On Almost Complex Structures on Six-dimensional Products of Spheres

Abstract

In this paper, we discuss almost complex structures on the sphere S6 and on the products of spheres S3 × S3, S1 × S5, and S2 × S4. 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 d𝜔. We show that through each point of a fiber of the twistor bundle over S6, a one-parameter family of Cayley structures passes. We describe the set of U(2) ×U(2)- invariant Hermitian metrics on S3 × S3 and find estimates of the sectional sectional curvature. We consider the space of left-invariant, almost complex structures on S3 × S3 = 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.