Curvature tensors and covariant derivatives

Summary The problems considered here are of two types.(i) What are implications of vanishing k-th covariant derivatives of curvature tensors?(ii) Under what conditions on curvature tensors, does the k-th covariant derivative ∇^kT=0 for a tensor T mean ∇T=0? Entrato in Redazione il 16 Maggio 1972.