A (1, 1)-tensor field φ on a pseudo-Riemannian manifold (M, g) determines a map φ: T M → T M, where T M denotes the tangent bundle of M. The main purpose of this paper is to determine a necessary and sufficient condition for φ: (T M, g~C) → (T M, g~C) to be harmonic; where g~C is the (pseudo-Riemannian) complete lift metric as introduced by Yano and Ishihara in [18]. Our main result is that the harmonicity condition is equivalent to ▽~* φ = 0, where ▽~* is the formal adjoint of the Levi Civita connection ▽ of (M, g).
展开▼
