On operators associated with tensor fields (Q641696)

The author considers a \(C^\infty\)-manifold \(M\) together with a \((1,1)\)-tensor field \(\phi\) on \(M\). An \((r,s)\)-tensor field \(t\) on \(M\) is said to be pure with respect to \(\phi\) if \[ t(\phi(X_1),\dots,X_s,\xi^1,\dots,\xi^r)=t(X_1,\dots, \phi(X_i),\dots,\xi^r)= t(X_1,\dots, {}'\phi(\xi_j),\dots,\xi^r) \] for all \(1\leq i \leq s\) and all \(1\leq j\leq r\). Here, \('\phi\) denotes the adjoint of \(\phi\). The main topic of the paper are operators which are applied to pure tensor fields. Among them are Tachibana, Vishnevsjii and Yano-Ako operators. We refer to the author's erratum [ibid. 99, No. 1--2, 179--181 (2010; Zbl 1229.53013)] for correct versions of Definition 2.1 and Theorem 2.11.