Classification of harmonic homomorphisms between Riemannian three-dimensional unimodular Lie groups

PurposeThe purpose of this study is to classify harmonic homomorphisms ϕ : (G, g) → (H, h), where G, H are connected and simply connected three-dimensional unimodular Lie groups and g, h are left-invariant Riemannian metrics.Design/methodology/approachThis study aims the classification up to conjugation by automorphism of Lie groups of harmonic homomorphism, between twodifferent non-abelian connected and simply connected three-dimensional unimodular Lie groups (G, g) and (H, h), where g and h are two left-invariant Riemannian metrics on G and H, respectively.FindingsThis study managed to classify some homomorphisms between two different non-abelian connected and simply connected three-dimensional uni-modular Lie groups.Originality/valueThe theory of harmonic maps into Lie groups has been extensively studied related homomorphism in compact Lie groups by many mathematicians, harmonic maps into Lie group and harmonics inner automorphisms of compact connected semi-simple Lie groups and intensively study harmonic and biharmonic homomorphisms between Riemannian Lie groups equipped with a left-invariant Riemannian metric.