Left-Invariant Einstein-like Metrics on Compact Lie Groups

An Wu and Huafei Sun
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An Wu: School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 102488, China
Huafei Sun: School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 102488, China

Mathematics, 2022, vol. 10, issue 9, 1-18

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 ⟨ · , · ⟩ t 1 , t 2 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 ⟨ · , · ⟩ t 1 , t 2 .

Keywords: homogeneous space; compact Lie groups; Einstein-like metric; \({\mathcal{A}}\)-metric; ?-metric (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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