Polars and antipodal sets for the outer $3$-symmetric space ${\mathbb{S}}^7\times {\mathbb{S}}^7$

Abstract

We determine the polar and the maximal antipodal set $P$ for the outer 3-symmetric space ${\mathbb{S}}^7\times {\mathbb{S}}^7={\mathrm{Spin}}_8/G_2$ where the 3-symmetric structure is given by the triality automorphism $\tau$ on ${\mathrm{Spin}}_8$. It turns out that $P$ has three elements. The 3-symmetric structure extends to a (non-abelian) $S_3$-structure which we also investigate.

8 pages