Connections which are harmonic with respect to general natural metrics

Abstract

We find all the general natural metrics and all the natural diagonal metrics on TM with respect to which any (non)linear connection on a (pseudo)-Riemannian manifold (M,g) (viewed as an almost product structure on TM) is harmonic. Moreover, we give necessary and sufficient conditions such that the Levi-Civita connection of g, the (non)linear connections on (M,g), and some special (1,1)-tensor fields on TM are harmonic with respect to any general natural metric G. We also study the harmonicity of these connections and of several (1,1)-tensor fields with respect to the natural diagonal metrics, and in particular, we obtain that the Levi-Civita connection of g is harmonic with respect to the Sasaki metric on TM.