Harmonic Almost Complex Structures with Respect to General Natural Metrics

Abstract

We continue here the study initiated in [9] on the harmonicity of certain geometric objects on the total space TM of the tangent bundle of a Riemannian space form (M(c), g). Precisely, in this paper we find all the general natural metrics on TM, with respect to which the canonical almost complex structure J on TM is harmonic. We also study the harmonicity of this tensor field with respect to the natural diagonal metrics. In particular, we obtain that J is harmonic with respect to the Sasaki metric on TM if and only if the base manifold is flat.