nabla }({\tilde{e}}_i,{\tilde{e}}_j){\tilde{e}}_k& =\left({\tilde{e}}_i\left({\mathrm{\Gamma }}_{jk}^l\right)+{\mathrm{\Gamma }}_{jk}^s{\mathrm{\Gamma }}_{is}^l-{\tilde{e}}_j\left({\mathrm{\Gamma }}_{ik}^l\right)-{\mathrm{\Gamma }}_{ik}^s{\mathrm{\Gamma }}_{js}^l-{\mathrm{\Gamma }}_{ij}^s{\mathrm{\Gamma }}_{sk}^l+{\mathrm{\Gamma }}_{ji}^s{\mathrm{\Gamma }}_{sk}^l\right){\tilde{e}}_l\hfill \\ \hfill R^{\nabla }(\xi ,\eta )\left(A\left(\zeta \right)\right)& =R^{\nabla }({\xi }^i{\tilde{e}}_i,{\eta }^j{\tilde{e}}_j)\left(u_{kt}{\zeta }^t{\tilde{e}}_k\right)={\xi }^i{\eta }^j{\zeta }^t{u}_{kt}{R}^{\nabla }({\tilde{e}}_i,{\tilde{e}}_j){\tilde{e}}_k\hfill \\ \hfill & ={\xi }^i{\eta }^j{\zeta }^t{u}_{kt}\left({\tilde{e}}_i\left({\mathrm{\Gamma }}_{jk}^l\right)+{\mathrm{\Gamma }}_{jk}^s{\mathrm{\Gamma }}_{is}^l-{\tilde{e}}_j\left({\mathrm{\Gamma }}_{ik}^l\right)-{\mathrm{\Gamma }}_{ik}^s{\mathrm{\Gamma }}_{js}^l-{\mathrm{\Gamma }}_{ij}^s{\mathrm{\Gamma }}_{sk}^l+{\mathrm{\Gamma }}_{ji}^s{\mathrm{\Gamma }}_{sk}^l\right){\tilde{e}}_l\hfill \\ \hfill & ={\xi }^i{\eta }^j{\zeta }^k{u}_{tk}\left({\tilde{e}}_i\left({\mathrm{\Gamma }}_{jt}^l\right)+{\mathrm{\Gamma }}_{jt}^s{\mathrm{\Gamma }}_{is}^l-{\tilde{e}}_j\left({\mathrm{\Gamma }}_{it}^l\right)-{\mathrm{\Gamma }}_{it}^s{\mathrm{\Gamma }}_{js}^l-{\mathrm{\Gamma }}_{ij}^s{\mathrm{\Gamma }}_{st}^l+{\mathrm{\Gamma }}_{ji}^s{\mathrm{\Gamma }}_{st}^l\right){\tilde{e}}_l\hfill \\ \hfill R^{\nabla }(\eta ,\zeta )\left(A\left(\xi \right)\right)& ={\xi }^i{\eta }^j{\zeta }^k{u}_{ti}\left({\tilde{e}}_j\left({\mathrm{\Gamma }}_{kt}^l\right)+{\mathrm{\Gamma }}_{kt}^s{\mathrm{\Gamma }}_{js}^l-{\tilde{e}}_k\left({\mathrm{\Gamma }}_{jt}^l\right)-{\mathrm{\Gamma }}_{jt}^s{\mathrm{\Gamma }}_{ks}^l-{\mathrm{\Gamma }}_{jk}^s{\mathrm{\Gamma }}_{st}^l+{\mathrm{\Gamma }}_{kj}^s{\mathrm{\Gamma }}_{st}^l\right){\tilde{e}}_l\hfill \\ \hfill R^{\nabla }(\zeta ,\xi )\left(A\left(\eta \right)\right)& ={\xi }^i{\eta }^j{\zeta }^k{u}_{tj}\left({\tilde{e}}_k\left({\mathrm{\Gamma }}_{it}^l\right)+{\mathrm{\Gamma }}_{it}^s{\mathrm{\Gamma }}_{ks}^l-{\tilde{e}}_i\left({\mathrm{\Gamma }}_{kt}^l\right)-{\mathrm{\Gamma }}_{kt}^s{\mathrm{\Gamma }}_{is}^l-{\mathrm{\Gamma }}_{ki}^s{\mathrm{\Gamma }}_{st}^l+{\mathrm{\Gamma }}_{ik}^s{\mathrm{\Gamma }}_{st}^l\right){\tilde{e}}_l.\hfill \end{array}$$

Thus, the proof is ended. □

Now, we obtain the local representation of ${\mathrm{d}}^{\nabla }A$ and ${\mathrm{d}}^{\nabla }\circ {\mathrm{d}}^{\nabla }A$, which can almost explain ${\mathrm{d}}^{\nabla }A\ne 0$ for a complex structure A and ${\mathrm{d}}^{\nabla }\circ {\mathrm{d}}^{\nabla }A\ne 0$ for $A\in {\mathrm{\Omega }}^1(M,TM)$.

From the proof of Lemma 3, we obtain the following lemma.

Lemma 4. 

If $A\in AC\left(M\right)$ with $d^{\nabla }\circ d^{\nabla }A=0$, then for any three vector fields ξ, η and ζ,

$$R^{\nabla }(\xi ,\eta )\left(A\left(\zeta \right)\right)+R^{\nabla }(\eta ,\zeta )\left(A\left(\xi \right)\right)+R^{\nabla }(\zeta ,\xi )\left(A\left(\eta \right)\right)=0.$$

3. The Curvature of Spheres

Let $S^n$ be the n-sphere with coordinate $(x^1,x^2,\cdots ,x^{n+1})$ in ⁿ⁺¹. In

$$\tilde{S}=S^n-\left\{(0,0,\cdots ,-1)\right\},$$

it has a local coordinate $({\xi }^1,{\xi }^2,\cdots ,{\xi }^n)$, where ${\xi }^i={\displaystyle \frac{x^i}{1+x^{n+1}}}$.

Conversely, if $a={\displaystyle \sum _{k=1}^n}{\left({\xi }^k\right)}^2$, then

$$x^i={\displaystyle \frac{2{\xi }^i}{1+a}},i=1,2,\cdots ,n,x^{n+1}={\displaystyle \frac{1-a}{1+a}},$$

and the embedding $h:\tilde{S}\to$ ⁿ⁺¹ is given by

$$h({\xi }^1,\cdots ,{\xi }^n)=(x^1,\cdots ,x^n,x^{n+1})=\left({\displaystyle \frac{2{\xi }^1}{1+a}},\cdots ,{\displaystyle \frac{2{\xi }^n}{1+a}},{\displaystyle \frac{1-a}{1+a}}\right).$$

Thus,

$$\begin{array}{cc}\hfill h^{\ast }\left({\displaystyle \frac{\partial }{\partial {\xi }^i}}\right)& =\sum _{k=1}^n{\displaystyle \frac{\partial x^k}{\partial {\xi }^i}}{\displaystyle \frac{\partial }{\partial x^k}}+{\displaystyle \frac{\partial x^{n+1}}{\partial {\xi }^i}}{\displaystyle \frac{\partial }{\partial x^{n+1}}}\hfill \\ \hfill & =\sum _{k=1}^n\left({\displaystyle \frac{2\mathrm{d}elt{a}_{ki}}{1+a}}-{\displaystyle \frac{4{\xi }^i{\xi }^k}{{(1+a)}^2}}\right){\displaystyle \frac{\partial }{\partial x^k}}-{\displaystyle \frac{4{\xi }^i}{{(1+a)}^2}}{\displaystyle \frac{\partial }{\partial x^{n+1}}}\hfill \\ \hfill & ={\displaystyle \frac{2}{1+a}}{\displaystyle \frac{\partial }{\partial x^i}}-\sum _{k=1}^n{\displaystyle \frac{4{\xi }^i{\xi }^k}{{(1+a)}^2}}{\displaystyle \frac{\partial }{\partial x^k}}-{\displaystyle \frac{4{\xi }^i}{{(1+a)}^2}}{\displaystyle \frac{\partial }{\partial x^{n+1}}},\hfill \\ \hfill h^{\ast }\left({\displaystyle \frac{\partial }{\partial {\xi }^j}}\right)& ={\displaystyle \frac{2}{1+a}}{\displaystyle \frac{\partial }{\partial x^j}}-\sum _{k=1}^n{\displaystyle \frac{4{\xi }^j{\xi }^k}{{(1+a)}^2}}{\displaystyle \frac{\partial }{\partial x^k}}-{\displaystyle \frac{4{\xi }^j}{{(1+a)}^2}}{\displaystyle \frac{\partial }{\partial x^{n+1}}},\hfill \\ \hfill g_{ij}& =〈h^{\ast }\left({\displaystyle \frac{\partial }{\partial {\xi }^i}}\right),h^{\ast }\left({\displaystyle \frac{\partial }{\partial {\xi }^j}}\right)〉\hfill \\ \hfill & ={\displaystyle \frac{4{\delta }_{ij}}{{(1+a)}^2}}+\sum _{k=1}^n{\displaystyle \frac{16{\xi }^i{\xi }^j{\left({\xi }^k\right)}^2}{{(1+a)}^4}}+{\displaystyle \frac{16{\xi }^i{\xi }^j}{{(1+a)}^4}}-{\displaystyle \frac{16{\xi }^i{\xi }^j}{{(1+a)}^3}}\hfill \\ \hfill & ={\displaystyle \frac{4{\delta }_{ij}}{{(1+a)}^2}}+{\displaystyle \frac{16a{\xi }^i{\xi }^j}{{(1+a)}^4}}+{\displaystyle \frac{16{\xi }^i{\xi }^j}{{(1+a)}^4}}-{\displaystyle \frac{16{\xi }^i{\xi }^j}{{(1+a)}^3}}\hfill \\ \hfill & ={\displaystyle \frac{4{\delta }_{ij}}{{(1+a)}^2}},\hfill \end{array}$$

which gives the Riemannian metric of $\tilde{S}$ induced by the Euclidean metric of ⁿ⁺¹,

$$g_{ij}=g\left({\displaystyle \frac{\partial }{\partial {\xi }^i}},{\displaystyle \frac{\partial }{\partial {\xi }^j}}\right)={\displaystyle \frac{4{\delta }_{ij}}{{\left(1+{\displaystyle \sum _{s=1}^n}{\left({\xi }^s\right)}^2\right)}^2}}:={\displaystyle \frac{4{\delta }_{ij}}{A^2}},$$

and its inverse $g^{ij}={\displaystyle \frac{A^2{\delta }_{ij}}{4}}$. Now, we obtain the connection coefficient of the Levi-Civita connection of g,

$$\begin{array}{cc}\hfill {\mathrm{\Gamma }}_{ij}^k& ={\displaystyle \frac{1}{2}}{g}^{kl}\left({\displaystyle \frac{\partial g_{il}}{\partial {\xi }^j}}+{\displaystyle \frac{\partial g_{lj}}{\partial {\xi }^i}}-{\displaystyle \frac{\partial g_{ij}}{\partial {\xi }^l}}\right)={\displaystyle \frac{1}{2}}·{\displaystyle \frac{A^2}{4}}\left({\displaystyle \frac{\partial g_{ik}}{\partial {\xi }^j}}+{\displaystyle \frac{\partial g_{kj}}{\partial {\xi }^i}}-{\displaystyle \frac{\partial g_{ij}}{\partial {\xi }^k}}\right)\hfill \\ \hfill & ={\displaystyle \frac{A^2}{8}}\left({\displaystyle \frac{4{\delta }_{ik}·(-2)·2{\xi }^j}{A^3}}+{\displaystyle \frac{-16{\delta }_{kj}{\xi }^i}{A^3}}-{\displaystyle \frac{-16{\delta }_{ij}{\xi }^k}{A^3}}\right)\hfill \\ \hfill & ={\displaystyle \frac{2}{A}}(-{\delta }_{ik}{\xi }^j-{\delta }_{kj}{\xi }^i+{\delta }_{ij}{\xi }^k).\hfill \end{array}$$

For $i\ne j$,

$$\begin{array}{cc}\hfill {\nabla }_{{\displaystyle \frac{\partial }{\partial {\xi }^j}}}{\displaystyle \frac{\partial }{\partial {\xi }^i}}& ={\mathrm{\Gamma }}_{ij}^k{\displaystyle \frac{\partial }{\partial {\xi }^k}}={\displaystyle \frac{2}{A}}(-{\delta }_{ik}{\xi }^j-{\delta }_{kj}{\xi }^i+{\delta }_{ij}{\xi }^k){\displaystyle \frac{\partial }{\partial {\xi }^k}}\hfill \\ \hfill & =-{\displaystyle \frac{2{\xi }^j}{A}}{\displaystyle \frac{\partial }{\partial {\xi }^i}}-{\displaystyle \frac{2{\xi }^i}{A}}{\displaystyle \frac{\partial }{\partial {\xi }^j}}.\hfill \\ \hfill {\nabla }_{{\displaystyle \frac{\partial }{\partial {\xi }^i}}}{\nabla }_{{\displaystyle \frac{\partial }{\partial {\xi }^j}}}{\displaystyle \frac{\partial }{\partial {\xi }^i}}& ={\nabla }_{{\displaystyle \frac{\partial }{\partial {\xi }^i}}}\left(-{\displaystyle \frac{2{\xi }^j}{A}}{\displaystyle \frac{\partial }{\partial {\xi }^i}}-{\displaystyle \frac{2{\xi }^i}{A}}{\displaystyle \frac{\partial }{\partial {\xi }^j}}\right)\hfill \\ \hfill & ={\displaystyle \frac{\partial }{\partial {\xi }^i}}\left(-{\displaystyle \frac{2{\xi }^j}{A}}\right){\displaystyle \frac{\partial }{\partial {\xi }^i}}-{\displaystyle \frac{2{\xi }^j}{A}}{\nabla }_{{\displaystyle \frac{\partial }{\partial {\xi }^i}}}{\displaystyle \frac{\partial }{\partial {\xi }^i}}-{\displaystyle \frac{\partial }{\partial {\xi }^i}}\left({\displaystyle \frac{2{\xi }^i}{A}}\right){\displaystyle \frac{\partial }{\partial {\xi }^j}}-{\displaystyle \frac{2{\xi }^i}{A}}{\nabla }_{{\displaystyle \frac{\partial }{\partial {\xi }^i}}}{\displaystyle \frac{\partial }{\partial {\xi }^j}}\hfill \\ \hfill & ={\displaystyle \frac{4{\xi }^i{\xi }^j}{A^2}}{\displaystyle \frac{\partial }{\partial {\xi }^i}}-{\displaystyle \frac{2{\xi }^j}{A}}\left(-{\displaystyle \frac{4{\xi }^i}{A}}{\displaystyle \frac{\partial }{\partial {\xi }^i}}+{\displaystyle \frac{2}{A}}{\xi }^k{\displaystyle \frac{\partial }{\partial {\xi }^k}}\right)-{\displaystyle \frac{2A-4{\left({\xi }^i\right)}^2}{A^2}}{\displaystyle \frac{\partial }{\partial {\xi }^j}}\hfill \\ \hfill & +{\displaystyle \frac{2{\xi }^i}{A}}\left({\displaystyle \frac{2{\xi }^i}{A}}{\displaystyle \frac{\partial }{\partial {\xi }^j}}+{\displaystyle \frac{2{\xi }^j}{A}}{\displaystyle \frac{\partial }{\partial {\xi }^i}}\right)\hfill \\ \hfill & ={\displaystyle \frac{-4{\left({\xi }^j\right)}^2-2A+8{\left({\xi }^i\right)}^2}{A^2}}{\displaystyle \frac{\partial }{\partial {\xi }^j}}+\mathrm{other}\mathrm{terms},\hfill \\ \hfill {\nabla }_{{\displaystyle \frac{\partial }{\partial {\xi }^j}}}{\nabla }_{{\displaystyle \frac{\partial }{\partial {\xi }^i}}}{\displaystyle \frac{\partial }{\partial {\xi }^i}}& ={\nabla }_{{\displaystyle \frac{\partial }{\partial {\xi }^j}}}\left(-{\displaystyle \frac{4{\xi }^i}{A}}{\displaystyle \frac{\partial }{\partial {\xi }^i}}+{\displaystyle \frac{2}{A}}{\xi }^k{\displaystyle \frac{\partial }{\partial {\xi }^k}}\right)\hfill \\ \hfill & =-{\displaystyle \frac{4{\xi }^i}{A}}{\nabla }_{{\displaystyle \frac{\partial }{\partial {\xi }^j}}}{\displaystyle \frac{\partial }{\partial {\xi }^i}}+{\displaystyle \frac{2A-4{\left({\xi }^j\right)}^2}{A^2}}{\displaystyle \frac{\partial }{\partial {\xi }^j}}+{\displaystyle \frac{2{\xi }^k}{A}}{\nabla }_{{\displaystyle \frac{\partial }{\partial {\xi }^j}}}{\displaystyle \frac{\partial }{\partial {\xi }^k}}+\mathrm{other}\mathrm{terms},\hfill \\ \hfill & ={\displaystyle \frac{8{\left({\xi }^i\right)}^2}{A^2}}{\displaystyle \frac{\partial }{\partial {\xi }^j}}+{\displaystyle \frac{2A-4{\left({\xi }^j\right)}^2}{A^2}}{\displaystyle \frac{\partial }{\partial {\xi }^j}}\hfill \\ \hfill & +{\displaystyle \frac{4{\xi }^k}{A^2}}\left(-{\delta }_{kj}{\xi }^j-{\delta }_{jj}{\xi }^k+{\delta }_{kj}{\xi }^j\right){\displaystyle \frac{\partial }{\partial {\xi }^j}}+\mathrm{other}\mathrm{terms},\hfill \\ \hfill & ={\displaystyle \frac{8{\left({\xi }^i\right)}^2}{A^2}}{\displaystyle \frac{\partial }{\partial {\xi }^j}}+{\displaystyle \frac{2A-4{\left({\xi }^j\right)}^2}{A^2}}{\displaystyle \frac{\partial }{\partial {\xi }^j}}-{\displaystyle \frac{4{\displaystyle \sum _{k=1}^n}{\left({\xi }^k\right)}^2}{A^2}}{\displaystyle \frac{\partial }{\partial {\xi }^j}}+\mathrm{other}\mathrm{terms},\hfill \end{array}$$

Take ${\tilde{e}}_i={\displaystyle \frac{A}{2}}{\displaystyle \frac{\partial }{\partial {\xi }^i}}$, and $\left\{{\tilde{e}}_i\right\}$ is a unit orthogonal basis with dual basis $e^i={\displaystyle \frac{2}{A}}\mathrm{d}{\xi }^i$. Since

$$\mathrm{d}{e}^i=\sum _j-{\displaystyle \frac{4{\xi }^j\mathrm{d}{\xi }^i\mathrm{d}{\xi }^i}{A^2}}=\sum _j-{\xi }^j{e}^j\wedge e^i=\sum _j{e}^j\wedge (-{\xi }^j{e}^i+{\xi }^i{e}^j),$$

and the connection forms $w_j^i={\xi }^i{e}^j-{\xi }^j{e}^i$, we obtain

$$\begin{array}{cc}\hfill {\nabla }_{{\tilde{e}}_j}{\tilde{e}}_k& =w_k^s\left({\tilde{e}}_j\right){\tilde{e}}_s=({\xi }^s{e}^k-{\xi }^k{e}^s)\left({\tilde{e}}_j\right){\tilde{e}}_s={\xi }^s{\delta }_{kj}{\tilde{e}}_s-{\xi }^k{\tilde{e}}_j\hfill \\ \hfill {\nabla }_{{\tilde{e}}_i}{\nabla }_{{\tilde{e}}_j}{\tilde{e}}_k& ={\nabla }_{{\tilde{e}}_i}({\xi }^s{\delta }_{kj}{\tilde{e}}_s-{\xi }^k{\tilde{e}}_j)\hfill \\ \hfill & =e^i\left({\xi }^s{\delta }_{kj}\right){\tilde{e}}_s+{\xi }^s{\delta }_{kj}{\nabla }_{{\tilde{e}}_i}{\tilde{e}}_s-{\tilde{e}}_i\left({\xi }^k\right){\tilde{e}}_j-{\xi }^k{\nabla }_{{\tilde{e}}_i}{\tilde{e}}_j\hfill \\ \hfill & ={\displaystyle \frac{A}{2}}{\delta }_{is}{\delta }_{kj}{\tilde{e}}_s+{\xi }^s{\delta }_{kj}({\xi }^t{\delta }_{si}{\tilde{e}}_t-{\xi }^s{\tilde{e}}_i)-{\displaystyle \frac{A}{2}}{\delta }_{ki}{\tilde{e}}_j-{\xi }^k({\xi }^s{\delta }_{ij}{\tilde{e}}_s-{\xi }^j{\tilde{e}}_i)\hfill \\ \hfill & ={\displaystyle \frac{A}{2}}{\delta }_{kj}{\tilde{e}}_i+{\xi }^i{\xi }^t{\delta }_{kj}{\tilde{e}}_t-\sum _s{\left({\xi }^s\right)}^2{\delta }_{kj}{\tilde{e}}_i-{\displaystyle \frac{A}{2}}{\delta }_{ki}{\tilde{e}}_j-{\xi }^k{\xi }^s{\delta }_{ij}{\tilde{e}}_s+{\xi }^k{\xi }^j{\tilde{e}}_i,\hfill \\ \hfill {\nabla }_{{\tilde{e}}_i}{\nabla }_{{\tilde{e}}_j}{\tilde{e}}_k& ={\displaystyle \frac{A}{2}}{\delta }_{ki}{\tilde{e}}_j+{\xi }^j{\xi }^t{\delta }_{ki}{\tilde{e}}_t-\sum _s{\left({\xi }^s\right)}^2{\delta }_{ki}{\tilde{e}}_j-{\displaystyle \frac{A}{2}}{\delta }_{kj}{\tilde{e}}_i-{\xi }^k{\xi }^s{\delta }_{ij}{\tilde{e}}_s+{\xi }^k{\xi }^i{\tilde{e}}_j,\hfill \\ \hfill {\tilde{e}}_i,{\tilde{e}}_j]& =\left[{\displaystyle \frac{A}{2}}{\displaystyle \frac{\partial }{\partial {\xi }^i}},{\displaystyle \frac{A}{2}}{\displaystyle \frac{\partial }{\partial {\xi }^j}}\right]={\displaystyle \frac{A}{2}}{\displaystyle \frac{\partial }{\partial {\xi }^i}}\left({\displaystyle \frac{A}{2}}\right){\displaystyle \frac{\partial }{\partial {\xi }^j}}-{\displaystyle \frac{A}{2}}{\displaystyle \frac{\partial }{\partial {\xi }^j}}\left({\displaystyle \frac{A}{2}}\right){\displaystyle \frac{\partial }{\partial {\xi }^i}}\hfill \\ \hfill & ={\displaystyle \frac{A}{2}}·{\xi }_i{\displaystyle \frac{\partial }{\partial {\xi }_j}}-{\displaystyle \frac{A}{2}}·{\xi }_j{\displaystyle \frac{\partial }{\partial {\xi }_i}}={\xi }^i{\tilde{e}}_j-{\xi }^j{\tilde{e}}_i,\hfill \\ \hfill {\nabla }_{[{\tilde{e}}_i,{\tilde{e}}_j]}{\tilde{e}}_k& ={\nabla }_{{\xi }^i{\tilde{e}}_j-{\xi }^j{\tilde{e}}_i}{\tilde{e}}_k={\xi }^i{\nabla }_{{\tilde{e}}_j}{\tilde{e}}_k-{\xi }^j{\nabla }_{{\tilde{e}}_i}{\tilde{e}}_k={\xi }^i({\xi }^s{\delta }_{kj}{\tilde{e}}_s-{\xi }^k{\tilde{e}}_j)-{\xi }^j({\xi }^s{\delta }_{ki}{\tilde{e}}_s-{\xi }^k{\tilde{e}}_i),\hfill \\ \hfill R_{ijkl}& ={\displaystyle \frac{A}{2}}{\delta }_{kj}{\delta }_{il}+{\xi }^i{\xi }^l{\delta }_{kj}-\sum _s{\left({\xi }^s\right)}^2{\delta }_{kj}{\delta }_{il}-{\displaystyle \frac{A}{2}}{\delta }_{ki}{\delta }_{jl}-{\xi }^k{\xi }^l{\delta }_{ij}-{\xi }^k{\xi }^j{\delta }_{il}\hfill \\ \hfill & -\left({\displaystyle \frac{A}{2}}{\delta }_{ki}{\delta }_{jl}+{\xi }^j{\xi }^l{\delta }_{ki}-\sum _s{\left({\xi }^s\right)}^2{\delta }_{ki}{\delta }_{jl}-{\displaystyle \frac{A}{2}}{\delta }_{kj}{\delta }_{il}-{\xi }^k{\xi }^l{\delta }_{ij}+{\xi }^k{\xi }^i{\delta }_{jl}\right)\hfill \\ \hfill & -({\xi }^i{\xi }^l{\delta }_{kj}-{\xi }^i{\xi }^k{\delta }_{jl}-{\xi }^j{\xi }^l{\delta }_{ki}+{\xi }^j{\xi }^k{\delta }_{il})\hfill \\ \hfill & =({\delta }_{kj}{\delta }_{il}-{\delta }_{ki}{\delta }_{jl})\left(A-\sum _s{\left({\xi }^s\right)}^2\right)=-{\delta }_{kj}{\delta }_{il}+{\delta }_{ki}{\delta }_{jl}.\hfill \end{array}$$

Hence, $R_{ijkl}=1$ if and only if $k=i\ne j=l$; $R_{ijkl}=-1$ if and only if $k=j\ne i=l$; and $R_{ijkl}=0$ for all other cases.

Proof of Main Theorem 2. 

Suppose that $S^6$ has a complex structure J with ${\mathrm{d}}^{\nabla }\circ {\mathrm{d}}^{\nabla }J=0$. According to Lemma 4, we obtain

$$R(\xi ,\eta ,J(\zeta ),\tau )+R(\eta ,\zeta ,J(\xi ),\tau )+R(\zeta ,\xi ,J(\eta ),\tau )=0$$

for any four vector fields $\eta ,\zeta ,\xi$ and $\tau$. □

With the notations above and for $(0,0,0,0,0,0,1)\in S^6$ and its neighborhood $\tilde{S}$, we obtain

$$R({\tilde{e}}_i,{\tilde{e}}_j,J\left({\tilde{e}}_k\right),{\tilde{e}}_i)+R({\tilde{e}}_j,{\tilde{e}}_k,J\left({\tilde{e}}_i\right),{\tilde{e}}_l)+R({\tilde{e}}_k,{\tilde{e}}_i,J\left({\tilde{e}}_j\right),{\tilde{e}}_i)=0$$

for $i,j,k=1,2,\cdots ,6$. If $J=J_{ij}{e}^i\otimes {\tilde{e}}_j$, it becomes

$$\sum _{p=1}^{6}R({\tilde{e}}_i,{\tilde{e}}_j,J_{kp}{\tilde{e}}_p,{\tilde{e}}_i)+R({\tilde{e}}_j,{\tilde{e}}_k,J_{ip}{\tilde{e}}_p,{\tilde{e}}_i)+R({\tilde{e}}_k,{\tilde{e}}_i,J_{jp}{\tilde{e}}_p,{\tilde{e}}_i)=0,$$

or

$$\sum _{p=1}^6\left(J_{kp}{R}_{ijpi}+J_{ip}{R}_{jkpi}+J_{jp}{R}_{kipi}\right)=0.$$

The only nonzero curvature coefficients above are $R_{ijji}=-1$ and $R_{kiki}=1$, which implies

$$J_{jk}=J_{kj},j,k=1,2,\cdots 6.$$

Namely, J is a symmetric matrix. A real symmetric matrix must have a real eigenvalue, which is a contradiction.

4. Some Examples

In 2001, S.M. Salamon [9] classified real six-dimensional nilpotent Lie algebras for which the corresponding Lie group had a left-invariant complex structure. He obtained the following theorems.

Proposition 1. 

Four-dimensional nilpotent Lie algebras admitting a complex structure are isomorphic to $(0,0,0,12)$ or $(0,0,0,0)$.

Proposition 2. 

Any six-dimensional nilpotent Lie algebras admitting a complex structure are isomorphic to one of the following 18 types.

Type $1:(0,0,12,13,23,14+25)$;		Type $2:(0,0,0,12,13,14)$;
Type $3:(0,0,0,12,13,23)$;		Type $4:(0,0,0,12,14,24)$;
Type $5:(0,0,0,12,13,24)$;		Type $6:(0,0,0,12,13+14,24)$;
Type $7:(0,0,0,12,13,14+23)$;		Type $8:(0,0,0,12,14,13+42)$;
Type $9:(0,0,0,12,13+42,14+23)$;		Type $10:(0,0,0,12,23,14-35)$;
Type $11:(0,0,0,0,12,14+25)$;		Type $12:(0,0,0,0,12,13)$;
Type $13:(0,0,0,0,13+42,14+23)$;		Type $14:(0,0,0,0,12,14+23)$;
Type $15:(0,0,0,0,12,34)$;		Type $16:(0,0,0,0,0,12+34)$;
Type $17:(0,0,0,0,0,12)$;		Type $18:(0,0,0,0,0,0)$.

The symbol $(0,0,0,12)$ refers to the Lie algebra whose dual has a basis for which $\mathrm{d}{e}^i=0$ for $i=1,2,3$ and $\mathrm{d}{e}^4=e^1\wedge e^2$; $(0,0,0,0)$ is simply an Abelian algebra. For the first type of six-dimensional nilpotent Lie algebras, $14+25$ means $\mathrm{d}{e}^6=e^1\wedge e^4+e^2\wedge e^5$.

We will compute ${\mathrm{d}}^{\nabla }J$ and ${\mathrm{d}}^{\nabla }\circ {\mathrm{d}}^{\nabla }J$ for a complex structure J, where M is a four-dimensional or six-dimensional nilpotent Lie algebra admitting a complex structure, and has a kind of standard metric such that the base forms an orthogonal basis. We show that they are not zero.

Proof of Cases in Proposition 1. 

The second four-dimensional nilpotent Lie algebra is isomorphic to $(0,0,0,12)$. This is the famous Kodaira–Thurston surface and was used in [22] to construct the first example of a simply connected compact holomorphic symplectic manifold that is (topologically) nonKähler. The other (the first) one is exactly the abelian variety or the complex torus. □

Choose a Riemannian metric g with $g({\tilde{e}}_i,{\tilde{e}}_j)={\delta }_{ij}$, $i,j=1,2,3,4$, and where ∇ is the Levi-Civita connection of g. Let $w^1=e^1+\sqrt{-1}{e}^2$ and $w^2=e^3+\sqrt{-1}{e}^4$. Then, $w^1$ and $w^2$ are a basis of ${\mathrm{\Lambda }}^{1,0}$. An almost-complex structure J on M induces a complex structure on ${\mathrm{\Lambda }}^{1,0}$ in the following way:

$$J{w}^1=\sqrt{-1}{w}^1,J{w}^2=\sqrt{-1}{w}^2,$$

or

$$J{e}^1=e^2,J{e}^2=-e^1,J{e}^3=e^4,J{e}^4=-e^3.$$

Hence, J acts on $TM$ as

$$J{\tilde{e}}_1=-{\tilde{e}}_2,J{\tilde{e}}_2={\tilde{e}}_1,J{\tilde{e}}_3=-{\tilde{e}}_4,J{\tilde{e}}_4={\tilde{e}}_3.$$

Since $\mathrm{d}{e}^4=e^1\wedge e^2$, we have

$$\mathrm{d}{e}^4({\tilde{e}}_1,{\tilde{e}}_2)=e^1\left({\tilde{e}}_1\right)e^2\left({\tilde{e}}_2\right)-e^2\left({\tilde{e}}_1\right)e^1\left({\tilde{e}}_2\right)=1,$$

and

$$\mathrm{d}{e}^4({\tilde{e}}_1,{\tilde{e}}_2)={\tilde{e}}_1\left(e^4\left({\tilde{e}}_2\right)\right)-{\tilde{e}}_2\left(e^4\left({\tilde{e}}_1\right)\right)-e^4\left([{\tilde{e}}_1,{\tilde{e}}_2]\right)=-e^4\left([{\tilde{e}}_1,{\tilde{e}}_2]\right).$$

It is easy to see that

$$e^1\left([{\tilde{e}}_1,{\tilde{e}}_2]\right)=e^2\left([{\tilde{e}}_1,{\tilde{e}}_2]\right)=e^3\left([{\tilde{e}}_1,{\tilde{e}}_2]\right)=0.$$

Thus, all bracket products $[{\tilde{e}}_i,{\tilde{e}}_j]=0$ except $[{\tilde{e}}_1,{\tilde{e}}_2]=-[{\tilde{e}}_2,{\tilde{e}}_1]=-{\tilde{e}}_4$. According to the Koszul Formula, the coefficients of connection ∇

$$\begin{array}{cc}\hfill {\mathrm{\Gamma }}_{ji}^k& =\langle {\nabla }_{{\tilde{e}}_j}{\tilde{e}}_i,{\tilde{e}}_k\rangle ={\displaystyle \frac{1}{2}}\left(-\langle {\tilde{e}}_j,[{\tilde{e}}_i,{\tilde{e}}_k]\rangle +\langle {\tilde{e}}_i,[{\tilde{e}}_k,{\tilde{e}}_j]\rangle +\langle {\tilde{e}}_k,[{\tilde{e}}_j,{\tilde{e}}_i]\rangle \right).\hfill \end{array}$$

We obtain the only nonzero coefficients

$${\mathrm{\Gamma }}_{41}^2={\displaystyle \frac{1}{2}},{\mathrm{\Gamma }}_{42}^1=-{\displaystyle \frac{1}{2}},{\mathrm{\Gamma }}_{24}^1=-{\displaystyle \frac{1}{2}},{\mathrm{\Gamma }}_{14}^2={\displaystyle \frac{1}{2}},{\mathrm{\Gamma }}_{12}^4=-{\displaystyle \frac{1}{2}},{\mathrm{\Gamma }}_{21}^4={\displaystyle \frac{1}{2}}.$$

If $J=u_{ij}{\tilde{e}}_i\otimes e^j$ with associate matrix $\mathit{J}=\mathrm{diag}({\mathit{J}}_0,{\mathit{J}}_0)$, where

$${\mathit{J}}_0=\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]$$

and $\xi ={\xi }^i{\tilde{e}}_i$, $\eta ={\eta }^j{\tilde{e}}_j$, $\zeta ={\zeta }^k{\tilde{e}}_k$, we have

$$\begin{array}{cc}\hfill \left({\mathrm{d}}^{\nabla }J\right)(\xi ,\eta )& ={\displaystyle \frac{1}{2}}\left({\xi }^1{\eta }^4-{\xi }^4{\eta }^1-{\xi }^2{\eta }^3+{\xi }^3{\eta }^2\right){\tilde{e}}_1+{\displaystyle \frac{1}{2}}\left(-{\xi }^4{\eta }^2+{\xi }^1{\eta }^3+{\xi }^2{\eta }^4-{\xi }^3{\eta }^1\right){\tilde{e}}_2\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}\left(-2{\xi }^1{\eta }^2+2{\xi }^2{\eta }^1\right){\tilde{e}}_3,\hfill \\ \hfill ({\mathrm{d}}^{\nabla }\circ {\mathrm{d}}^{\nabla }J)(\xi ,\eta ,\zeta )& =(-{\xi }^1{\eta }^3{\zeta }^4+{\xi }^1{\eta }^4{\zeta }^3+{\xi }^3{\eta }^1{\zeta }^4-{\xi }^3{\eta }^4{\zeta }^1-{\xi }^4{\eta }^1{\zeta }^3+{\xi }^4{\eta }^3{\zeta }^1){\tilde{e}}_1+\hfill \\ \hfill & (-{\xi }^2{\eta }^3{\zeta }^4+{\xi }^2{\eta }^4{\zeta }^3+{\xi }^3{\eta }^2{\zeta }^4-{\xi }^3{\eta }^4{\zeta }^2-{\xi }^4{\eta }^2{\zeta }^3+{\xi }^4{\eta }^3{\zeta }^2){\tilde{e}}_2+\hfill \\ \hfill & (-2{\xi }^1{\eta }^2{\zeta }^4+2{\xi }^1{\eta }^4{\zeta }^2+2{\xi }^2{\eta }^1{\zeta }^4-2{\xi }^2{\eta }^4{\zeta }^1-2{\xi }^4{\eta }^1{\zeta }^2+2{\xi }^4{\eta }^2{\zeta }^1){\tilde{e}}_4.\hfill \end{array}$$

In particular, we obtain

$$\left({\mathrm{d}}^{\nabla }J\right)({\tilde{e}}_2,{\tilde{e}}_1)={\tilde{e}}_3,\mathrm{and}\left(({\mathrm{d}}^{\nabla }\circ {\mathrm{d}}^{\nabla })J\right)({\tilde{e}}_4,{\tilde{e}}_2,{\tilde{e}}_1)=2{\tilde{e}}_4.$$

Hence, ${\mathrm{d}}^{\nabla }J\not\equiv 0$ and $({\mathrm{d}}^{\nabla }\circ {\mathrm{d}}^{\nabla })J\not\equiv 0$.

Proof of the Cases in Proposition 2. 

First, we deal with the fifteenth in the list. In this case, the six-dimensional nilpotent Lie algebra is isomorphic to $(0,0,0,0,12,34)$. Let $J=u_{ij}{\tilde{e}}_i\otimes e^j$ with associate matrix $\mathit{J}=\mathrm{diag}({\mathit{J}}_0,{\mathit{J}}_0,{\mathit{J}}_0)$, and $\xi ={\xi }^i{\tilde{e}}_i$, $\eta ={\eta }^j{\tilde{e}}_j$, $\zeta ={\zeta }^k{\tilde{e}}_k$. In the same way as above, we compute the nonzero bracket products:

$$[{\tilde{e}}_1,{\tilde{e}}_2]=-{\tilde{e}}_5,[{\tilde{e}}_3,{\tilde{e}}_4]=-{\tilde{e}}_6$$

and the nonzero coefficients:

$${\mathrm{\Gamma }}_{12}^5=1,{\mathrm{\Gamma }}_{15}^2=1,{\mathrm{\Gamma }}_{21}^5=-1,{\mathrm{\Gamma }}_{25}^1=-1,$$

$${\mathrm{\Gamma }}_{34}^6=1,{\mathrm{\Gamma }}_{36}^4=1,{\mathrm{\Gamma }}_{43}^6=-1,{\mathrm{\Gamma }}_{46}^3=-1,$$

$${\mathrm{\Gamma }}_{51}^2=1,{\mathrm{\Gamma }}_{52}^1=-1,{\mathrm{\Gamma }}_{63}^4=1,{\mathrm{\Gamma }}_{64}^3=-1.$$

Then, we obtain

$$\begin{array}{cc}\hfill \left({\mathrm{d}}^{\nabla }J\right)(\xi ,\eta )& =({\xi }^1{\eta }^5+{\xi }^2{\eta }^6-{\xi }^5{\eta }^1-{\xi }^6{\eta }^2){\tilde{e}}_1+(-{\xi }^1{\eta }^6+{\xi }^2{\eta }^5-{\xi }^5{\eta }^2+{\xi }^6{\eta }^1){\tilde{e}}_2\hfill \\ \hfill & +({\xi }^3{\eta }^6-{\xi }^4{\eta }^5+{\xi }^5{\eta }^4-{\xi }^6{\eta }^3){\tilde{e}}_3+({\xi }^3{\eta }^5+{\xi }^4{\eta }^6-{\xi }^5{\eta }^3-{\xi }^6{\eta }^4){\tilde{e}}_4\hfill \\ \hfill & +(-2{\xi }^3{\eta }^4+2{\xi }^4{\eta }^3){\tilde{e}}_5+(2{\xi }^1{\eta }^2-2{\xi }^2{\eta }^1){\tilde{e}}_6\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \left(({\mathrm{d}}^{\nabla }\circ {\mathrm{d}}^{\nabla })J\right)(\xi ,\eta ,\zeta )& =(-{\xi }_1{\eta }_5{\zeta }_6+{\xi }_1{\eta }_6{\zeta }_5+{\xi }_5{\eta }_1{\zeta }_6-{\xi }_5{\eta }_6{\zeta }_1-{\xi }_6{\eta }_1{\zeta }_5+{\xi }_6{\eta }_5{\zeta }_1){\tilde{e}}_1\hfill \\ \hfill & +(-{\xi }_2{\eta }_5{\zeta }_6+{\xi }_2{\eta }_6{\zeta }_5+{\xi }_5{\eta }_2{\zeta }_6-{\xi }_5{\eta }_6{\zeta }_2-{\xi }_6{\eta }_2{\zeta }_5+{\xi }_6{\eta }_5{\zeta }_2){\tilde{e}}_2\hfill \\ \hfill & +(-{\xi }_3{\eta }_5{\zeta }_6+{\xi }_3{\eta }_6{\zeta }_5+{\xi }_5{\eta }_3{\zeta }_6-{\xi }_5{\eta }_6{\zeta }_3-{\xi }_6{\eta }_3{\zeta }_5+{\xi }_6{\eta }_5{\zeta }_3){\tilde{e}}_3\hfill \\ \hfill & +(-{\xi }_4{\eta }_5{\zeta }_6+{\xi }_4{\eta }_6{\zeta }_5+{\xi }_5{\eta }_4{\zeta }_6-{\xi }_5{\eta }_6{\zeta }_4-{\xi }_6{\eta }_4{\zeta }_5+{\xi }_6{\eta }_5{\zeta }_4){\tilde{e}}_4\hfill \\ \hfill & +(-2{\xi }_1{\eta }_2{\zeta }_5+2{\xi }_1{\eta }_5{\zeta }_2+2{\xi }_2{\eta }_1{\zeta }_5-2{\xi }_2{\eta }_5{\zeta }_1-2{\xi }_5{\eta }_1{\zeta }_2+2{\xi }_5{\eta }_2{\zeta }_1){\tilde{e}}_5\hfill \\ \hfill & +(-2{\xi }_3{\eta }_4{\zeta }_6+2{\xi }_3{\eta }_6{\zeta }_4+2{\xi }_4{\eta }_3{\zeta }_6-2{\xi }_4{\eta }_6{\zeta }_3-2{\xi }_6{\eta }_3{\zeta }_4+2{\xi }_6{\eta }_4{\zeta }_3){\tilde{e}}_6.\hfill \end{array}$$

In particular,

$$\left({\mathrm{d}}^{\nabla }J\right)({\tilde{e}}_1,{\tilde{e}}_2)=\left(({\mathrm{d}}^{\nabla }\circ {\mathrm{d}}^{\nabla })J\right)({\tilde{e}}_6,{\tilde{e}}_4,{\tilde{e}}_3)=2{\tilde{e}}_6.$$

For the other 16 non-Kähler types, both ${\mathrm{d}}^{\nabla }J$ and $({\mathrm{d}}^{\nabla }\circ {\mathrm{d}}^{\nabla })J$ do not vanish through the following equalities:

Type 1:	$(0,0,12,13,23,14+25),$	$\left(({\mathrm{d}}^{\nabla }\circ {\mathrm{d}}^{\nabla })J\right)({\tilde{e}}_6,{\tilde{e}}_5,{\tilde{e}}_4)={\tilde{e}}_4+{\tilde{e}}_5$;
Type 2:	$(0,0,0,12,13,14),$	$\left(({\mathrm{d}}^{\nabla }\circ {\mathrm{d}}^{\nabla })J\right)({\tilde{e}}_6,{\tilde{e}}_5,{\tilde{e}}_4)=2{\tilde{e}}_4$;
Type 3:	$(0,0,0,12,13,23),$	$\left(({\mathrm{d}}^{\nabla }\circ {\mathrm{d}}^{\nabla })J\right)({\tilde{e}}_6,{\tilde{e}}_5,{\tilde{e}}_3)=2{\tilde{e}}_3$;
Type 4:	$(0,0,0,12,14,24),$	$\left(({\mathrm{d}}^{\nabla }\circ {\mathrm{d}}^{\nabla })J\right)({\tilde{e}}_6,{\tilde{e}}_5,{\tilde{e}}_4)=2{\tilde{e}}_4$;
Type 5:	$(0,0,0,12,13,24),$	$\left(({\mathrm{d}}^{\nabla }\circ {\mathrm{d}}^{\nabla })J\right)({\tilde{e}}_6,{\tilde{e}}_5,{\tilde{e}}_4)={\tilde{e}}_4$;
Type 6:	$(0,0,0,12,13+14,24),$	$\left(({\mathrm{d}}^{\nabla }\circ {\mathrm{d}}^{\nabla })J\right)({\tilde{e}}_6,{\tilde{e}}_5,{\tilde{e}}_4)={\tilde{e}}_3+2{\tilde{e}}_4$;
Type 7:	$(0,0,0,12,13,14+23),$	$\left(({\mathrm{d}}^{\nabla }\circ {\mathrm{d}}^{\nabla })J\right)({\tilde{e}}_6,{\tilde{e}}_5,{\tilde{e}}_4)=2{\tilde{e}}_4$;
Type 8:	$(0,0,0,12,14,13+42),$	$\left(({\mathrm{d}}^{\nabla }\circ {\mathrm{d}}^{\nabla })J\right)({\tilde{e}}_6,{\tilde{e}}_5,{\tilde{e}}_4)={\tilde{e}}_4$;
Type 9:	$(0,0,0,12,13+42,14+23),$	$\left(({\mathrm{d}}^{\nabla }\circ {\mathrm{d}}^{\nabla })J\right)({\tilde{e}}_6,{\tilde{e}}_5,{\tilde{e}}_4)=4{\tilde{e}}_4$;
Type 10:	$(0,0,0,12,23,14-35),$	$\left(({\mathrm{d}}^{\nabla }\circ {\mathrm{d}}^{\nabla })J\right)({\tilde{e}}_6,{\tilde{e}}_5,{\tilde{e}}_4)={\tilde{e}}_4$;
Type 11:	$(0,0,0,0,12,14+25),$	$\left(({\mathrm{d}}^{\nabla }\circ {\mathrm{d}}^{\nabla })J\right)({\tilde{e}}_6,{\tilde{e}}_5,{\tilde{e}}_4)={\tilde{e}}_4$;
Type 12:	$(0,0,0,0,12,13),$	$\left(({\mathrm{d}}^{\nabla }\circ {\mathrm{d}}^{\nabla })J\right)({\tilde{e}}_6,{\tilde{e}}_5,{\tilde{e}}_4)=-{\tilde{e}}_2$;
Type 13:	$(0,0,0,0,13+42,14+23),$	$\left(({\mathrm{d}}^{\nabla }\circ {\mathrm{d}}^{\nabla })J\right)({\tilde{e}}_6,{\tilde{e}}_5,{\tilde{e}}_4)=4{\tilde{e}}_4$;
Type 14:	$(0,0,0,0,12,14+23),$	$\left(({\mathrm{d}}^{\nabla }\circ {\mathrm{d}}^{\nabla })J\right)({\tilde{e}}_6,{\tilde{e}}_5,{\tilde{e}}_4)={\tilde{e}}_1+{\tilde{e}}_4$;
Type 16:	$(0,0,0,0,0,12+34),$	$\left(({\mathrm{d}}^{\nabla }\circ {\mathrm{d}}^{\nabla })J\right)({\tilde{e}}_6,{\tilde{e}}_5,{\tilde{e}}_4)={\tilde{e}}_4$;
Type 17:	$(0,0,0,0,0,12),$	$\left(({\mathrm{d}}^{\nabla }\circ {\mathrm{d}}^{\nabla })J\right)({\tilde{e}}_6,{\tilde{e}}_5,{\tilde{e}}_2)={\tilde{e}}_2$.

Thus, Main Theorem 1 is proved. □

Moreover, we obtain:

Theorem 1. 

The complex structures in the Salamon’s paper is semi-weakly Kähler with a standard metric such that the basis is orthogonal if and only if the related nil-manifold is abelian.