ON ALMOST (PARA)COMPLEX CAYLEY STRUCTURES ON SPHERES S-2,S-4 AND S-3,S-3

It is well known that almost complex structures exist on the six-dimensional sphere S-6 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 S-6 is the Cayley structure which is obtained by means of the vector product in the space R-7 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,R-4 with a vector product. In the space R-3,R-4, there are two types of spheres: pseudospheres S-2,S-4 of real radius and pseudo sphere S-3,S-3 of imaginary radius. In this paper, we study the Cayley structures on these pseudo-Riemannian spheres. On the first sphere S-2,S-4, the Cayley structure defines an orthogonal almost complex structure J; on the second sphere, S-3,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 S-6, there are (integrable) complex structures on S-2,S-4 and para-complex structures on S-3,S-3.