# Hopf Problem by an Algebraic Approach

Jun Ling

► To cite this version:

Jun Ling. Hopf Problem by an Algebraic Approach. 2025. hal-05380624

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# HOPF PROBLEM BY AN ALGEBRAIC APPROACH

JUN LING

**ABSTRACT.** We construct a differential 2-form from any almost complex structure on a smooth manifold. If this 2-form is not vanishing, then the almost complex structure is not integrable. We show that for the classical almost complex structure $J_0$ on $S^6$ from the multiplication in Octonions, its 2-form is not vanish therefore this gives a new proof for non-integrability of $J_0$. Therefore this 2-form is not trivial, nor empty. This 2-form suggests a relation between the integrability of almost complex structures and the three cohomology groups $H^1(S^6) = 0$, $H^2(S^6) = 0$ and $H^3(S^6) = 0$ of the underlying manifold $S^6$. Applying this 2-form to $S^6$ Hopf problem, we prove that all almost complex structures on $S^6$ are not integrable. Therefore $S^6$ is not a complex manifold. Our approach is algebraic.

## 1. INTRODUCTION

Integability of almost complex structure on differential manifold, in particular on high dimensional spheres has been studied extensively. If an almost complex structure is integrable, then it is a complex structure that makes the underlying differential manifold a complex manifold. Conversely, the complex structure of a complex manifold induces a (an integrable) almost complex structure. There have been lot of great work in this field, for example, Hopf [14], Borel and Serre [5], Ehresmann [9], Kirchhoff [15], Eckmann and Frölicher [8], Ehressmann and Libermann [10], LeBrun [17], Atiyah [2], Bryant and S.S. Chern [3] [4], Sullivan [21] Cirici-Wilson [6], and the author [19], and etc to name a few. For spheres, it is known that all high dimensional (dimension greater than 2) spheres are not complex manifolds except for $S^6$. Whether $S^6$ is a complex manifold or not is still a well-known open problem, known as Hopf Problem. See Hirzebruch [13] in 1954, Libermann [18] in 1955 and Yau [22] in 1990. Almost complex structures do exist on $S^6$. It is known that none of known almost complex structures on $S^6$ is integrable, therefore none of them is a complex structure that makes $S^6$ a complex manifold. See [1] for a survey.

Date: November 10, 2025.

2010 Mathematics Subject Classification. Primary 53C15; Secondary 32Q60.

Key words and phrases. Hopf Problem, $S^6$, almost complex structures.

Thank Princeton University and Cornell University for their hospitality.
In this paper we study integrability of an almost complex structure in a new approach. We construct a differential 2-form from any almost complex structure on a smooth manifold. If the 2-form is not vanishing, the almost complex structure is not integrable. We show that for the classic almost complex structure $J_0$ on $S^6$ from the multiplication in Octonions, its 2-form is not vanish therefore this gives a new proof for non-integrability of $J_0$. Therefore this 2-form is not trivial, nor empty. This 2-form suggests a relation between the integrability of the almost complex structures and the three cohomology groups $H^1(M)$, $H^2(M)$ and $H^3(M)$ of the underlying manifold $M^n$ ($2 < n \le 6$), as we see from examples of complex manifolds $M = \mathbb{CP}^3$, $M = S^3 \times S^3$ and $M = S^1 \times S^3$, comparing with $H^1(S^6) = 0$, $H^2(S^6) = 0$ and $H^3(S^6) = 0$. We then show that any almost complex structure $J$ on $S^6$ is not integrable through the 2-form $\ell$ of $J$. Therefore $S^6$ is not a complex manifold.

Our approach is algebraic.

Let $M$ be a differential $n$-manifold and $J$ any almost complex structure on $M$. An almost complex structure $J$ on differential manifold $M$ is a $(1, 1)$ tensor field on the tangent bundle $T(M)$ of $M$ with $J^2 = -1$. Nijenhuis tensor $N = N_J$ of almost complex structure $J$ is defined by

$$N(X, Y) = [JX, JY] - J[X, JY] - J[JX, Y] - [X, Y]$$

for vector fields $X, Y$, where $[\cdot, \cdot]$ is Lie bracket.

We define the quantity $\ell = \ell_J$ from an almost complex structure $J$ where $N = N_J$ is defined above by $J$.

$$(1.1) \qquad \ell(X, Y) := \operatorname{trace}\{JN_J(N_J(X, \cdot), Y)\}$$

for vector fields $X, Y$ on $M$.

We have the following results.

**Theorem 1.1.** Let $J$ be any almost complex structure on a smooth manifold $M^n$. Then $\ell = \ell_J$ defined by (1.1) is a 2-form on $M$.

**Corollary 1.2.** If form $\ell = \ell_J$ is not vanishing, then $J$ is not integrable.

*Proof.* If $\ell = \ell_J$ is not vanishing then Nijenhuis $N_J$ is not vanishing. Theorem follows from Newlander-Nirenberg Theorem [20]. □

We apply the above 2-form and theorems to a classical case on $S^6$, show the 2-form is not trivial nor empty and re-prove a known classical result.

View $S^6$ as the unit sphere in the imaginary part of Octonians. There is a well-known classical almost complex structure $J_0$ define on the tangent bundle of $S^6$ by multiplication in Octonions. We call it Left-multiplication almost complex structure $J_0$. Please refer to Section 3, [16], or [7] for more details about Left-multiplication almost complex structure $J_0$. We re-prove the following known result.
**Theorem 1.3.** Form $\ell_0 = \ell_{J_0}$ is not vanishing for the Left-multiplication almost complex structure $J_0$ on $S^6$. So it is not integrable.

We apply the 2-form to $S^6$ Hopf problem and have the following result.

**Theorem 1.4.** Any almost complex structure on $S^6$ is not integrable. $S^6$ is not a complex manifold.

In the following, Section 2 is for proving Theorem 1.1. Section 3 is for proving Theorem 1.3. Section 4 is for proving Theorem 1.4.

In [19] author proved the double trace of any almost structure vanishes. Along this line, the author later studied single trace during the visits to Princeton Mathematics Department and to Cornell Mathematics Department, and afterwards. The author sincerely appreciate hospitality of Princeton University and Cornell University.

## 2. A DIFFERENTIAL 2-FORM

We take a local coordinates $\{x^i\}_{i=1}^n$ at point $x \in M^n$. For $i, j, k = 1, \dots, n$, let $\partial_i = \frac{\partial}{\partial x^i}$, $J_i \frac{\partial}{\partial x^i} = \sum_{j=1}^n J_j^j \frac{\partial}{\partial x^j}$, $J_i = \sum_{p=1}^n J_i^p \frac{\partial}{\partial x^p}$, and $N_{ij}^k$ components of Nijenhuis tensor $N$, where

$$N(\frac{\partial}{\partial x^i}, \frac{\partial}{\partial x^j}) := \sum_{k=1}^{n} N_{ij}^{k} \frac{\partial}{\partial x^k}.$$

It is easy to verify

$$N_{ij}^{k} = \sum_{p=1}^{n} \left\{ J_{i}^{p} (\partial_{p} J_{j}^{k} - \partial_{j} J_{i}^{k}) - J_{j}^{p} (\partial_{p} J_{i}^{k} - \partial_{i} J_{p}^{k}) \right\}$$

for $i, j, k = 1, \dots, n$.

*Proof of Theorem 1.1.* The component form of Theorem 1.1 is the following component version Theorem 2.1 in local. we prove it.

**Theorem 2.1.** $\ell$ can be written in the following form in local coordinates:

$$\ell = \sum_{i,j=1}^{n} \ell_{ij} dx^i \wedge dx^j,$$
where for $i,j = 1,2,\dots,n$,

$$ (2.1) $$

$$ \begin{aligned} \ell_{ij} = & \sum_{k,l,r,p,q=1}^{n} -J_k^l J_i^p \partial_p J_l^r J_j^q \partial_q J_r^k + \sum_{k,l,r=1}^{n} J_k^l \partial_i J_r^k \partial_j J_l^r \\ & + \sum_{k,l,r=1}^{n} \left\{ -J_i^r \partial_r J_k^l \partial_j J_l^k + J_j^r \partial_r J_k^l \partial_i J_l^k \right\} + \sum_{k,l,r=1}^{n} 2 \left\{ J_i^r \partial_r J_k^l \partial_l J_j^k - J_j^r \partial_r J_k^l \partial_l J_i^k \right\} \\ & + \sum_{k,l,r=1}^{n} 2 \left\{ -J_k^r \partial_r J_i^l \partial_l J_j^k + J_j^r \partial_r J_i^l \partial_l J_i^k \right\} + \sum_{k,l,r=1}^{n} 2 \left\{ J_k^r \partial_r J_i^l \partial_j J_l^k - J_k^r \partial_r J_j^l \partial_i J_l^k \right\}, \end{aligned} $$

and $\ell_{ij} = -\ell_{ji}$.

In the following calculations, we adopt Einstein's sum convention: take sum for any repeating index, unless otherwise stated. Let $J_i = J_i^p \frac{\partial}{\partial x^p} = J_i^p \partial_p$. Then (2.1) can be written as

$$ \begin{aligned} \ell_{ij} = & -J_k^l \cdot J_i J_l^r \cdot J_j J_r^k + J_k^l \partial_i J_r^k \partial_j J_l^r + \{-J_i J_k^l \partial_j J_l^k + J_j J_k^l \partial_i J_l^k\} \\ & + 2\{J_i J_k^l \partial_l J_j^k - J_j J_k^l \partial_l J_i^k\} + 2\{-J_k J_i^l \partial_l J_j^k + J_j J_k^l \partial_l J_i^k\} + 2\{J_k J_i^l \partial_j J_l^k - J_k J_j^l \partial_i J_l^k\}. \end{aligned} \quad (2.2) $$

We need only to show matrix $[\ell_{ij}]$ is skew-symmetric at every $x \in M$. By (1.1), we have

$$ \begin{aligned} \ell_{ij} &= N_{ik}^r N_{rj}^s J_s^k = J_s^k N_{ik}^r N_{rj}^s \\ &= J_s^k \{J_i^p (\partial_p J_k^r - \partial_k J_p^r) - J_k^p (\partial_p J_i^r - \partial_i J_p^r)\} \{J_r^q (\partial_q J_j^s - \partial_j J_q^s) - J_j^q (\partial_q J_r^s - \partial_r J_q^s)\} \\ &= \{J_s^k J_i^p (\partial_p J_k^r - \partial_k J_p^r) - J_s^k J_k^p (\partial_p J_i^r - \partial_i J_p^r)\} \{J_r^q (\partial_q J_j^s - \partial_j J_q^s) - J_j^q (\partial_q J_r^s - \partial_r J_q^s)\} \\ &= \{J_s^k J_i^p \partial_p J_k^r - J_s^k J_i^p \partial_k J_p^r + \delta_s^p (\partial_p J_i^r - \partial_i J_p^r)\} \{J_r^q (\partial_q J_j^s - \partial_j J_q^s) - J_j^q (\partial_q J_r^s - \partial_r J_q^s)\} \\ &= \{J_s^k J_i^p \partial_p J_k^r - J_s^k J_i^p \partial_k J_p^r + \delta_s^p (\partial_p J_i^r - \partial_i J_p^r)\} \{J_r^q \partial_q J_j^s - J_r^q \partial_j J_q^s - J_j^q \partial_q J_r^s + J_j^q \partial_r J_q^s\} \\ &= \{J_s^k J_i^p \partial_p J_k^r - J_s^k J_i^p \partial_k J_p^r + \delta_s^p (\partial_p J_i^r - \partial_i J_p^r)\} \{J_r^q \partial_q J_j^s - J_r^q \partial_j J_q^s - J_j^q \partial_q J_r^s + J_j^q \partial_r J_q^s\}. \end{aligned} $$

Therefore

$$ \begin{aligned} \ell_{ij} = & J_s^k \cdot J_i J_k^r \cdot J_r J_j^s - J_r^q J_s^k \cdot J_i J_k^r \cdot \partial_j J_q^s - J_s^k \cdot J_i J_k^r \cdot J_j J_r^s + J_j^q J_s^k \cdot J_i J_k^r \cdot \partial_r J_q^s \\ & - J_i^p \cdot J_s J_p^r \cdot J_r J_j^s + J_r^q J_i^p \cdot J_s J_p^r \cdot \partial_j J_q^s + J_i^p \cdot J_s J_p^r \cdot J_j J_r^s - J_j^q J_i^p \cdot J_s J_p^r \cdot \partial_r J_q^s \\ & + J_r J_j^s \cdot \partial_s J_i^r - J_r^q \cdot \partial_j J_q^s \cdot \partial_s J_i^r - J_j J_r^s \cdot \partial_s J_i^r + J_j^q \cdot \partial_r J_q^s \cdot \partial_s J_i^r \\ & - J_r J_j^s \cdot \partial_i J_s^r + J_r^q \cdot \partial_i J_s^r \cdot \partial_j J_q^s + J_j J_r^s \cdot \partial_i J_s^r - J_j^q \cdot \partial_r J_q^s \cdot \partial_i J_s^r \end{aligned} $$

$$ \begin{aligned} \stackrel{\text{def}}{=} & a(1) + a(2) + a(3) + a(4) \\ & + b(1) + b(2) + b(3) + b(4) \\ & + c(1) + c(2) + c(3) + c(4) \\ & + d(1) + d(2) + d(3) + d(4) \end{aligned} $$

where and below we use facts $J^2 = -1$ and $\text{tr}(J) = 0$.
$$a(1) = J_s^k \cdot J_i J_k^r \cdot J_r J_j^s = J_k^l \cdot J_i J_l^r \cdot J_r J_j^k,$$

and

$$b(3) = J_i^p \cdot J_s J_p^r \cdot J_j J_r^s = -J_p^r \cdot J_s J_i^p \cdot J_j J_r^s = -J_k^l \cdot J_s J_i^k \cdot J_j J_l^s = -J_k^l \cdot J_j J_l^r \cdot J_r J_i^k.$$

Therefore we have

$$a(1) + b(3) = J_k^l \cdot J_i J_l^r \cdot J_r J_j^k - J_k^l \cdot J_j J_l^r \cdot J_r J_i^k.$$

Furthermore,

$$\begin{aligned}
a(1)+b(3) &= J_k^l \cdot J_i J_l^r \cdot J_r J_j^k - J_k^l \cdot J_j J_l^r \cdot J_r J_i^k = J_k^l \cdot J_i^p \partial_p J_l^r \cdot J_r^s \partial_s J_j^k - J_k^l \cdot J_j^p \partial_p J_l^r \cdot J_r^s \partial_s J_i^k \\
&= -\partial_p J_k^l \cdot J_i^p J_l^r J_r^s \partial_s J_j^k + \partial_p J_k^l \cdot J_j^p J_l^r J_r^s \partial_s J_i^k = \partial_p J_k^l \cdot J_i^p \partial_l J_j^k - \partial_p J_k^l \cdot J_j^p \partial_l J_i^k \\
&= J_i J_k^l \partial_l J_j^k - J_j J_k^l \partial_l J_i^k = a(4) + c(3),
\end{aligned}$$

$$a(2) = -J_r^q J_s^k \cdot J_i J_k^r \partial_j J_q^s = J_q^s J_s^k \cdot J_i J_k^r \partial_j J_r^q = -J_i J_k^r \partial_j J_r^k = -J_i J_k^l \partial_j J_l^k,$$

and

$$d(3) = J_j J_r^s \cdot \partial_i J_s^r = \partial_i J_p^k \cdot J_j J_k^p = J_j J_k^l \partial_i J_l^k.$$

Therefore we have

$$a(2) + d(3) = -J_i J_k^l \partial_j J_l^k + J_j J_k^l \partial_i J_l^k.$$

$$a(3) = -J_l^p \cdot J_i J_p^k \cdot J_j J_k^l = -J_k^l \cdot J_i J_l^r \cdot J_j J_r^k = -J_t^p \cdot J_i J_p^k \cdot J_j J_k^l = -J_t^p \cdot J_i J_p^l \cdot J_j J_l^t.$$

On the ther hand

$$\begin{aligned}
a(3) &= -J_l^p \cdot J_i J_p^k \cdot J_j J_k^l = -J_k^l \cdot J_i J_k^l \cdot J_j J_l^t = -J_k^l \cdot J_i^p \partial_p J_k^l \cdot J_j^q \partial_q J_l^t \\
&= -\partial_p(J_l^k J_i^p J_k^l J_j^q \partial_q J_l^t) + \partial_p J_k^l \cdot J_i^p J_k^l J_j^q \partial_q J_l^t + J_k^l \partial_p J_i^p \cdot J_k^l J_j^q \partial_q J_l^t \\
&\quad + J_l^k J_i^p J_k^l \partial_p J_j^q \partial_q J_l^t + J_l^k J_i^p J_k^l J_j^q \partial_p \partial_q J_l^t \\
&= \partial_p(J_i^p \cdot J_j^q \partial_q J_l^t) + J_i J_k^l \cdot J_k^l \cdot J_j J_l^t - \partial_p J_i^p \cdot J_j^q \partial_q J_l^t \\
&\quad - J_i^p \cdot \partial_p J_j^q \partial_q J_l^t - J_i^p \cdot J_j^q \partial_p \partial_q J_l^t \\
&= J_l^p \cdot J_j J_p^k \cdot J_i J_k^l = J_k^l \cdot J_i J_r^k \cdot J_j J_l^t = J_k^l \cdot J_j J_l^r \cdot J_i J_r^k.
\end{aligned}$$

Therefore

$$a(3) = -J_k^l \cdot J_i J_l^r \cdot J_j J_r^k = J_k^l \cdot J_j J_l^r \cdot J_i J_r^k.$$

$$a(4) = J_j^q J_s^k \cdot J_i J_k^r \partial_r J_q^s = -J_q^s J_s^k \cdot J_i J_k^r \partial_r J_j^q = J_i J_k^r \partial_r J_j^k = J_i J_k^l \partial_l J_j^k,$$

and

$$c(3) = -J_j J_r^s \partial_s J_i^r = -J_j J_k^l \partial_l J_i^k.$$

Therefore we have

$$a(4) + c(3) = J_i J_k^l \partial_l J_j^k - J_j J_k^l \partial_l J_i^k = a(1) + b(3).$$
$$ 
\begin{aligned}
b(1) &= -J_i^p \cdot J_s J_p^r \cdot J_r J_j^s = J_s J_i^p \cdot J_p^r \cdot J_r J_j^s = J_s J_i^p \cdot J_p^r J_r^t \partial_t J_j^s = -J_s J_i^p \partial_p J_j^s \\
&= -J_k J_i^l \partial_l J_j^k,
\end{aligned}
$$

and

$$ 
\begin{aligned}
b(4) &= -J_j^q J_i^p \cdot J_s J_p^r \partial_r J_q^s = -J_i^p \cdot J_s J_p^r \cdot J_j^q \partial_r J_q^s = J_i^p \cdot J_s J_p^r \cdot J_q^s \partial_r J_j^q \\
&= J_i^p \cdot J_s^t \partial_t J_p^r \cdot J_q^s \partial_r J_j^q = -J_i^p \partial_q J_p^r \cdot \partial_r J_j^q = J_p^r \partial_q J_i^p \cdot \partial_r J_j^q = \partial_q J_i^p \cdot J_p J_j^q \\
&= \partial_k J_i^l \cdot J_l J_j^k = J_k J_j^l \partial_l J_i^k = c(1),
\end{aligned}
$$

that is,

$$ b(4) = c(1). $$

Therefore

$$ b(1) + b(4) = (b(1) + c(1)) = -J_k J_i^l \partial_l J_j^k + J_k J_j^l \partial_l J_i^k. $$

$$ 
\begin{aligned}
b(2) &= J_r^q J_i^p \cdot J_s J_p^r \partial_j J_q^s = -J_i^p J_q^s \cdot J_s J_p^r \partial_j J_r^q = -J_i^p J_q^s \cdot J_s^k \partial_k J_p^r \partial_j J_r^q \\
&= J_i^p \partial_k J_p^r \partial_j J_r^k = -J_p^r \partial_k J_i^p \partial_j J_r^k = J_r^k \partial_k J_i^p \partial_j J_p^r = J_r J_i^p \partial_j J_p^r \\
&= J_k J_i^l \partial_l J_j^k,
\end{aligned}
$$

and

$$ 
\begin{aligned}
d(4) &= -J_j^q \partial_r J_q^s \partial_i J_s^r = J_q^s \partial_r J_j^q \partial_i J_s^r = -J_s^r \partial_r J_j^q \cdot \partial_i J_q^s \\
&= -J_s J_j^q \partial_i J_q^s = -J_k J_j^l \partial_i J_l^k = -J_l J_j^k \partial_i J_l^l = -\partial_i J_k^l \cdot J_l J_j^k.
\end{aligned}
$$

Therefore we have

$$ b(2) + d(4) = J_k J_i^l \partial_l J_j^k - J_k J_j^l \partial_l J_i^k. $$

$$ c(1) = \partial_k J_i^l \cdot J_l J_j^k, $$

and

$$ c(4) = J_j^q \partial_k J_q^s \partial_s J_i^k = -J_q^s \partial_r J_j^q \partial_s J_i^r = -J_q J_i^r \partial_r J_j^q = -J_k J_i^l \partial_l J_j^k = b(1). $$

Therefore we have

$$ c(4) + c(1) = -J_k J_i^l \partial_l J_j^k + J_k J_j^l \partial_l J_i^k. $$

and

$$ 
\begin{aligned}
b(1) + b(4) &= c(4) + c(1) = -J_k J_i^l \partial_l J_j^k + J_k J_j^l \partial_l J_i^k \\
&= -J_k^p \partial_p J_i^l \partial_l J_j^k + J_k^p \partial_p J_j^l \partial_l J_i^k
\end{aligned}
$$

$$ 
\begin{aligned}
c(2) &= -J_r^q \partial_j J_q^s \cdot \partial_s J_i^r = \partial_j J_r^q \cdot J_q^s \partial_s J_i^r = J_q^s \partial_s J_i^r \partial_j J_r^q = J_q J_i^r \partial_j J_r^q \\
&= J_k J_i^l \partial_l J_j^k,
\end{aligned}
$$

and

$$ d(1) = -J_r J_j^s \partial_i J_s^r = -J_k J_j^l \partial_i J_l^k. $$
Therefore

$$c(2) + d(1) = J_k J_i^l \partial_j J_l^k - J_k J_j^l \partial_i J_l^k.$$

$$d(2) = J_r^q \partial_i J_s^r \partial_j J_q^s = J_k^l \partial_i J_r^k \partial_j J_l^k.$$

On the other hand

$$\begin{aligned} d(2) &= J_r^q \partial_i J_s^r \partial_j J_q^s \\ &= \partial_j (J_r^q \partial_i J_s^r \cdot J_q^s) - \partial_j J_r^q \partial_i J_s^r \cdot J_q^s - J_r^q \partial_j \partial_i J_s^r \cdot J_q^s \\ &= -\partial_j (\partial_i J_r^r) - \partial_j J_r^q \partial_i J_s^r \cdot J_q^s + \partial_j \partial_i J_r^r \\ &= -\partial_j J_r^q \partial_i J_s^r \cdot J_q^s = -J_k^l \partial_i J_l^k \partial_j J_r^k = -J_k^l \partial_j J_r^k \partial_i J_l^r \end{aligned}$$

Therefore

$$d(2) = J_k^l \partial_i J_r^k \partial_j J_l^r = -J_k^l \partial_j J_r^k \partial_i J_l^r.$$

In summary, we have the following equations

$$\begin{aligned} a(1) + b(3) &= a(4) + c(3) = J_i J_k^l \partial_l J_j^k - J_j J_k^l \partial_l J_i^k, \\ a(2) + d(3) &= -J_i J_k^l \partial_j J_l^k + J_j J_k^l \partial_i J_l^k, \\ a(3) &= -J_k^l \cdot J_i J_l^r \cdot J_j J_r^k = J_k^l \cdot J_j J_l^r \cdot J_i J_r^k = J_k^l \cdot J_i J_r^k \cdot J_j J_l^r, \\ a(4) + c(3) &= J_i J_k^l \partial_l J_j^k - J_j J_k^l \partial_l J_i^k = a(1) + b(3). \\ b(1) + b(4) &= -J_k J_l^l \partial_l J_j^k + J_k J_l^l \partial_l J_i^k, \\ b(2) + d(4) &= J_k J_l^l \partial_j J_l^k - J_k J_l^l \partial_i J_l^k, \\ c(1) &= b(4), \quad c(4) = b(1), \\ c(1) + c(4) &= -J_k J_i^l \partial_l J_j^k + J_k J_j^l \partial_l J_i^k = b(1) + b(4), \\ c(2) &= b(2), \quad d(1) = d(4), \\ c(2) + d(1) &= J_k J_i^l \partial_j J_l^k - J_k J_j^l \partial_i J_l^k = b(2) + d(4), \end{aligned}$$

and

$$d(2) = J_k^l \partial_i J_r^k \partial_j J_l^r = -J_k^l \partial_j J_r^k \partial_i J_l^r.$$

Note that all terms above are sew-symmetric on i and j. Therefore,

$$\begin{aligned} \ell_{ij} &= \{a(1)+b(3)\}+\{a(2)+d(3)\}+a(3)+\{a(4)+c(3)\}+\{b(1)+b(4)\}+\{b(2)+d(4)\} \\ &\quad +\{c(1)+c(4)\} + \{c(2)+d(1)\} + d(2) \\ &= 2\{a(4)+c(3)\}+\{a(2)+d(3)\}+a(3)+2\{b(1)+b(4)\}+2\{b(2)+d(4)\}+d(2) \\ &= a(3)+d(2)+\{a(2)+d(3)\}+2\{a(4)+c(3)\}+2\{b(1)+b(4)\}+2\{b(2)+d(4)\} \\ &= -J_k^l \cdot J_i J_l^r \cdot J_j J_r^k + J_k^l \partial_i J_r^k \partial_j J_l^r + \{-J_i J_k^l \partial_j J_l^k + J_j J_k^l \partial_i J_l^k\} \\ &\quad +2\{J_i J_k^l \partial_l J_j^k - J_j J_k^l \partial_l J_i^k\}+2\{-J_k J_i^l \partial_l J_j^k + J_k J_j^l \partial_l J_i^k\}+2\{J_k J_i^l \partial_j J_l^k - J_k J_j^l \partial_i J_l^k\}, \\ \text{and } \ell_{ij} \text{ is sew-symmetric on } i \text{ and } j. \end{aligned}$$

□
### 3. NONVANISHING OF $\ell_0$ FOR LEFT-MULTIPLICATION ON $S^6$

Let $\mathcal{O}$ be octonions. Every octonion $x$ is a real linear combination of the unit octonions

$$x = x_0 e_0 + x_1 e_1 + x_2 e_2 + x_3 e_3 + x_4 e_4 + x_5 e_5 + x_6 e_6 + x_7 e_7, \quad e_0 = 1.$$

The product of $x, y \in \mathcal{O}$ is denoted by $x \cdot y$. If $x$ and $y$ are perpendicular, then $\mathrm{Im}(x \cdot y) = x \cdot y$.

$$S^6 = \{x \in \mathrm{Im}(\mathcal{O}) \mid |X|^2 = 1\}.$$

Take a point $e_1 \in S^6$, and $y$ in tangent space of $S^6$: $Jv = e_1 \cdot v$ defines a linear transformation on the tangent space $T_{e_1}S^6$ with $J^2 = -1$. So it is an almost complex structure on $S^6$. We call it the Left-multiplication $J = J_0$. For convenience, we use $J$ for $J_0$ and $\ell$ for $\ell_0 = \ell_{J_0}$ (1.1), $N$ for $N_{J_0}$ in this section. There are more details in [16] and [7] and many other presentations about this $J_0$.

Consider $X, Y: S^6 \to \mathbb{R}^7$ as mappings, and $dX$ and $dY$ are differentials of mappings, Ch. [7]. Therefore

$$\begin{align*}
& [X, Y] = dY(X) - dX(Y) \\
d(JY)(JX) &= (JdY)(JX) + (dJY)(JX) = JdY(JX) + JX \cdot Y \\
d(JY)(X) &= (JdY)(X) + (dJY)(X) = JdY(X) + X \cdot Y \\
& [X, Y] = dY(X) - dX(Y)
\end{align*}$$

$$\begin{align*}
[JX, JY] &= d(JY)(JX) - d(JX)(JY) = JdY(JX) + JX \cdot Y - JdX(JY) - JY \cdot X \\
J[X, JY] &= Jd(JY)(X) - JdX(JY) = -dY(X) + J(X \cdot Y) - JdX(JY) \\
J[JX, Y] &= J(dY)(JX) - Jd(JX)(Y) = JdY(JX) + dX(Y) - J(Y \cdot X)
\end{align*}$$

$$\begin{align*}
N(X,Y) &= [JX, JY] - J[X, JY] - J[JX, Y] - [X, Y] \\
&= JdY(JX) + JX \cdot Y - JdX(JY) - JY \cdot X \\
&\quad +dY(X) - J(X \cdot Y) + JdX(JY) - JdY(JX) - dX(Y) + J(Y \cdot X) - [X, Y] \\
&= JX \cdot Y - JY \cdot X - J(X \cdot Y) + J(Y \cdot X)
\end{align*}$$

Let $N_1$ be Nijenhuis tensor at point $e_1 \in S^6$. We have the following.

$$\begin{align*}
N_1(2,3) &= (e_1 \cdot e_2) \cdot e_3 - (e_1 \cdot e_3) \cdot e_2 - e_1 \cdot (e_2 \cdot e_3) + e_1 \cdot (e_3 \cdot e_2) \\
&= e_4 \cdot e_3 - e_7 \cdot e_2 - e_1 \cdot e_5 + e_1 \cdot (-e_5) \\
&= e_4 \cdot e_3 - e_7 \cdot e_2 - 2e_1 \cdot e_5 \\
&= -e_6 - e_6 - 2e_6 = -4e_6.
\end{align*}
\[
N_1(2,4) = (e_1 \cdot e_2) \cdot e_4 - (e_1 \cdot e_4) \cdot e_2 - e_1 \cdot (e_2 \cdot e_4) + e_1 \cdot (e_4 \cdot e_2)
\]

$$N_1(2,4) = (e_1 \cdot e_2) \cdot e_4 - (e_1 \cdot e_4) \cdot e_2 - e_1 \cdot (e_2 \cdot e_4) + e_1 \cdot (e_4 \cdot e_2)$$
$$ 
\begin{aligned}
&= e_4 \cdot e_4 + e_2 \cdot e_2 - 2e_1 \cdot e_1. \\
&N_1(2, 4) = 0.
\end{aligned}
$$

$$ 
\begin{aligned}
N_1(2, 5) &= (e_1 \cdot e_2) \cdot e_5 - (e_1 \cdot e_5) \cdot e_2 - e_1 \cdot (e_2 \cdot e_5) + e_1 \cdot (e_5 \cdot e_2) \\
&= e_4 \cdot e_5 - e_6 \cdot e_2 + 2e_1 \cdot e_3 \\
&= e_7 + e_7 + 2e_7. \\
&N_1(2, 5) = 4e_7.
\end{aligned}
$$

$$ 
\begin{aligned}
N_1(2, 6) &= (e_1 \cdot e_2) \cdot e_6 - (e_1 \cdot e_6) \cdot e_2 - e_1 \cdot (e_2 \cdot e_6) + e_1 \cdot (e_6 \cdot e_2) \\
&= e_4 \cdot e_6 + e_5 \cdot e_2 - 2e_1 \cdot e_7 \\
&= e_3 + e_3 + 2e_3. \\
&N_1(2, 6) = 4e_3.
\end{aligned}
$$

$$ 
\begin{aligned}
N_1(2, 7) &= (e_1 \cdot e_2) \cdot e_7 - (e_1 \cdot e_7) \cdot e_2 - e_1 \cdot (e_2 \cdot e_7) + e_1 \cdot (e_7 \cdot e_2) \\
&= e_4 \cdot e_7 + e_3 \cdot e_2 + 2e_1 \cdot e_6 \\
&= -e_5 - e_5 - 2e_5. \\
&N_1(2, 7) = -4e_5.
\end{aligned}
$$

$$ 
\begin{aligned}
N_1(3, 4) &= (e_1 \cdot e_3) \cdot e_4 - (e_1 \cdot e_4) \cdot e_3 - e_1 \cdot (e_3 \cdot e_4) + e_1 \cdot (e_4 \cdot e_3) \\
&= e_7 \cdot e_4 - (-e_2) \cdot e_3 - 2e_1 \cdot e_6 \\
&= e_7 \cdot e_4 + e_2 \cdot e_3 - 2e_1 \cdot e_6 \\
&= e_5 + e_5 + 2e_5. \\
&N_1(3, 4) = 4e_5.
\end{aligned}
$$

$$ 
\begin{aligned}
N_1(3, 5) &= (e_1 \cdot e_3) \cdot e_5 - (e_1 \cdot e_5) \cdot e_3 - e_1 \cdot (e_3 \cdot e_5) + e_1 \cdot (e_5 \cdot e_3) \\
&= e_7 \cdot e_5 - e_6 \cdot e_3 - 2e_1 \cdot e_2 \\
&= -e_4 - e_4 - 2e_4. \\
&N_1(3, 5) = -4e_4.
\end{aligned}
$$

$$ 
\begin{aligned}
N_1(3, 6) &= (e_1 \cdot e_3) \cdot e_6 - (e_1 \cdot e_6) \cdot e_3 - e_1 \cdot (e_3 \cdot e_6) + e_1 \cdot (e_6 \cdot e_3) \\
&= e_7 \cdot e_6 + e_5 \cdot e_3 + 2e_1 \cdot e_4 \\
&= -e_2 - e_2 - 2e_2. \\
&N_1(3, 6) = -4e_2.
\end{aligned}
$$

$$ 
\begin{aligned}
N_1(3, 7) &= (e_1 \cdot e_3) \cdot e_7 - (e_1 \cdot e_7) \cdot e_3 - e_1 \cdot (e_3 \cdot e_7) + e_1 \cdot (e_7 \cdot e_3) \\
&= e_7 \cdot e_7 + e_3 \cdot e_3 - 2e_1 \cdot e_1.
\end{aligned}
$$
$$N_1(3, 7) = 0.$$

$$
\begin{aligned}
N_1(4, 5) &= (e_1 \cdot e_4) \cdot e_5 - (e_1 \cdot e_5) \cdot e_4 - e_1 \cdot (e_4 \cdot e_5) + e_1 \cdot (e_5 \cdot e_4) \\
&= -e_2 \cdot e_5 - e_6 \cdot e_4 - 2e_1 \cdot e_7 \\
&= e_3 + e_3 + 2e_3.
\end{aligned}
$$
$$N_1(4, 5) = 4e_3.$$

$$
\begin{aligned}
N_1(4, 6) &= (e_1 \cdot e_4) \cdot e_6 - (e_1 \cdot e_6) \cdot e_4 - e_1 \cdot (e_4 \cdot e_6) + e_1 \cdot (e_6 \cdot e_4) \\
&= -e_2 \cdot e_6 + e_5 \cdot e_4 - 2e_1 \cdot e_3 \\
&= -e_7 - e_7 - 2e_7.
\end{aligned}
$$
$$N_1(4, 6) = -4e_7.$$

$$
\begin{aligned}
N_1(4, 7) &= (e_1 \cdot e_4) \cdot e_7 - (e_1 \cdot e_7) \cdot e_4 - e_1 \cdot (e_4 \cdot e_7) + e_1 \cdot (e_7 \cdot e_4) \\
&= -e_2 \cdot e_7 + e_3 \cdot e_4 + 2e_6 \\
&= e_6 + e_6 + 2e_6.
\end{aligned}
$$
$$N_1(4, 7) = 4e_6.$$

$$
\begin{aligned}
N_1(5, 6) &= (e_1 \cdot e_5) \cdot e_6 - (e_1 \cdot e_6) \cdot e_5 - e_1 \cdot (e_5 \cdot e_6) + e_1 \cdot (e_6 \cdot e_5) \\
&= e_6 \cdot e_6 + e_5 \cdot e_5 - 2e_1 \cdot e_1 \\
&= -2 + 2.
\end{aligned}
$$
$$N_1(5, 6) = 0.$$

$$
\begin{aligned}
N_1(5, 7) &= (e_1 \cdot e_5) \cdot e_7 - (e_1 \cdot e_7) \cdot e_5 - e_1 \cdot (e_5 \cdot e_7) + e_1 \cdot (e_7 \cdot e_5) \\
&= e_2 + e_2 + 2e_2.
\end{aligned}
$$
$$N_1(5, 7) = 4e_2.$$

$$
\begin{aligned}
N_1(6, 7) &= (e_1 \cdot e_6) \cdot e_7 - (e_1 \cdot e_7) \cdot e_6 - e_1 \cdot (e_6 \cdot e_7) + e_1 \cdot (e_7 \cdot e_6) \\
&= -e_5 \cdot e_7 + e_3 \cdot e_6 - 2e_1 \cdot e_2 \\
&= -e_4 - e_4 - 2e_4.
\end{aligned}
$$
$$N_1(6, 7) = -4e_4.$$

In summary we have

$$
\begin{aligned}
N_1(2, 3) &= -4e_6, N_1(2, 4) = 0, N_1(2, 5) = 4e_7, N_1(2, 6) = 4e_3, N_1(2, 7) = -4e_5, \\
N_1(3, 4) &= 4e_5, N_1(3, 5) = -4e_4, N_1(3, 6) = -4e_2, N_1(3, 7) = 0, \\
N_1(4, 5) &= 4e_3, N_1(4, 6) = -4e_7, N_1(4, 7) = 4e_6, \\
N_1(5, 6) &= 0, N_1(5, 7) = 4e_2,
\end{aligned}
$$
$$N_1(6, 7) = -4e_4,$$

Let $e_1 = f_7, e_2 = f_1, e_3 = f_2, e_4 = f_3, e_5 = f_4, e_6 = f_5, e_7 = f_6$. In the following we denote $N(i, j) := N_1(f_i, f_j)$ for $i, j = 1, 2, 3, 4, 5, 6$. At $f_7 = e_1 \in S^6$, we have

$$ 
\begin{aligned}
& N(1, 2) = -4f_5, N(1, 3) = 0, N(1, 4) = 4f_6, N(1, 5) = 4f_2, N_1(1, 6) = -4f_4, \\
& N(2, 3) = 4f_4, N(2, 4) = -4f_3, N(2, 5) = -4f_1, N(2, 6) = 0, \\
& N(3, 4) = 4f_2, N(3, 5) = -4f_6, N(3, 6) = 4f_5, \\
& N(4, 5) = 0, N(4, 6) = 4f_1, \\
& N(5, 6) = -4f_3.
\end{aligned}
$$

From matrix

$$ (3.1) \qquad \frac{1}{4}[N(f_i, f_j)] = \begin{bmatrix} 0 & -f_5 & 0 & f_6 & f_2 & -f_4 \\ f_5 & 0 & f_4 & -f_3 & -f_1 & 0 \\ 0 & -f_4 & 0 & f_2 & -f_6 & f_5 \\ -f_6 & f_3 & -f_2 & 0 & 0 & f_1 \\ -f_2 & f_1 & f_6 & 0 & 0 & -f_3 \\ f_4 & 0 & -f_5 & -f_1 & f_3 & 0 \end{bmatrix} $$

we have

$$ 
\begin{aligned}
\frac{1}{4}N_{11}^r &= 0, \quad \frac{1}{4}N_{12}^r = -\delta_5^r, \quad \frac{1}{4}N_{13}^r = 0, \quad \frac{1}{4}N_{14}^r = \delta_6^r, \quad \frac{1}{4}N_{15}^r = \delta_2^r, \quad \frac{1}{4}N_{16}^r = -\delta_4^r, \\
\frac{1}{4}N_{21}^r &= \delta_5^r, \quad \frac{1}{4}N_{22}^r = 0, \quad \frac{1}{4}N_{23}^r = \delta_4^r, \quad \frac{1}{4}N_{24}^r = -\delta_3^r, \quad \frac{1}{4}N_{25}^r = -\delta_1^r, \quad \frac{1}{4}N_{26}^r = 0, \\
\frac{1}{4}N_{31}^r &= 0, \quad \frac{1}{4}N_{32}^r = -\delta_4^r, \quad \frac{1}{4}N_{33}^r = 0, \quad \frac{1}{4}N_{34}^r = \delta_2^r, \quad \frac{1}{4}N_{35}^r = -\delta_6^r, \quad \frac{1}{4}N_{36}^r = \delta_5^r, \\
\frac{1}{4}N_{41}^r &= -\delta_6^r, \quad \frac{1}{4}N_{42}^r = \delta_3^r, \quad \frac{1}{4}N_{43}^r = -\delta_2^r, \quad \frac{1}{4}N_{44}^r = 0, \quad \frac{1}{4}N_{45}^r = 0, \quad \frac{1}{4}N_{46}^r = \delta_1^r, \\
\frac{1}{4}N_{51}^r &= -\delta_2^r, \quad \frac{1}{4}N_{52}^r = \delta_1^r, \quad \frac{1}{4}N_{53}^r = \delta_6^r, \quad \frac{1}{4}N_{54}^r = 0, \quad \frac{1}{4}N_{55}^r = 0, \quad \frac{1}{4}N_{56}^r = -\delta_3^r, \\
\frac{1}{4}N_{61}^r &= \delta_4^r, \quad \frac{1}{4}N_{62}^r = 0, \quad \frac{1}{4}N_{63}^r = -\delta_5^r, \quad \frac{1}{4}N_{64}^r = -\delta_1^r, \quad \frac{1}{4}N_{65}^r = \delta_3^r, \quad \frac{1}{4}N_{66}^r = 0.
\end{aligned}
$$

Similarly we may calculate $J_i^j$ at point $f_7 = e_1$. Note $J : T_{f_7}S^6 \to T_{f_7}S^6$.

$$ [J_i^j] = \begin{bmatrix} 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ -1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -1 & 0 & 0 \\ 0 & -1 & 0 & 0 & 0 & 0 \end{bmatrix} $$

$$ J_1^3 = J_2^6 = J_4^5 = 1, J_3^1 = J_5^4 = J_6^2 = -1, $$

other $J_i^j = 0$. Use these we have

$$ -\ell_{ij} = -N_{ik}^r N_{rj}^s J_s^k = N_{ik}^r N_{jr}^s J_s^k $$
$$
\begin{align*}
&= N_{ik}^r N_{jr}^1 J_1^k + N_{ik}^r N_{jr}^2 J_2^k + N_{ik}^r N_{jr}^3 J_3^k + N_{ik}^r N_{jr}^4 J_4^k + N_{ik}^r N_{jr}^5 J_5^k + N_{ik}^r N_{jr}^6 J_6^k \\
&= N_{i3}^r N_{jr}^1 + N_{i6}^r N_{jr}^2 - N_{i1}^r N_{jr}^3 + N_{i5}^r N_{jr}^4 - N_{i4}^r N_{jr}^5 - N_{i2}^r N_{jr}^6 \\
&= N_{i3}^1 N_{j1}^1 + N_{i6}^1 N_{j1}^2 - N_{i1}^1 N_{j1}^3 + N_{i5}^1 N_{j1}^4 - N_{i4}^1 N_{j1}^5 - N_{i2}^1 N_{j1}^6 \\
&\quad + N_{i3}^2 N_{j2}^1 + N_{i6}^2 N_{j2}^2 - N_{i1}^2 N_{j2}^3 + N_{i5}^2 N_{j2}^4 - N_{i4}^2 N_{j2}^5 - N_{i2}^2 N_{j2}^6 \\
&\quad + N_{i3}^3 N_{j3}^1 + N_{i6}^3 N_{j3}^2 - N_{i1}^3 N_{j3}^3 + N_{i5}^3 N_{j3}^4 - N_{i4}^3 N_{j3}^5 - N_{i2}^3 N_{j3}^6 \\
&\quad + N_{i3}^4 N_{j4}^1 + N_{i6}^4 N_{j4}^2 - N_{i1}^4 N_{j4}^3 + N_{i5}^4 N_{j4}^4 - N_{i4}^4 N_{j4}^5 - N_{i2}^4 N_{j4}^6 \\
&\quad + N_{i3}^5 N_{j5}^1 + N_{i6}^5 N_{j5}^2 - N_{i1}^5 N_{j5}^3 + N_{i5}^5 N_{j5}^4 - N_{i4}^5 N_{j5}^5 - N_{i2}^5 N_{j5}^6 \\
&\quad + N_{i3}^6 N_{j6}^1 + N_{i6}^6 N_{j6}^2 - N_{i1}^6 N_{j6}^3 + N_{i5}^6 N_{j6}^4 - N_{i4}^6 N_{j6}^5 - N_{i2}^6 N_{j6}^6 \\
&= 0 + N_{i6}^1 N_{j1}^2 + 0 + N_{i5}^1 N_{j1}^4 - N_{i4}^1 N_{j1}^5 - N_{i2}^1 N_{j1}^6 \\
&\quad + N_{i3}^2 N_{j2}^1 + 0 - N_{i1}^2 N_{j2}^3 + N_{i5}^2 N_{j2}^4 - N_{i4}^2 N_{j2}^5 - 0 \\
&\quad + 0 + N_{i6}^3 N_{j3}^2 - 0 + N_{i5}^3 N_{j3}^4 - N_{i4}^3 N_{j3}^5 - N_{i2}^3 N_{j3}^6 \\
&\quad + N_{i3}^4 N_{j4}^1 + N_{i6}^4 N_{j4}^2 - N_{i1}^4 N_{j4}^3 + 0 - 0 - N_{i2}^4 N_{j4}^6 \\
&\quad + N_{i3}^5 N_{j5}^1 + N_{i6}^5 N_{j5}^2 - N_{i1}^5 N_{j5}^3 + 0 - 0 - N_{i2}^5 N_{j5}^6 \\
&\quad + N_{i3}^6 N_{j6}^1 + 0 - N_{i1}^6 N_{j6}^3 + N_{i5}^6 N_{j6}^4 - N_{i4}^6 N_{j6}^5 - 0
\end{align*}
$$

Now take $i=4, j=5$. Use values of $N_{ij}^k$ listed after (3.1), we have

$$
\begin{align*}
& -\ell_{45} \\
&= 0 + N_{46}^1 N_{51}^2 + 0 + N_{45}^1 N_{51}^4 - N_{44}^1 N_{51}^5 - N_{42}^1 N_{51}^6 \\
&\quad + N_{43}^2 N_{52}^1 + 0 - N_{41}^2 N_{52}^3 + N_{45}^2 N_{52}^4 - N_{44}^2 N_{52}^5 - 0 \\
&\quad + 0 + N_{46}^3 N_{53}^2 - 0 + N_{45}^3 N_{53}^4 - N_{44}^3 N_{53}^5 - N_{42}^3 N_{53}^6 \\
&\quad + N_{43}^4 N_{54}^1 + N_{46}^4 N_{54}^2 - N_{41}^4 N_{54}^3 + 0 - 0 - N_{42}^4 N_{54}^6 \\
&\quad + N_{43}^5 N_{55}^1 + N_{46}^5 N_{55}^2 - N_{41}^5 N_{55}^3 + 0 - 0 - N_{42}^5 N_{55}^6 \\
&\quad + N_{43}^6 N_{56}^1 + 0 - N_{41}^6 N_{56}^3 + N_{45}^6 N_{56}^4 - N_{44}^6 N_{56}^5 - 0 \\
&= 0 - 16 + 0 + 0 - 0 - 0 \\
&\quad - 16 + 0 - 0 + 0 - 0 - 0 \\
&\quad + 0 + 0 - 0 + 0 - 0 - 16 \\
&\quad + 0 + 0 - 0 + 0 - 0 - 0 \\
&\quad + 0 + 0 - 0 + 0 - 0 - 0 \\
&\quad + 0 + 0 - 16 + 0 - 0 - 0 \\
&= -64
\end{align*}
$$

Therefore $\ell_{45} = 64 \neq 0$.

□
## 4. $S^6$ HOPF PROBLEM

*Proof of Theorem 1.4.* Let $J$ be any almost complex structure on $S^6$, $\ell = \ell_J$ the 2-form of $J$ defined by (1.1). We show $\ell \neq 0$. This implies Nijenhuis tensor $N_J \neq 0$. By Newlander-Nirenberg Theorem [20], $J$ is not integrable.

Take local coordinates $\{x^i\}_{i=1}^6$ at $x \in S^6$.

(i) If $d\ell \neq 0$, then $\ell \neq 0$.

(ii) Suppose $d\ell \equiv 0$. Since $H_{\text{DeRham}}^2(S^6) = 0$, $\ell$ is exact. So there exist 1-form $\eta$ such that

$$ (4.1) \qquad \ell = d\eta, \quad \text{where} \quad \eta = \sum_{j=1}^{6} \eta_j dx^j. $$

We show $\ell = d\eta \neq 0$.

If othersie

$$ (4.2) \qquad \ell = d\eta \equiv 0, $$

We look closely at $\ell_{ij}$ and rewrite its (2.1) with highlights on terms containing $\frac{\partial}{\partial x^i} = \partial_i = \partial_i$ and $\frac{\partial}{\partial x^j} = \partial_j = \partial_j$, grouped inside $\{\}$, with $n=6$:

$$ (4.3) \qquad \begin{aligned} \ell_{ij} = & \sum_{k,l,r,p,q=1}^{n} -J_k^l J_i^p J_j^q \partial_p J_l^r \partial_q J_r^k + \sum_{k,l,r=1}^{n} \{(J_k^l) \cdot \partial_i J_r^k \cdot \partial_j J_l^r\} \\ & + \sum_{k,l,r=1}^{n} \{(-J_i^r \partial_r J_k^l) \cdot \partial_j J_l^k + (J_j^r \partial_r J_k^l) \cdot \partial_i J_l^k\} + \sum_{k,l,r=1}^{n} 2(J_i^r \partial_r J_k^l \partial_l J_j^k - J_j^r \partial_r J_k^l \partial_l J_i^k) \\ & + \sum_{k,l,r=1}^{n} 2(-J_k^r \partial_r J_i^l \partial_l J_j^k + J_k^r \partial_r J_j^l \partial_l J_i^k) + \sum_{k,l,r=1}^{n} 2\{(J_k^r \partial_r J_i^l) \cdot \partial_j J_l^k - (J_k^r \partial_r J_j^l) \cdot \partial_i J_l^k\}. \end{aligned} $$

$H_{\text{DeRham}}^3(S^6) = 0$ and Lemma 4.2 imply that the first term in the right hand side of the above equation vanishes.

(4.1) implies that terms $\ell_{ij}$ should come as partial derivatives with respect to $x^i$ or partial derivatives with respect to $x^j$. So in (4.3), terms other than ones with $\frac{\partial}{\partial x^i} = \partial_i = \partial_i$ and $\frac{\partial}{\partial x^j} = \partial_j = \partial_j$ (inside $\{\}$) vanish. Therefore we have

$$ \frac{1}{2}(\partial_i \eta^j - \partial_j \eta_i) = \ell_{ij} = \sum_{k,l,r=1}^{6} \{(J_k^l) \cdot \partial_i J_r^k \cdot \partial_j J_l^r\} + \sum_{k,l,r=1}^{6} \{(-J_i^r \partial_r J_k^l) \cdot \partial_j J_l^k + (J_j^r \partial_r J_k^l) \cdot \partial_i J_l^k\} \\ + \sum_{k,l,r=1}^{6} 2\{(J_k^r \partial_r J_i^l) \cdot \partial_j J_l^k - (J_k^r \partial_r J_j^l) \cdot \partial_i J_l^k\}. $$
Since $H^1_{\text{DeRham}}(S^6) = 0$, we have $\eta = df$ for some smooth function on $S^6$, so $d\eta$ expression (4.4) consists of terms of containing both partial derivatives with respect to $x^i$ and partial derivatives with respect to $x^j$, so (4.4) becomes

$$ (4.5) \qquad \ell_{ij} = \sum_{p,q,r=1}^{6} \partial_i J_p^r J_r^q \partial_j J_q^p =: \xi_{ij}. $$

Let

$$ (4.6) \qquad \xi = \sum_{i,j=1}^{6} \xi_{ij} dx^i \wedge dx^j = \sum_{i,j=1}^{6} \left( \sum_{p,q,r=1}^{6} \partial_i J_p^r J_r^q \partial_j J_q^p \right) dx^i \wedge dx^j $$

(4.2) –(4.6) imply $\xi = \ell = d\eta \equiv 0$. This contradicts Lemma 4.1. Therefore $\ell = d\eta \neq 0$. $\square$

**Lemma 4.1.** Let $\{x^i\}_{i=1}^6$ be local coordinates of $S^6$ at $x$, $J$ an almost complex structure on $S^6$. $J(\partial_i) = \sum_{j=1}^6 J_i^j \partial_j$, $\partial_i = \frac{\partial}{\partial x^i}$ for $i, j = 1, \dots, 6$. $\xi$ is defined by (4.6). Then $\xi \neq 0$ on $S^6$.

*Proof.* It is easy to check $\xi_{ij} = -\xi_{ji}$ and $\xi_{ij} = \text{tr}\{\partial_i J J \partial_j J\}$, for $i, j = 1, \dots, 6$.

We prove $\xi \neq 0$. At the local coordinates at $x \in S^6$. Let $B$ be the Jordan matrix of $J$ and $P(x)$ is continuous (therefore smooth) at the local coordinates with $P^{-1}JP = B$. Note the eigenvalues of $J$ are constants $\pm\sqrt{-1}$ in pairs.

$$
\begin{align*}
\xi_{ij} &= \text{tr}\{\partial_i J J \partial_j J\} = \text{tr}\{\partial_i J P B P^{-1} \partial_j J\} \\
&= \text{tr}\{\partial_i (P B P^{-1}) P B P^{-1} \partial_j (P B P^{-1})\} \\
&= \text{tr}\{(\partial_i P B P^{-1} + P B \partial_i P^{-1}) P B P^{-1} (\partial_j P B P^{-1} + P B \partial_j P^{-1})\} \\
&= \text{tr}\{\partial_i P B P^{-1} P B P^{-1} \partial_j P B P^{-1}\} + \text{tr}\{\partial_i P B P^{-1} P B P^{-1} P B \partial_j P^{-1}\} \\
&\quad + \text{tr}\{P B \partial_i P^{-1} P B P^{-1} \partial_j P B P^{-1}\} + \text{tr}\{P B \partial_i P^{-1} P B P^{-1} P B \partial_j P^{-1}\} \\
&= -\text{tr}\{\partial_i P P^{-1} \partial_j P B P^{-1}\} - \text{tr}\{\partial_i P B \partial_j P^{-1}\} \\
&\quad + \text{tr}\{B \partial_i P^{-1} P B P^{-1} \partial_j P B\} - \text{tr}\{P B \partial_i P^{-1} P \partial_j P^{-1}\} \\
&= \text{tr}\{P \partial_i P^{-1} \partial_j P B P^{-1}\} - \text{tr}\{\partial_i P B \partial_j P^{-1}\} \\
&\quad - \text{tr}\{B^{-1} \partial_i P^{-1} P B P^{-1} \partial_j P B\} + \text{tr}\{P B \partial_i P^{-1} \partial_j P P^{-1}\} \\
&= \text{tr}\{\partial_i P^{-1} \partial_j P B\} - \text{tr}\{\partial_i P B \partial_j P^{-1}\} \\
&\quad - \text{tr}\{P^{-1} \partial_i P B \partial_j P^{-1} P\} + \text{tr}\{B \partial_i P^{-1} \partial_j P\} \\
&= \text{tr}\{B \partial_i P^{-1} \partial_j P\} - \text{tr}\{\partial_i P B \partial_j P^{-1}\} \\
&\quad - \text{tr}\{\partial_i P B \partial_j P^{-1}\} + \text{tr}\{B \partial_i P^{-1} \partial_j P\} \\
&= 2\text{tr}\{B \partial_i P^{-1} \partial_j P\} - 2\text{tr}\{\partial_i P B \partial_j P^{-1}\} \\
&= 2\text{tr}\{B \partial_i P^{-1} \partial_j P\} - 2\text{tr}\{B \partial_j P^{-1} \partial_i P\} \\
&= 2\text{tr}\{B(\partial_i P^{-1} \partial_j P - \partial_j P^{-1} \partial_i P)\} \\
&\neq 0,
\end{align*}
$$
since $\partial_i P^{-1} \partial_j P$ and $\partial_j P^{-1} \partial_i P$ do not commute

$$ \partial_i P^{-1} \partial_j P - \partial_j P^{-1} \partial_i P \neq 0 $$

on $S^6$ for $i \neq j, i, j = 1, \dots, 6$ at all $x \in S^6$, and $B$ is Jordan matrix with main diagonal entries $\pm\sqrt{-1}$ at $x \in S^6$. $\square$

**Lemma 4.2.** On $S^6$ for any $k = 1, 2, 3, 4, 5, 6$, if $d\ell \equiv 0$ then the first term of the right hand side of equation (4.3)

$$ \sum_{k,l,r,p,q=1}^{n} -J_{k}^{l}J_{i}^{p}J_{j}^{q}\partial_{p}J_{l}^{r}\partial_{q}J_{r}^{k} $$

vanishes.

*Proof.* We take $k = 6$. The other cases are similar. $\ell \wedge dx^6$ is a 3-form. $d(\ell \wedge dx^6) = (d\ell) \wedge dx^6 \equiv 0$. Since $H_{\text{DeRham}}^3(S^6) = 0$, $\ell \wedge dx^6 = d\phi$ for some 2-form $\phi$. Note that

$$ \ell \wedge dx^6 = \sum_{i,j=1}^{5} \ell_{ij} dx^i \wedge dx^j \wedge dx^6. $$

Suppose

$$ \phi = \sum_{i,j=1}^{6} \phi_{ij} dx^i \wedge dx^j. $$

Then

$$ \begin{aligned} d\phi &= \sum_{i,j,k=1}^{6} \partial_k \phi_{ij} dx^k \wedge dx^i \wedge dx^j \\ &= \sum_{j,k=1}^{5} \partial_k \phi_{6j} dx^k \wedge dx^6 \wedge dx^j + \sum_{i,k=1}^{5} \partial_k \phi_{i6} dx^k \wedge dx^i \wedge dx^6 \\ &\quad + \sum_{i,j=1}^{5} \partial_6 \phi_{ij} dx^6 \wedge dx^i \wedge dx^j + \sum_{i,j,k=1}^{5} \partial_k \phi_{ij} dx^k \wedge dx^i \wedge dx^j \\ &= \sum_{i,j=1}^{5} (-\partial_i \phi_{6j} - \partial_j \phi_{i6} + \partial_6 \phi_{ij}) dx^i \wedge dx^j \wedge dx^6 + \sum_{i,j,k=1}^{5} \partial_k \phi_{ij} dx^k \wedge dx^i \wedge dx^j \\ \ell \wedge dx^6 &= d\phi \text{ yields} \\ \sum_{i,j=1}^{5} \ell_{ij} dx^i \wedge dx^j \wedge dx^6 &= \sum_{i,j=1}^{5} (-\partial_i \phi_{6j} - \partial_j \phi_{i6} + \partial_6 \phi_{ij}) dx^i \wedge dx^j \wedge dx^6, \end{aligned} $$

$$ \sum_{i,j=1}^{5} \ell_{ij} dx^i \wedge dx^j \wedge dx^6 = \sum_{i,j=1}^{5} (-\partial_i \phi_{6j} - \partial_j \phi_{i6} + \partial_6 \phi_{ij}) dx^i \wedge dx^j \wedge dx^6, $$

$$ \ell_{ij} = -\partial_i \phi_{6j} - \partial_j \phi_{i6} + \partial_6 \phi_{ij}, \quad \text{for } i, j = 1, 2, 3, 4, 5. $$

Therefore $\ell_{ij}$ consists of terms with factors of partial derivatives with respect to $x^i$, $\partial_i$ or with respect to $x^j$, $\partial_j$;

or terms with factors of partial derivatives of component functions with
index *i* or *j*, with respect to $x^6$, $\partial_6$. Similar argument to the above *k* = 6 applies for *k* = 1, 2, 3, 4, 5. Therefore

Therefore $\ell_{ij}$ consists of terms with factors of partial derivatives with respect to $x^i$, $\partial_i$ or with respect to $x^j$, $\partial_j$ or terms with factors of partial derivatives of component functions with index *i* or *j*, with respect to $x^k$, $\partial_k$ for $k \neq i, k \neq j$, in, $k = 1, 2, 3, 4, 5, 6$. Similar argument to the above $k = 6$ applies for $k = 1, 2, 3, 4, 5$.

Therefore the first term of in the right hand side of equation (4.3) vanishes. □

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DEPARTMENT OF MATHEMATICS, UTAH VALLEY UNIVERSITY, OREM, UTAH 84058, USA

Email address: lingju@uvu.edu