Product structures and decompositions of $\mathbb{Z}_2\times\mathbb{Z}_2$-symmetric spaces

RESUMO

In this paper, we establish the decomposition and product theory of $\Gamma$-symmetric spaces, where $\Gamma$ is isomorphic to $\mathbb{Z}_2\times\mathbb{Z}_2$. These results lead to the local classification of semisimple Riemannian $\mathbb{Z}_2\times\mathbb{Z}_2$-symmetric spaces of nontrivial type. Furthermore, we identify notable phenomena specific to $\mathbb{Z}_2\times\mathbb{Z}_2$-symmetric spaces: the existence of distinct product structures and that of an example with the trivial infinitesimal isotropy representation.