Left-Invariant Einstein-like Metrics on Compact Lie Groups - An Wu; Huafei Sun (Mathematics, 2022)
In this paper, we study left-invariant Einstein-like metrics on the compact Lie group G. Assume that there exist two subgroups, <semantics>H⊂K⊂G</semantics>, such that <semantics>G/K</semantics> is a compact, connected, irreducible, symmetric space, and the isotropy representation of <semantics>G/H</semantics> has exactly two inequivalent, irreducible summands. We prove that the left metric <semantics>⟨·,·⟩t1,t2</semantics> on G defined by the first equation, must be an <semantics>A</semantics>-metric. Moreover, we prove that compact Lie groups do not admit non-naturally reductive left-invariant <semantics>B</semantics>-metrics, such as <semantics>⟨·,·⟩t1,t2</semantics>.