Harmonic endomorphism fields
A (1,1)-tensor field \(\phi\) on a pseudo-Riemannian manifold \((M,g)\) determines a map \(\phi:TM\to TM\), where \(TM\) denotes the tangent bundle of \(M\). The main purpose of this paper is to determine a necessary and sufficient condition for \(\phi:(TM,g^C)\to (TM,g^C)\) to be harmonic, where \(g^C\) is the pseudo-Riemannian complete lift metric introduced by Yano and Ishihara. The main result is that the harmonicity condition is equivalent to \(\nabla^*\phi=0\), where \(\nabla^*\) is the formal adjoint of the Levi-Civita connection \(\nabla\) of \((M,g)\). The authors also dicuss the relationship between harmonic endomorphism fields and the notion of harmonic (0,2)-tensors introduced and investigated in [\textit{B. Y. Chen} and \textit{T. Nagano}, J. Math. Soc. Japan 36, 295-313 (1984; Zbl 0543.58015)].
scientific article; zbMATH DE number 1028999