On Almost Complex Structures on 6-dimensional Products of Spheres

In this article, almost complex structures on the sphere $S^6$ and on the products of spheres $S^1\times S^5$, $S^2\times S^4$, and $S^3\times S^3$ which naturally arise at their embeddings in the algebra of Cayley numbers are considered. It is shown that all of them are nonintegrable. Expressions of the fundamental form $\omega$ and the Nijenhuis tensor for each case are obtained. It is also shown that the form $d\omega$ is nondegenerate. New special almost complex structures on products of spheres are constructed.