On almost (para)complex Cayley structures on spheres $\mathbf{S}^{2,4}$ and $\mathbf{S}^{3,3}$

N. K. Smolentsev

Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 53 (2018), pp. 22-38

Résumé

It is well known that almost complex structures exist on the six-dimensional sphere $\mathbf{S}^6$ 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 $\mathbf{S}^6$ is the Cayley structure which is obtained by means of the vector product in the space $\mathbf{R}^7$ of the purely imaginary octaves of Cayley $\mathbf{Ca}$. There is another, split Cayley algebra $\mathbf{Ca'}$, which has a pseudo-Euclidean scalar product of signature $(4,4)$. The space of purely imaginary split octonions is the pseudo-Euclidean space $\mathbf{R}^{3,4}$ with a vector product. In the space $\mathbf{R}^{3,4}$, there are two types of spheres: pseudospheres $\mathbf{S}^{2,4}$ of real radius and pseudo sphere $\mathbf{S}^{3,3}$ of imaginary radius. In this paper, we study the Cayley structures on these pseudo-Riemannian spheres. On the first sphere $\mathbf{S}^{2,4}$, the Cayley structure defines an orthogonal almost complex structure $J$; on the second sphere, $\mathbf{S}^{3,3}$, 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 $\mathbf{S}^6$, there are (integrable) complex structures on $\mathbf{S}^{2,4}$ and para-complex structures on $\mathbf{S}^{3,3}$.

Détail

Keywords: Cayley algebra, split Cayley algebra, split-octonions, vector product, almost complex structure, almost para-complex structure, six-dimensional pseudo-Riemannian spheres.
Mots-clés : $G2$ group