Almost Complex Structures on the Direct Product of Three-Dimensional Spheres

We study some almost complex structures on $S^3\times S^3$. In the first part of the article, the product $S^3\times S^3$ is considered as the homogeneous space $U(2)/U(1)\times U(2)/U(1)$. We describe the set of all invariant almost complex structures on $U(2)/U(1)\times U(2)/U(1)$ and the metrics associated with them. We list some results on the properties of these metrics. In the second part of the article, the product of 3d̄imensional spheres is considered as the Lie group $SU(2)\times SU(2)$. In this case, the set of invariant structures is much wider that the set of the structures of the first part. We describe the class of all left-invariant complex structures on $SU(2)\times SU(2)$. Among structures orthogonal with respect to the Killing–Cartan metric, we distinguish the class at which the maximum of the norm of the Nijenhuis tensor is attained. We study some properties of the positively associated almost complex structures on $SU(2)\times SU(2)$.