Homogeneous Riemannian structures in dimension three

In this note, we determine all the homogeneous structures on non-symmetric three-dimensional Riemannian Lie groups. We show that a non-symmetric three-dimensional Riemannian Lie group admits a non-canonical homogeneous structure if and only if its isometry group has dimension four.

Automated summary

In this study, we identify all the homogeneous structures on non-symmetric three-dimensional Riemannian Lie groups and prove that a non-symmetric three-dimensional Riemannian Lie group has a non-canonical homogeneous structure if and only if its isometry group has dimension four.