Orthogonal almost complex structure and its Nijenhuis tensor
Zizhou Tang, Wenjiao Yan
TL;DR
The paper addresses the existence and rigidity of orthogonal almost complex structures on curved manifolds by using the twistor bundle to pull back the Kähler form and study a derived 2-form $\varphi$ on an almost Hermitian manifold. The main method constructs the twistor framework, derives an explicit expression for $\varphi$ in terms of connection data, and proves a sharp $|N|^2$-dependent nondegeneracy criterion: $\varphi$ is non-degenerate if $|N|^2<\frac{64}{5}$ for $n\ge 3$ (and $<16$ when $n=2$). This leads to a corollary generalizing Blanchard–LeBrun: on the standard sphere $(S^6, ds_0^2)$ there is no orthogonal almost complex structure with $|N|^2<\frac{64}{5}$ everywhere. The result provides a quantitative obstruction to certain almost complex structures on $S^6$ and showcases the utility of twistor methods in almost Hermitian geometry.
Abstract
In this paper, we demonstrate that on an almost Hermitian manifold $(M^{2n}, J, ds^2)$, a 2-form $\varphi=S^*Φ$, the pulling back of the Kähler form $Φ$ on the twistor bundle over $M^{2n}$, is non-degenerate if the squared norm $|N|^2$ of the Nijenhuis tensor is less than $\frac{64}{5}$ when $n\geq 3$ or less than $16$ when $n=2$. As a corollary, there exists no orthogonal almost complex structure on the standard sphere $(S^6, ds_0^2)$ with $|N|^2<\frac{64}{5}$ everywhere.