On invariant operations on a manifold with connection or metric
The paper starts with a readable survey of the classical reduction theorems for some natural operators on connections and Riemannian metrics and of the nonlinear Peetre theorem, which implies the finiteness of the order in several interesting cases. Let \(Q_ \tau P^ 1\), \(S_ +^ 2 T^*\), \(T^{(0,r)}\) or \(\Lambda T^*\) denote the natural bundle of symmetric linear connections, of Riemannian metrics, of \(r\)-times contravariant tensor fields or of exterior forms, respectively. The author first deduces that all natural operators \(Q_ \tau P^ 1\times T^{(0,r)}\to\Lambda T^*\), \(r>0\), are generated by the Chern forms, the alternation and the exterior differentiation. Then it is proved that there is no natural operator \(S_ +^ 2 T^*\times T^{(0,r)}\to\Lambda T^*\), \(r>0\), with positive weight and all conformal operators of this type are generated by the Pontryagin forms, the alternation and the exterior differentiation. The cases \(r=0\) are discussed separately.
scientific article; zbMATH DE number 146254