Left-Invariant Einstein-like Metrics on Compact Lie Groups

Abstract

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

Keywords

• homogeneous space
• compact Lie groups
• Einstein-like metric
• \({\mathcal{A}}\)-metric
• ℬ-metric