Invariant connections on non-irreducible symmetric spaces with simple Lie group

Abstract:

Consider a symmetric space $G/H$ with simple Lie group $G$. We demonstrate that when $G/H$ is not irreducible, it is necessarily even dimensional and noncompact. Furthermore, the subgroup $H$ is also both noncompact and non-semisimple. Additionally, we establish that the only $G$-invariant connection on $G/H$ is the canonical connection. On the other hand, we show that if $G/H$ has an odd dimension, it must be irreducible, and the subgroup $H$ must be semisimple. Finally, we present an explicit example, and we show that there exists no other torsion-free $G$-invariant connection on a symmetric space $G/H$ with semisimple Lie group $G$ which has the same curvature as the canonical one.