Conformal invariant operators on Riemmannian manifolds

In this paper we study the decompositions problem, introducing a (r, r) -tensor algebra, r > 2. of conformal invariant operators, interpreting its elements P like endomorphisms on the vector space of (1, r - 1) -tensors. For geometrical reasons, the emphasis is on the family of conformal projections and on conformal splittings of tensors, generalizing the decomposition of Singer-Thorpe-Nomizu, given for the space of the curvature tensors. The extension of P to the set of (1, r - 1) -tensor fields and to the affine space $\mathcal{A}^1$_r-1$(M)$, generated by the parallel affine spaces of geometrical object fields of type (1, r - 1), having the difference or the skew symmetric part with respect to a pair of indices a (1, r - 1) -tensor field, leads to the problem of conformal decompositions of tensor fields and connections. As application, we study a closed diagram via conformal projections, which reflects the conformal gauge invariance of the exotic conformal Thomas connection and exotic Weyl conformal curvature tensor field. Are obtained invariants for some transformations of geometrical object fields, in particular connections, generalizing the known properties of the Weyl conformal curvature tensor field. Key words: conformal invariant tensor algebra, conformal decompositions, conformal projections, gauge invariance. MSC 2000: 53A55, 53B10, 53C99, 15A72