Ricci-corrected derivatives and invariant differential operators

Abstract

We introduce the notion of Ricci-corrected differentiation in parabolic geometry, which is a modification of covariant differentiation with better transformation properties. This enables us to simplify the explicit formulae for standard invariant operators given in work of Cap, Slovak and Soucek, and at the same time extend these formulae from the context of AHS structures (which include conformal and projective structures) to the more general class of all parabolic structures (including CR structures).Comment: Substantially revised, shortened and simplified, with new treatment of Weyl structures; 24 page

Conformally invariant powers of the Dirac operator in Clifford analysis

“…[5], and a lot of different special constructions in the general case [1, 6--8]. It is often possible to write down explicit formulae for lower-order corrections in the homogeneous case and there is a broad class of operators in the general case, where an algorithm for the computation of lower-order corrections is available (see Reference [9]). Powers of the Laplace and the Dirac operator are very difficult to construct in general case, it is even possible to prove that in even dimensions, there is a critical power (depending on dimension) such that no higher power of the Laplace operator exists on a general conformal manifold [10].…”

Higher symmetries of the Laplacian

“…The use of Ricci-corrected covariant derivatives in this regard, as in (23) and (25), has a long history (the idea appears in [55]; see also [12], [22]) and has recently been formalised in [9]. For example, if we set w = s−1 in the formulae for τ k , only τ s is nonzero and (24) implies that τ s is conformally invariant.…”

On Nurowski’s conformal structure associated to a generic rank two distribution in dimension five

“…To obtain a formula forΦ(γ ) − Φ(γ ), we have to take (11), then subtract the analogous terms with ξ and η exchanged and further subtract (12). Using (13) and (14), we obtain…”