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

摘要

Book [1] contains one of the most recent surveys on Riemannian metrics with harmonic Weyl tensor. In the three-dimensional case, the Weyl tensor is trivial; its role is played by the Schouten–Weyl (Cotton) tensor. This paper studies three-dimensional Lie groups with left-invariant Riemannian metric and almost harmonic (i.e., having curl and divergence zero) Schouten–Weyl tensor. Moreover, by using the convolution of the Schouten–Weyl tensor in the direction of any vector, we define an antisymmetric 2-tensor and study the structure of the three-dimensional Lie groups and algebras with left-invariant Riemannian metric for which this tensor is harmonic.