On almost (para)complex Cayley structures on spheres S^2,4 and S^3,3

It is well known that almost complex structures exist on the six-dimensional sphere S⁶ but the question of the existence of complex (ie, integrable) structures has not been solved so far. The most known almost complex structure on the sphere S⁶ is the Cayley structure which is obtained by means of the vector product in the space R⁷ of the purely imaginary octaves of Cayley Ca. There is another, split Cayley algebra Ca', which has a pseudo-Euclidean scalar product of signature (4,4). The space of purely imaginary split octonions is the pseudo-Euclidean space R^3,4 with a vector product. In the space R^3,4, there are two types of spheres: pseudospheres S^2,4 of real radius and pseudo sphere S^3,3 of imaginary radius. In this paper, we study the Cayley structures on these pseudo-Riemannian spheres. On the first sphere S²^, the Cayley structure defines an orthogonal almost complex structure J; on the second sphere, S^³, the Cayley structure defines an almost para-complex structure P. It is shown that J and P are nonintegrable. The main characteristics of the structures J and P are calculated: the Nijenhuis tensors, as well as fundamental forms and their differentials. It is shown that, in contrast to the usual Riemann sphere S⁶, there are (integrable) complex structures on S^2,4 and para-complex structures on S^3,3.

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