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 × S 3 , S 1 × S 5 , and S 2 × S 4 . We prove that all almost complex Cayley structures that naturally appear from their embeddings into the Cayley octave algebra Ca are nonintegrable. We obtain expressions for the Nijenhuis tensor and the fundamental form ω for each gauge of the space Ca and prove the nondegeneracy of the form d ω. 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) × U (2)- invariant Hermitian metrics on S 3 × S 3 and find estimates of the sectional sectional curvature. We consider the space of left-invariant, almost complex structures on S 3 × S 3 = SU (2) × 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.