Orthogonal almost complex structure and its Nijenhuis tensor

In this paper, we demonstrate that on an almost Hermitian manifold $(M^{2n},J,d{s}^2)$, a 2-form $\phi =S^{\ast }\mathrm{\Phi }$, the pulling back of the Kähler form $\mathrm{\Phi }$ on the twistor bundle over $M^{2n}$, is non-degenerate if the squared norm $\mathrm{\mid }N{\mathrm{\mid }}^2$ of the Nijenhuis tensor is less than $\frac{64}{5}$ when $n\ge 3$ or less than $16$ when $n=2$. As a corollary, there exists no orthogonal almost complex structure on the standard sphere $(S^6,d{s}_0^2)$ with $\mathrm{\mid }N{\mathrm{\mid }}^2<\frac{64}{5}$ everywhere.