# Hopf Problem of Six-Sphere

Jun Ling

► To cite this version:

Jun Ling. Hopf Problem of Six-Sphere. 2026. hal-05464059

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# HOPF PROBLEM OF SIX-SPHERE

JUN LING

**ABSTRACT.** We solve Hopf problem of $S^6$ and prove that $S^6$ is not a complex manifold via an analysis approach, that is, we prove all almost complex structures on $S^6$ are not integrable by Newlander-Nirenberg Theorem. Therefore $S^6$ is not a complex manifold. Arguments and calculations in the proof are of algebra, differential topology, differential geometry, and analysis, based on the underlying topological structures of $S^6$. Namely, we construct a differential 2-form for any almost complex structure on any smooth manifold, and we reduce the 2-form for any almost complex structure on $S^6$ by triviality of three cohomology groups $H^1, H^2, H^3$ of $S^6$. The arguments and calculations in the proof are algebraic, geometric and analytic. The differential 2-form is not trivial nor empty, as it gives a new proof for non-integrability of classic almost complex structure $J_0$ on $S^6$. It can be applied for the study of integrability of other smooth manifolds with trivial $H^1, H^2, H^3$. It suggests a relation between the integrability of the almost complex structures and the three cohomology groups $H^1, H^2, H^3$ of the underlying manifold $M^n$ ($2 < n \le 6$), as we see from examples of complex manifolds $M = \mathbb{C}P^3$, $\bar{M} = S^3 \times S^3$, $M = \mathbb{C}P^2$ and $\bar{M} = S^1 \times S^3$.

## 1. INTRODUCTION

In this paper, we solve Hopf problem of $S^6$ and prove that $S^6$ is not a complex manifold via an analysis approach, that is, we prove all almost complex structures on $S^6$ are not integrable by Newlander-Nirenberg Theorem. Therefore $S^6$ is not a complex manifold. Arguments and calculations in the proof are of algebra, differential topology, differential geometry, and analysis, based on the underlying topological structures of $S^6$. Namely, we construct a differential 2-form for any almost complex structure on any smooth manifold, and we reduce the 2-form for any almost complex structure on $S^6$ by triviality of three cohomology

<footnote>Date: November 10, 2025.</footnote>

<footnote n="2010">2010 Mathematics Subject Classification. Primary 53C15; Secondary 32Q60.</footnote>

<footnote>Key words and phrases. Hopf Problem, $S^6$, almost complex structures.</footnote>

<footnote>Thank Princeton University and Cornell University for their hospitality.</footnote>
groups $H^1, H^2, H^3$ of $S^6$. The arguments and calculations in the proof are algebraic, geometric and analytic.

The problem of whether a smooth manifold is a complex manifold or not is interesting and important, it has a long history. If a manifold is a complex manifold then it has a canonical almost complex structure. If a smooth manifold has an integrable almost complex structure then the smooth manifold is a complex manifold. One analysis approach to this problem is to study integrability of almost complex structures of the manifolds.

Integrability of almost complex structure on smooth manifold, in particular on high dimensional spheres has been studied extensively. 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], Ehresmann 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.

In this paper we first construct a differential 2-form for any almost complex structure on any 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 $M = \mathbb{CP}^3$, $M = S^3 \times S^3$, $M = \mathbb{CP}^2$ 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.

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 Hopf problem of $S^6$ 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 $\partial_i = \frac{\partial}{\partial x^i}$, $J = \frac{\partial}{\partial x^i} = \sum_{j=1}^n J_j^j \frac{\partial}{\partial x^j}$, $J_i = \sum_{p=1}^n J_i^p \frac{\partial}{\partial x^p}$, 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

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

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 $i, j = 1, 2, \dots, n$,

$$\begin{align} \ell_{ij} = & \sum_{k,l,r,p,q=1}^{n} -J_k^l J_i^p \partial_p J_l^r J_j^q \partial_q J_r^k \nonumber \\ & + \sum_{k,l,r=1}^{n} J_k^l \partial_i J_r^k \partial_j J_l^r \nonumber \\ & + \sum_{k,l,r=1}^{n} \left\{ -J_i^r \partial_r J_k^l \partial_j J_l^k + J_j^r \partial_r J_k^l \partial_i J_l^k \right\} \tag{2.1} \\ & + \sum_{k,l,r=1}^{n} 2 \left\{ J_i^r \partial_r J_k^l \partial_l J_j^k - J_j^r \partial_r J_k^l \partial_l J_i^k \right\} \nonumber \\ & + \sum_{k,l,r=1}^{n} 2 \left\{ -J_k^r \partial_r J_i^l \partial_l J_j^k + J_k^r \partial_r J_j^l \partial_l J_i^k \right\} \nonumber \\ & + \sum_{k,l,r=1}^{n} 2 \left\{ J_k^r \partial_r J_i^l \partial_j J_l^k - J_k^r \partial_r J_j^l \partial_i J_l^k \right\}, \nonumber \end{align}$$

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

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

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

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

Therefore

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

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

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,

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

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

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

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

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

that is,

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

Therefore

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

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

and

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

Therefore we have

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

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

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

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

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

and

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

Therefore

$$c(2) + d(1) = J_k J_i^l \partial_j J_l^k - J_k J_j^l \partial_i J_l^k.$$
$$d(2) = J_r^q \partial_i J_s^r \partial_j J_q^s = J_k^l \partial_i J_r^k \partial_j J_l^r.$$

On the other hand

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

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

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

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

$$x = x_0e_0 + x_1e_1 + x_2e_2 + x_3e_3 + x_4e_4 + x_5e_5 + x_6e_6 + x_7e_7, \quad e_0 = 1.$$

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

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

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.

$$ \begin{align*} N_1(2, 3) &= (e_1 \cdot e_2) \cdot e_3 - (e_1 \cdot e_3) \cdot e_2 - e_1 \cdot (e_2 \cdot e_3) + e_1 \cdot (e_3 \cdot e_2) \\ &= e_4 \cdot e_3 - e_7 \cdot e_2 - e_1 \cdot e_5 + e_1 \cdot (-e_5) \\ &= e_4 \cdot e_3 - e_7 \cdot e_2 - 2e_1 \cdot e_5 \\ &= -e_6 - e_6 - 2e_6 = -4e_6. \\ N_1(2, 3) &= -4e_6. \end{align*} $$
$$
\begin{align*}
N_1(2, 4) &= (e_1 \cdot e_2) \cdot e_4 - (e_1 \cdot e_4) \cdot e_2 - e_1 \cdot (e_2 \cdot e_4) + e_1 \cdot (e_4 \cdot e_2) \\
&= e_4 \cdot e_4 + e_2 \cdot e_2 - 2e_1 \cdot e_1. \\
N_1(2, 4) &= 0.
\end{align*}
$$

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

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

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

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

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

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

$$N_1(3, 6) = -4e_2.$$

$$N_1(3, 7) = (e_1 \cdot e_3) \cdot e_7 - (e_1 \cdot e_7) \cdot e_3 - e_1 \cdot (e_3 \cdot e_7) + e_1 \cdot (e_7 \cdot e_3)$$
$$= e_7 \cdot e_7 + e_3 \cdot e_3 - 2e_1 \cdot e_1.$$

$$N_1(3, 7) = 0.$$

$$N_1(4, 5) = (e_1 \cdot e_4) \cdot e_5 - (e_1 \cdot e_5) \cdot e_4 - e_1 \cdot (e_4 \cdot e_5) + e_1 \cdot (e_5 \cdot e_4)$$
$$= -e_2 \cdot e_5 - e_6 \cdot e_4 - 2e_1 \cdot e_7$$
$$= e_3 + e_3 + 2e_3.$$

$$N_1(4, 5) = 4e_3.$$

$$N_1(4, 6) = (e_1 \cdot e_4) \cdot e_6 - (e_1 \cdot e_6) \cdot e_4 - e_1 \cdot (e_4 \cdot e_6) + e_1 \cdot (e_6 \cdot e_4)$$
$$= -e_2 \cdot e_6 + e_5 \cdot e_4 - 2e_1 \cdot e_3$$
$$= -e_7 - e_7 - 2e_7.$$

$$N_1(4, 6) = -4e_7.$$

$$N_1(4, 7) = (e_1 \cdot e_4) \cdot e_7 - (e_1 \cdot e_7) \cdot e_4 - e_1 \cdot (e_4 \cdot e_7) + e_1 \cdot (e_7 \cdot e_4)$$
$$= -e_2 \cdot e_7 + e_3 \cdot e_4 + 2e_6$$
$$= e_6 + e_6 + 2e_6.$$

$$N_1(4, 7) = 4e_6.$$

$$N_1(5, 6) = (e_1 \cdot e_5) \cdot e_6 - (e_1 \cdot e_6) \cdot e_5 - e_1 \cdot (e_5 \cdot e_6) + e_1 \cdot (e_6 \cdot e_5)$$
$$= e_6 \cdot e_6 + e_5 \cdot e_5 - 2e_1 \cdot e_1$$
$$= -2 + 2.$$

$$N_1(5, 6) = 0.$$

$$N_1(5, 7) = (e_1 \cdot e_5) \cdot e_7 - (e_1 \cdot e_7) \cdot e_5 - e_1 \cdot (e_5 \cdot e_7) + e_1 \cdot (e_7 \cdot e_5)$$
$$= e_2 + e_2 + 2e_2.$$

$$N_1(5, 7) = 4e_2.$$

$$N_1(6, 7) = (e_1 \cdot e_6) \cdot e_7 - (e_1 \cdot e_7) \cdot e_6 - e_1 \cdot (e_6 \cdot e_7) + e_1 \cdot (e_7 \cdot e_6)$$
$$= -e_5 \cdot e_7 + e_3 \cdot e_6 - 2e_1 \cdot e_2$$
$$= -e_4 - e_4 - 2e_4.$$

$$N_1(6, 7) = -4e_4.$$

In summary we have

$$N_1(2, 3) = -4e_6, N_1(2, 4) = 0, N_1(2, 5) = 4e_7, N_1(2, 6) = 4e_3, N_1(2, 7) = -4e_5,$$

$$N_1(3, 4) = 4e_5, N_1(3, 5) = -4e_4, N_1(3, 6) = -4e_2, N_1(3, 7) = 0,$$

$$N_1(4, 5) = 4e_3, N_1(4, 6) = -4e_7, N_1(4, 7) = 4e_6,$$

$$N_1(5, 6) = 0, N_1(5, 7) = 4e_2,$$

$$N_1(6, 7) = -4e_4.$$

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

$$N(1, 2) = -4f_5, N(1, 3) = 0, N(1, 4) = 4f_6, N(1, 5) = 4f_2, N_1(1, 6) = -4f_4,$$

$$N(2, 3) = 4f_4, N(2, 4) = -4f_3, N(2, 5) = -4f_1, N(2, 6) = 0,$$

$$N(3, 4) = 4f_2, N(3, 5) = -4f_6, N(3, 6) = 4f_5,$$

$$N(4, 5) = 0, N(4, 6) = 4f_1,$$

$$N(5, 6) = -4f_3.$$

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

$$\frac{1}{4}N_{11}^r = 0, \frac{1}{4}N_{12}^r = -\delta_5^r, \frac{1}{4}N_{13}^r = 0, \frac{1}{4}N_{14}^r = \delta_6^r, \frac{1}{4}N_{15}^r = \delta_2^r, \frac{1}{4}N_{16}^r = -\delta_4^r,$$

$$\frac{1}{4}N_{21}^r = \delta_5^r, \frac{1}{4}N_{22}^r = 0, \frac{1}{4}N_{23}^r = \delta_4^r, \frac{1}{4}N_{24}^r = -\delta_3^r, \frac{1}{4}N_{25}^r = -\delta_1^r, \frac{1}{4}N_{26}^r = 0,$$

$$\frac{1}{4}N_{31}^r = 0, \frac{1}{4}N_{32}^r = -\delta_4^r, \frac{1}{4}N_{33}^r = 0, \frac{1}{4}N_{34}^r = \delta_2^r, \frac{1}{4}N_{35}^r = -\delta_6^r, \frac{1}{4}N_{36}^r = \delta_5^r,$$

$$\frac{1}{4}N_{41}^r = -\delta_6^r, \frac{1}{4}N_{42}^r = \delta_3^r, \frac{1}{4}N_{43}^r = -\delta_2^r, \frac{1}{4}N_{44}^r = 0, \frac{1}{4}N_{45}^r = 0, \frac{1}{4}N_{46}^r = \delta_1^r,$$

$$\frac{1}{4}N_{51}^r = -\delta_2^r, \frac{1}{4}N_{52}^r = \delta_1^r, \frac{1}{4}N_{53}^r = \delta_6^r, \frac{1}{4}N_{54}^r = 0, \frac{1}{4}N_{55}^r = 0, \frac{1}{4}N_{56}^r = -\delta_3^r,$$

$$\frac{1}{4}N_{61}^r = \delta_4^r, \frac{1}{4}N_{62}^r = 0, \frac{1}{4}N_{63}^r = -\delta_5^r, \frac{1}{4}N_{64}^r = -\delta_1^r, \frac{1}{4}N_{65}^r = \delta_3^r, \frac{1}{4}N_{66}^r = 0.$$
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

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

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

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

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

□

### 4. HOPF PROBLEM OF $S^6$

*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

$$ (4.1) \qquad \ell \neq 0. $$

This implies Nijenhuis tensor $N_J \neq 0$. By Newlander-Nirenberg Theorem [20], $J$ is not integrable. Theorem 1.4 follows.

If $d\ell \neq 0$, then (4.1) holds. We are done.

Otherwise,

$$ (4.2) \qquad d\ell \equiv 0 $$

$\ell$ is closed, $\ell \in H_{\text{DeRham}}^2(S^6) = 0$

Recall (2.1): Therefore $\ell$ is exact. There exists 1-form $\eta$ such that

$$ (4.3) \qquad \ell = d\eta, $$

where

$$ (4.4) \qquad \eta = \sum_{j=1}^{6} \eta_j dx^j, \quad \ell_{ij} = \partial_i \eta_j = -\partial_j \eta_i $$

For fixed $i, j, i \neq j$, $(\ell_{ii} = \ell_{jj})$, in the expression (2.1) of $\ell_{ij}$. The sum of terms without $\partial_i$ or $\partial_j$ vanishes. We write $i$ in $\partial_i$ by $\mathbf{i}$, write $j$ in $\partial_j$ by $\mathbf{j}$, while write $i, j$ in other locations normally. Note

$$ \ell_{ij} = \partial_i \eta_{jl} = -\partial_{jl} \eta_i \qquad \partial_i \eta_i = 0 = \partial_j \eta_j $$

In the following, we show

$$ \ell = d\eta \neq 0, $$
then (4.1) holds, the proof is completed.

Suppose, contrary to the above,

$$ (4.5) \qquad \ell = d\eta \equiv 0, $$

then we will have a contradiction.

The above equation shows $\eta$ is closed 1-form on $S^6$. $H_{\text{DeRham}}^1(S^6) = 0$ implies, for some smooth function on $S^6$,

$$ (4.6) \qquad \eta = df. $$

{

We fix $i$ and $j$ and reduce terms in $\ell_{ij}$. We emphasize $i, j$ in $\partial_i, \partial_j$ by write $\partial_i$, $\partial_j$ respectively. (4.3)-(4.6) imply that $\ell_{ij}$ consists of terms of partial derivatives with respect both to $x^i$ and to $x^j$. The sum of all terms in $\ell_{ij}$ without factors $\partial_i$ and $\partial_j$ vanishes. The sum of all terms in $\ell_{ij}$ with factors both $\partial_i$ and $\partial_j$ vanishes. The sum of all terms in $\ell_{ij}$ with factors $\partial_j$ and $\partial_i$ vanishes.

We write "$\equiv$ mod" means module those terms whose total sum in $\ell_{ij}$ vanish, as described by above statements and the following lemma.

**Lemma 4.1.** Suppose $d\ell = d\ell_J \equiv 0$ for $J$ on $S^6$. For pairwise distinct $i, j, k$, $i, j, k = 1, 2, 3, 4, 5, 6$, all terms of $\ell_{ij}$ other than ones in the form $\partial_{i_1}\psi_{i_2i_3}$ with set $\{i_1, i_2, i_3\} = \{i, j, k\}$ for some functions $\psi_{kl}$, $k, l = 1, 2, 3, 4, 5, 6$, vanish on $S^6 \setminus \cup_{i=1}^7 \{\pm P_i\}$. In particular, terms in form $\partial_{i_1}\psi_{i_2i_3}$ with a repeating index on set $\{i_1, i_2, i_3\} = \{i, j, k\}$ vanish as well on $S^6 \setminus \cup_{i=1}^7 \{\pm P_i\}$, where $\pm P_i$, $i = 1, 2, 3, 4, 5, 6, 7$ are intersection points of $S^6$ and coordinate axes of $\mathbb{R}^7$.

The proof is at the end of paper.

We reduce terms of $\ell_{ij}$ on $S^6 \setminus \cup_{i=1}^7 \{\pm P_i\}$.

Note $u\partial_i v = \partial_i w$ and $u\partial_j v = \partial_j y$ for some $w, y$ via the chain rule.

In the following reduction of terms of $\ell_{ij}$, "$=$" means "$\equiv$ mod".

}}

Applying all things in $\{ \}$, we have the following.

The first line in (2.1) is

$$ \sum_{k,l,r,p,q=1}^{6} -J_k^l J_i^p J_j^q \partial_p J_l^r \partial_q J_r^k. $$
$$
\begin{align*}
&= \sum_{k,l,r,q=1}^{6} \left\{ -J_k^l J_i^i J_j^q \partial_i J_l^r \partial_q J_r^k - J_k^l J_i^j J_j^q \partial_j J_l^r \partial_q J_r^k \right\} \\
&\qquad - \sum_{k,l,r=1}^{6} -J_k^l J_i^i J_j^q \partial_i J_l^r \partial_j J_r^k + \sum_{k,l,r=1}^{6} -J_k^l J_i^j J_j^q \partial_j J_l^r \partial_i J_r^k \\
&= J_i^i J_j^j \sum_{k,l,r=1}^{6} -J_k^l \partial_i J_l^r \partial_j J_r^k + J_i^j J_j^i \sum_{k,l,r=1}^{6} -J_k^l \partial_j J_l^r \partial_i J_r^k \\
&= J_i^i J_j^j \sum_{k,l,r=1}^{6} J_k^l \partial_i J_r^k \partial_j J_l^r + J_i^j J_j^i \sum_{k,l,r=1}^{6} -J_k^l \partial_i J_r^k \partial_j J_l^r \\
&= (J_i^i J_j^j - J_i^j J_j^i) \sum_{k,l,r=1}^{6} J_k^l \partial_i J_r^k \partial_j J_l^r
\end{align*}
$$

The second line in (2.1) is

$$
\sum_{k,l,r=1}^{6} J_k^l \partial_i J_r^k \partial_j J_l^r .
$$

The third line in (2.1) becomes

$$
\begin{align*}
& \sum_{k,l,r=1}^{6} \left\{ -J_i^r \partial_r J_k^l \partial_j J_l^k + J_j^r \partial_r J_k^l \partial_i J_l^k \right\} \\
&= \sum_{k,l,r=1}^{6} -J_i^r \partial_r J_k^l \partial_j J_l^k + \sum_{k,l,r=1}^{6} J_j^r \partial_r J_k^l \partial_i J_l^k \\
&= \sum_{l,r=1}^{6} -J_i^r \partial_r J_i^l \partial_j J_l^i + \sum_{l,r=1}^{6} J_j^r \partial_r J_j^l \partial_i J_l^j \\
&= \sum_{k,l=1}^{6} -J_i^i \partial_i J_k^l \partial_j J_l^k + \sum_{k,l=1}^{6} J_j^i \partial_j J_k^l \partial_i J_l^k \\
&= \sum_{l=1,k=i}^{6} -J_i^l \partial_i J_k^l \partial_j J_l^k + \sum_{l=1,k=i}^{6} J_j^l \partial_j J_k^l \partial_i J_l^k \\
&\quad + \sum_{l=1,k=j}^{6} -J_i^l \partial_i J_k^l \partial_j J_l^k + \sum_{l=1,k=j}^{6} J_j^l \partial_j J_k^l \partial_i J_l^k
\end{align*}
$$
$$
\begin{align*}
&+ \sum_{k=1, l=i}^{6} -J_i^i \partial_i J_k^l \partial_j J_l^k + \sum_{k=1, l=i}^{6} J_j^j \partial_j J_k^l \partial_i J_l^k \\
&+ \sum_{k=1, l=j}^{6} -J_i^i \partial_i J_k^l \partial_j J_l^k + \sum_{k=1, l=j}^{6} J_j^j \partial_j J_k^l \partial_i J_l^k \\
&= \sum_{l=1}^{6} -J_i^l \partial_i J_k^l \partial_j J_l^k + \sum_{l=1}^{6} J_j^l \partial_j J_k^l \partial_i J_l^k \\
&\quad + \sum_{l=1}^{6} -J_i^i \partial_i J_k^l \partial_j J_l^j + \sum_{l=1}^{6} J_j^i \partial_j J_k^l \partial_i J_l^j \\
&\quad + \sum_{k=1}^{6} -J_i^i \partial_i J_k^l \partial_j J_l^k + \sum_{k=1}^{6} J_j^i \partial_j J_k^l \partial_i J_l^k \\
&\quad + \sum_{k=1}^{6} -J_i^l \partial_i J_k^j \partial_j J_l^k + \sum_{k=1}^{6} J_j^l \partial_j J_k^j \partial_i J_l^k \\
&= 0.
\end{align*}
$$

The fourth line in (2.1) is

$$
\begin{align*}
& \sum_{k,l,r=1}^{6} 2 J_i^r \partial_r J_k^l \partial_l J_j^k - \sum_{k,l,r=1}^{6} 2 J_j^r \partial_r J_k^l \partial_l J_i^k \\
&= \sum_{k=1}^{6} 2 J_i^k \partial_i J_k^l \partial_j J_j^k + \sum_{k=1}^{6} 2 J_j^k \partial_j J_k^l \partial_i J_i^k - \sum_{k=1}^{6} 2 J_j^l \partial_j J_k^l \partial_i J_i^k - \sum_{k=1}^{6} 2 J_j^i \partial_i J_k^l \partial_j J_j^k \\
&= \sum_{k=1}^{6} 2 \partial_i (J_i^k J_k^l \partial_j J_j^k) - \sum_{k=1}^{6} 2 \partial_j (J_j^k J_k^l \partial_i J_i^k) - \sum_{k=1}^{6} 2 J_i^k J_k^l \partial_i \partial_j J_j^k \\
&\quad + \sum_{k=1}^{6} 2 \partial_j (J_i^k J_k^l \partial_i J_j^k) - \sum_{k=1}^{6} 2 \partial_i (J_i^k J_k^l \partial_j J_j^k) - \sum_{k=1}^{6} 2 J_i^l J_k^l \partial_i \partial_j J_j^k \\
&\quad - \sum_{k=1}^{6} 2 \partial_i (J_j^k J_k^l \partial_j J_i^k) + \sum_{k=1}^{6} 2 \partial_j (J_i^k J_k^l \partial_j J_i^k) + \sum_{k=1}^{6} 2 J_j^l J_k^l \partial_i \partial_j J_j^k \\
&\quad - \sum_{k=1}^{6} 2 \partial_j (J_j^l J_k^l \partial_i J_i^k) + \sum_{k=1}^{6} 2 \partial_i (J_j^l J_k^l \partial_j J_i^k) + \sum_{k=1}^{6} 2 J_j^k J_k^l \partial_i \partial_j J_j^k \\
&= \sum_{k=1}^{6} 2 \left\{ \partial_i (J_i^k J_k^l \partial_j J_j^k) + \partial_j (J_j^k J_k^l \partial_i J_i^k) - \partial_i (J_j^l J_k^l \partial_j J_i^k) - \partial_j (J_j^l J_k^l \partial_i J_i^k) \right\}
\end{align*}
$$
$$ = \sum_{k=1}^{6} 2\{0 + \partial_{\mathbf{j}}(J_{i}^{j} J_{k}^{i} \partial_{\mathbf{i}} J_{j}^{k}) - \partial_{\mathbf{i}}(J_{j}^{i} J_{k}^{j} \partial_{\mathbf{j}} J_{i}^{k}) - 0\}, $$

We show the remaining two terms in the above equation vanish.

$$ \sum_{k=1}^{6} J_{k}^{i} \partial_{\mathbf{i}} J_{i}^{k} = \sum_{k=1}^{6} \partial_{\mathbf{i}}(J_{k}^{i} J_{i}^{k}) - \sum_{k=1}^{6} \partial_{\mathbf{i}} J_{k}^{i} \partial_{\mathbf{i}} J_{i}^{k} = \sum_{k=1}^{6} \partial_{\mathbf{i}}(1) - \sum_{k=1}^{6} \partial_{\mathbf{i}} J_{k}^{i} \partial_{\mathbf{i}} J_{i}^{k} = 0. $$

Therefore

$$ \partial_{\mathbf{j}}(J_{i}^{j} J_{k}^{i} \partial_{\mathbf{i}} J_{j}^{k}) = 0. $$

$$ \sum_{k=1}^{6} J_{k}^{j} \partial_{\mathbf{j}} J_{i}^{k} = \sum_{k=1}^{6} \partial_{\mathbf{j}}(J_{k}^{j} J_{i}^{k}) - \sum_{k=1}^{6} \partial_{\mathbf{j}} J_{k}^{j} J_{i}^{k} = \sum_{k=1}^{6} \partial_{\mathbf{j}}(0) - \sum_{k=1}^{6} 0 = 0. $$

Therefore

$$ -\partial_{\mathbf{i}}(J_{j}^{i} J_{k}^{j} \partial_{\mathbf{j}} J_{i}^{k}) = 0. $$

We have showed the fourth line vanishes.

The fifth line in (2.1) is

$$ \begin{aligned} & \sum_{k,l,r=1}^{6} 2\{-J_k^r \partial_r J_i^l \partial_l J_j^k + J_k^r \partial_r J_j^l \partial_l J_i^k\} \\ &= \sum_{k,l,r=1}^{6} 2\{-J_k^r \partial_r J_i^l \partial_l J_j^k + J_k^r \partial_r J_j^l \partial_l J_i^k\} \\ &= \sum_{k=1}^{6} -2J_k^i \partial_{\mathbf{i}} J_i^j \partial_{\mathbf{j}} J_j^k + \sum_{k=1}^{6} -2J_k^j \partial_{\mathbf{j}} J_i^l \partial_{\mathbf{i}} J_j^k + \sum_{k=1}^{6} 2J_k^i \partial_{\mathbf{i}} J_j^l \partial_{\mathbf{j}} J_i^k + \sum_{k=1}^{6} 2J_k^j \partial_{\mathbf{j}} J_i^l \partial_{\mathbf{i}} J_j^k \\ &= 0 + 0 + 0 + 0 = 0. \end{aligned} $$

The sixth line in (2.1) is

$$ \begin{aligned} & \sum_{k,l,r=1}^{6} 2\{J_k^r \partial_r J_i^l \partial_{\mathbf{j}} J_l^k - J_k^r \partial_r J_j^l \partial_{\mathbf{i}} J_l^k\}, \\ &= \sum_{k,l,r=1}^{6} 2J_k^r \partial_r J_i^l \partial_{\mathbf{j}} J_l^k - \sum_{k,l,r=1}^{6} 2J_k^r \partial_r J_j^l \partial_{\mathbf{i}} J_l^k \\ &= \sum_{k,l=1}^{6} 2J_k^i \partial_{\mathbf{i}} J_i^l \partial_{\mathbf{j}} J_l^k - \sum_{k,l=1}^{6} 2J_k^j \partial_{\mathbf{j}} J_j^l \partial_{\mathbf{i}} J_l^k \\ &= 0 + 0 = 0. \end{aligned} $$
(2.1) is now

$$ \ell_{ij} = (J_i^i J_j^j - J_i^j J_j^i) \sum_{k,l,r=1}^{6} J_k^l \partial_i J_r^k \partial_j J_l^r + \sum_{k,l,r=1}^{6} J_k^l \partial_i J_r^k \partial_j J_l^r, $$

that is,

$$ (4.7) \qquad \ell_{ij} = (m_{ij} + 1) \xi_{ij}, $$

where $m_{ij}$ denotes principal minor of order 2 of $J$.

$$ m_{ij} = \det \begin{bmatrix} J_i^i & J_i^j \\ J_j^i & J_j^j \end{bmatrix} = J_i^i J_j^j - J_i^j J_j^i. $$

and $\xi$ denotes 2-form

$$ (4.8) \qquad \xi = \sum_{i,j=1}^{6} \xi_{ij} dx^i \wedge dx^j = \sum_{i,j=1}^{6} \left( \sum_{p,q,r=1}^{6} \partial_i J_p^r J_r^q \partial_j J_q^p \right) dx^i \wedge dx^j. $$

Note that

$$ \xi_{ij} = \sum_{p,q,r=1}^{6} \partial_i J_p^r J_r^q \partial_j J_q^p = \mathrm{tr}\{\partial_i J J \partial_j J\}. $$

Note for $J = [J_i^j]_{6 \times 6}$ is a 6 × 6 real matrix at every point $x$ of $S^6$, with $J^2 = -\mathrm{Identity}$. At $x \in W$, all eigenvalues of $J$ are $\pm\sqrt{-1}$: For eigenvalue $\lambda$ and eigenvector $X$

$$ JX = \lambda X, \quad -X = J^2 X = \lambda^2 X. $$

So $\lambda^2 = -1$ and $\lambda = \pm\sqrt{-1}$. Let $I$ be the identity matrix at $x \in S^6$. Since $J$ is real, characteristic equation $\det(\lambda I - J) = 0$ of $J$ is a equations with real number coefficients, so complex conjugate of a zero of the equation is also a zero, a non-real zero with its conjugate have the same zero multiplicity. So characteristic equation of $J$ is

$$ (\lambda - \sqrt{-1})(\lambda + \sqrt{-1})(\lambda - \sqrt{-1})(\lambda + \sqrt{-1})(\lambda - \sqrt{-1})(\lambda + \sqrt{-1}) = 0, $$
that is,

$$ (\lambda^2 + 1)^3 = \lambda^6 + 3\lambda^4 + 3\lambda^2 + 1 = 0. $$

The sum of principal minors of order 2 of $J$ is 3 at any point $x \in S^6$. So at least one minor, say $m_{ij}(x)$, is positive at $x \in S^6$. By continuity, the minor $m_{ij}$ is positive in an open neighborhood $U \ni x$ in $S^6$, $m_{ij}|_U > 0$.

For such $i, j, i \neq j$, $\ell_{ij} = (m_{ij} + 1) \neq 0$ if and only if on $\ell_{ij} \neq 0$ on $U$.

By Lemma 4.2, $\ell_{ij} \neq 0$ on $U$, so $\ell_{ij} \neq 0$ on $U$ and $\ell \neq 0$ on $S^6$. By Lemma 4.2, $\ell \neq 0$ on $S^6$.

On the other hand (4.5) –(4.8) imply $\ell \equiv 0$ on $S^6$, a contradiction.
□

**Lemma 4.2.** 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 $i,j = 1, \cdots, 6$.

$\xi$ is defined by (4.8). Then $\xi_{ij} \neq 0$ on any open set $U$ in $S^6$ for all $i \neq j$, $i, j = 1, 2, 3, 4, 5, 6$, and $\xi \neq 0$ on $S^6$.

*Proof.* $J$ is a tensor field on $S^6$, it cannot be globally constant.

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

For non empty open set $U$ of $S^6$, $\cup_{i=1}^7 \{\pm P_i\}$, where $\pm P_i$, $1 \le i \le 7$ are intersection points of $S^6$ and coordinate axes of $\mathbb{R}^7$, take a point $x \in U$ with its local coordinates $V, \{x^i\}_{i=1}^6$, $V \subseteq U$ non-empty open. on $V$, $P(x)$ is continuous (therefore smooth) with $P^{-1}JP = B$, where $B$ is the Jordan matrix of $J$. Note the eigenvalues of $J$ are constants $\pm\sqrt{-1}$ in pairs. We prove $\xi \neq 0$ on $V$.

$$
\begin{align*}
\xi_{ij} &= \operatorname{tr}\{\partial_i J J \partial_j J\} = \operatorname{tr}\{\partial_i J P B P^{-1} \partial_j J\} \\
&= \operatorname{tr}\{\partial_i (P B P^{-1}) P B P^{-1} \partial_j (P B P^{-1})\} \\
&= \operatorname{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})\} \\
&= \operatorname{tr}\{\partial_i P B P^{-1} P B P^{-1} \partial_j P B P^{-1}\} + \operatorname{tr}\{\partial_i P B P^{-1} P B \partial_j P^{-1}\} \\
&\quad + \operatorname{tr}\{P B \partial_i P^{-1} P B P^{-1} \partial_j P B P^{-1}\} + \operatorname{tr}\{P B \partial_i P^{-1} P B P^{-1} P B \partial_j P^{-1}\} \\
&= -\operatorname{tr}\{\partial_i P P^{-1} \partial_j P B P^{-1}\} - \operatorname{tr}\{\partial_i P B \partial_j P^{-1}\} \\
&\quad + \operatorname{tr}\{B \partial_i P^{-1} P B P^{-1} \partial_j P B\} - \operatorname{tr}\{P B \partial_i P^{-1} P \partial_j P^{-1}\} \\
&= \operatorname{tr}\{P \partial_i P^{-1} \partial_j P B P^{-1}\} - \operatorname{tr}\{\partial_i P B \partial_j P^{-1}\} \\
&\quad - \operatorname{tr}\{B^{-1} \partial_i P^{-1} P B P^{-1} \partial_j P B\} + \operatorname{tr}\{P B \partial_i P^{-1} \partial_j P P^{-1}\} \\
&= \operatorname{tr}\{\partial_i P^{-1} \partial_j P B\} - \operatorname{tr}\{\partial_i P B \partial_j P^{-1}\} \\
&\quad - \operatorname{tr}\{P^{-1} \partial_i P B \partial_j P^{-1} P\} + \operatorname{tr}\{B \partial_i P^{-1} \partial_j P\} \\
&= \operatorname{tr}\{B \partial_i P^{-1} \partial_j P\} - \operatorname{tr}\{\partial_i P B \partial_j P^{-1}\} \\
&\quad - \operatorname{tr}\{\partial_i P B \partial_j P^{-1}\} + \operatorname{tr}\{B \partial_i P^{-1} \partial_j P\} \\
&= 2\operatorname{tr}\{B \partial_i P^{-1} \partial_j P\} - 2\operatorname{tr}\{\partial_i P B \partial_j P^{-1}\} \\
&= 2\operatorname{tr}\{B \partial_i P^{-1} \partial_j P\} - 2\operatorname{tr}\{B \partial_j P^{-1} \partial_i P\} \\
&= 2\operatorname{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 $V \subseteq S^6$ for $i \neq j$, $i, j = 1, \cdots, 6$, where $B$ is Jordan matrix of $J$ on $V \subseteq U \subseteq S^6$. □
Proof of Lemma 4.1. The other cases are similar. Let $S^6$ be the unit sphere in $\mathbb{R}^7$, and $h$ is the hight function on $S^6$, that is, $h(x) = 7th$ coordinate in $\mathbb{R}^7$ of $x \in S^6$. $dh$ is a global 1-form on $S^6$. $\ell \wedge \wedge dh$ is a 3-form. $d(\ell \wedge dh) = d\ell \wedge dh \equiv 0$. Since $H_{\text{DeRham}}^3(S^6) = 0$, $\ell \wedge dh$ is exact,

$$ (4.9) \qquad \ell \wedge dh = d\phi $$

for some 2-form $\phi$.

We take $k = 6$. Except for two points $\pm P_7 = (0,0,0,0,0,0,\pm 1)$, take $h = x^6$ in the local coordinate $\{x^i\}_{i=1}^6$ of $S^6$. Therefore (4.9) is

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

on $S^6 \setminus \{\pm P_7\}$. Suppose

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

Then

$$ \begin{aligned} d\phi &= \sum_{i,j,k=1}^{6} \partial_k \phi_{ij} dx^k \wedge dx^i \wedge dx^j \\ &= \sum_{j,k=1}^{5} \partial_k \phi_{6j} dx^k \wedge dx^6 \wedge dx^j + \sum_{i,k=1}^{5} \partial_k \phi_{i6} dx^k \wedge dx^i \wedge dx^6 \\ &\quad + \sum_{i,j=1}^{5} \partial_6 \phi_{ij} dx^6 \wedge dx^i \wedge dx^j + \sum_{i,j,k=1}^{5} \partial_k \phi_{ij} dx^k \wedge dx^i \wedge dx^j \\ &= \sum_{i,j=1}^{5} (-\partial_i \phi_{6j} - \partial_j \phi_{i6} + \partial_6 \phi_{ij}) dx^i \wedge dx^j \wedge dx^6 + \sum_{i,j,k=1}^{5} \partial_k \phi_{ij} dx^k \wedge dx^i \wedge dx^j \\ &\qquad \ell \wedge dx^6 = d\phi \quad \text{on } S^6 \setminus \{\pm P_7\} \text{ yields} \\ \sum_{i,j=1}^{5} \ell_{ij} dx^i \wedge dx^j \wedge dx^6 &= \sum_{i,j=1}^{5} (-\partial_i \phi_{6j} - \partial_j \phi_{i6} + \partial_6 \phi_{ij}) dx^i \wedge dx^j \wedge dx^6, \\ \ell_{ij} &= -\partial_i \phi_{6j} - \partial_j \phi_{i6} + \partial_6 \phi_{ij}, \quad \text{for } i,j = 1,2,3,4,5, i \neq j \\ \text{on } S^6 \setminus \{\pm P_7\}. \end{aligned} $$

$$ \sum_{i,j=1}^{5} \ell_{ij} dx^i \wedge dx^j \wedge dx^6 = \sum_{i,j=1}^{5} (-\partial_i \phi_{6j} - \partial_j \phi_{i6} + \partial_6 \phi_{ij}) dx^i \wedge dx^j \wedge dx^6, $$
$$ \ell_{ij} = -\partial_i \phi_{6j} - \partial_j \phi_{i6} + \partial_6 \phi_{ij}, \quad \text{for } i,j = 1,2,3,4,5, i \neq j $$
on $S^6 \setminus \{\pm P_7\}$.

Therefore $i, j = 1, 2, 3, 4, 5, i \neq j$, among terms of $\ell_{ij}$, terms other than ones containing $\partial_i \psi_{6j}$, or $\partial_i \psi_{j6}$, or $\partial_j \psi_{6i}$, or $\partial_j \psi_{i6}$, $\partial_6 \psi_{ij}$, and $\partial_6 \psi_{ji}$ for some functions $\psi_{kl}$, $k, l = 1, 2, 3, 4, 5, 6$, vanish on $S^6 \setminus \{\pm P_7\}$;
and terms containing $\partial_{i_1}\psi_{i_2i_3}$ with a repeating index on set $\{i_1, i_2, i_3\} = \{i, j, 6\}$ vanish as well on $S^6 \setminus \{\pm P_7\}$.

Similar argument to the above $k=6$ applies for $k=1, 2, 3, 4, 5,$

Therefore for pairwise distinct $i, j, k$, $i, j, k = 1, 2, 3, 4, 5, 6$, among terms of $\ell_{ij}$, terms other than ones in the form $\partial_{i_1}\psi_{i_2i_3}$ with set $\{i_1, i_2, i_3\} = \{i, j, k\}$, for some functions $\psi_{kl}$, $k, l = 1, 2, 3, 4, 5, 6$, vanish on $S^6 \setminus \cup_{i=1}^7 \{\pm P_i\}$. In particular, terms in form $\partial_{i_1}\psi_{i_2i_3}$ with a repeating index on set $\{i_1, i_2, i_3\} = \{i, j, k\}$ vanish as well. $\square$

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

Email address: lingju@uvu.edu