# A new differential form and 6-Shere

Jun Ling

► To cite this version:

Jun Ling. A new differential form and 6-Shere. 2025. hal-05060123v2

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Preprint submitted on 25 Jun 2025

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# A NEW DIFFERENTIAL FORM AND 6-SHERE

JUN LING

ABSTRACT. This is a research announcement. We construct a new differential form from an arbitrary almost complex structure on smooth manifolds. As an application of the form, we show that $S^6$ does not have a complex structure [4].

This is a research announcement. We construct a new differential form from an arbitrary almost complex structure on smooth manifolds. As an application of the form, we show that $S^6$ does not have a complex structure [4]. Namely, we have the following results.

We construct A new differential 2-form $\ell$ from an arbitrary almost complex structure $J$ and the related Nijenhuis tensor $N$. This form yields a topological obstruction on the existence of complex structures and establishes relation between triviality of second cohomology and the nonexistence of complex structures in high dimensions, which is verified by examples of complex manifolds in high dimensions and their second cohomology $H^2(\mathrm{CP}^2; \mathbb{Z})$, $H^2(\mathrm{CP}^3; \mathbb{Z})$, $H^2(\mathrm{CP}^4; \mathbb{Z})$, and so on. This new differential form can be applied to any almost complex manifold to study existence of complex structures.

$S^6$ has an almost complex structure whose 2-form is nonvanishing.

As a consequence of this and the fact $H^2(S^6) = 0$, we show that

$S^6$ is not a complex manifold.

In [4], we define $\ell$ of almost complex structure $J$ on $M$,

$$ \ell(X, Y) := \mathrm{trace}\{JN(N(X, \cdot), Y)\} $$

for smooth vector fields $X$ and $Y$, where $[\cdot, \cdot]$ is Lie bracket, and $N$ is the well-known Nijenhuis tensor of $J$ given by

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

The following results are proved in [4].

<footnote>Date: May 7, 2025.</footnote>

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

<footnote>Key words and phrases. Nijenhuis Tensor, almost complex structures, cohomology.</footnote>

<footnote>Thank Princeton University and Cornell University for their hospitality.</footnote>
**Theorem 0.1.** $\ell$ is a differential 2-form on $M$.

**Corollary 0.2.** If the $\ell$ is not closed, then $J$ is not integrable.

To state the next result, we express $\ell$ in local coordinates $\{x^i\}_{i=1}^n$ at point $x \in M$. Let $\partial_i = \frac{\partial}{\partial x^i}$, $J = \frac{\partial}{\partial x^i} = \sum_j J_j^j \frac{\partial}{\partial x^j}$, $J_i = \sum_p J_p^p \frac{\partial}{\partial x^p}$, and components of Nijenhuis tensor $N$ are

$$N_{ij}^k = \sum_p \left\{ J_i^p (\partial_p J_j^k - \partial_j J_i^k) - J_j^p (\partial_p J_i^k - \partial_i J_j^k) \right\}, \quad \text{where}$$

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

**Theorem 0.3.** In local coordinates:

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

with

$$l_{ij} = -l_{ji}$$

and

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

In the following, we use Einstein's sum convention: take sum for a repeating indices, unless otherwise stated.

View $S^6$ as the unit sphere in the imaginary part of the Octonions. There is a well-known canonical almost complex structure defined on the tangent bundle of $S^6$ by multiplication in the Octonions. We call it left-multiplication almost complex structure. Please refer to [2], or [1] for more details about left-multiplication almost complex structure.

**Theorem 0.4.** Form $\ell$ of left-multiplication almost complex structure on $S^6$ is nonvanishing.

By the above theorems and $H^2(S^6) = 0$, we have the following theorem.

**Theorem 0.5.** No almost complex structure on $S^6$ is integrable. Therefore $S^6$ is not a complex manifold.

In [3] the author proved above the trace of $\ell$ vanishes at any point, among other results. The author obtained differential form $\ell$ in July of 2024, which includes traceless result in [3] since a skew-symmetric matrix is hollow therefore is traceless. The author obtain Theorem 0.4 and Theorem 0.5 in [4] in January of 2025.
REFERENCES

[1] L. Díaz, *A note on Kirchhoff's theorem for almost complex spheres I*, https://arxiv.org/pdf/1804.05794.

[2] P.Konstantis and M. Parton, *Almost complex structures on spheres*, Differential Geometry and its Applications, **57**(2018), 10–12.

[3] J. Ling *The square of Nijenhuis tensor and its vanishing results*, Asian-European Journal of Mathematics, Asian-European Journal of Mathematics, **15**, no 8(2022).

[4] J. Ling *Integrability and cohomology of six-sphere*, preprint, Jan 29, submit 6169314 on Jan 31, 2025. https://hal.science/hal-05027695v2/document, April 9, 2025.

DEPARTMENT OF MATHEMATICS, UTAH VALLEY UNIVERSITY, OREM, UTAH 84058
*Email address: lingju@uvu.edu*