Almost complex and almost para-complex Cayley structures on six-dimensional pseudo-Riemannian spheres

Abstract: In this paper we study almost complex and almost para-complex Cayley structures on six-dimensional pseudo-Riemannian spheres in the space of purely imaginary octaves of the split Cayley algebra $\mathbf{Ca}&#39;$. It is shown that the Cayley structures are non-integrable, their basic geometric characteristics are calculated. In contrast to the usual Riemann sphere $\mathbb{S}<sup>6$, there exist (integrable) complex structures and para-complex structures on the pseudospheres under consideration.