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

Abstract

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