On harmonic tensors on three-dimensional Lie groups with left-invariant Riemannian metric

On half conformally flat four-dimensional Lie algebras

“…It follows from (6) and (7) that the components R ijkt of the curvature tensor, the components r sm of the Ricci tensor, the components W pqef of the Weyl tensor, the elements of the matrices , and the scalar cur vature s are functions of the structure constants and the components g ij of the metric tensor (see [2,6]). = Dα, = -= -D < 0.…”

“…In ᒄ, we fix a basis E 1 , E 2 , E 3 , E 4 of left invariant vector fields on G and set (6) where the { } are the structure constants of the Lie algebra ᒄ, the {g ij } are the components of the metric tensor, and the are the Christoffel symbols of the second kind. Setting c ijs = g ks , we obtain the follow ing expressions for the Christoffel symbols of the first and the second kind:…”

About the Schouten-Weyl tensor on 3-dimensional Lorenzian Lie groups

“…The previous paper was a continuation of J. Milnor's paper [3] about the left-invariant Riemannian metrics on three-dimensional Lie groups. A classification of the three-dimensional metric Lie algebras of the Lie groups with a left-invariant Riemannian metric and trivial divergence and curl of the Schouten-Weyl tensor is given in [4].…”