On Almost Complex Structures on Six-Dimensional Products of Spheres

In this paper, we discuss almost complex structures on the sphere $S^6$ and on the products of spheres $S^3\times S^3$, $S^1\times S^5$, and $S^2\times S^4$. We prove that all almost complex Cayley structures that naturally appear from their embeddings into the Cayley octave algebra $\mathbb{C}\mathrm{a}$ are nonintegrable. We obtain expressions for the Nijenhuis tensor and the fundamental form $\omega$ for each gauge of the space $\mathbb{C}\mathrm{a}$ and prove the nondegeneracy of the form $d\omega$. We show that through each point of a fiber of the twistor bundle over $S^6$, a one-parameter family of Cayley structures passes. We describe the set of $U(2)\times U(2)$-invariant Hermitian metrics on $S^3\times S^3$ and find estimates of the sectional sectional curvature. We consider the space of left-invariant, almost complex structures on $S^3\times S^3=SU(2)\times SU(2)$ and prove the properties of left-invariant structures that yield the maximal value of the norm of the Nijenhuis tensor on the set of left-invariant, orthogonal, almost complex structures.