On the relation between conformally invariant operators and some geometric tensors

Abstract

In this note we introduce and study some new tensors on general Riemannian manifolds which provide a link between the geometry of the underlying manifold and conformally invariant operators (up to order four). We study some of their properties and their relations with well-known geometric objects, such as the scalar curvature, the $Q$-curvature, the Paneitz operator and the Schouten tensor, and with the elementary conformal tensors $\{T_{m,\alpha }^u\}$ and $\{X_{m,\mu }^u\}$ on Euclidean space introduced in [7] and [6].