Sur l'existence d'une prescription d'ordre naturelle projectivement invariante

Abstract

P.Lecomte has proposed to take into account the covariant derivatives used to build ordering prescriptions for the naturality of transformation properties and has conjectured that there exists an natural ordering prescription for differential operators of any orders between density bundles which in addition is invariant under projective changes of the covariant derivatives. We prove this conjecture by constructing a projectively invariant lift of a torsion-free connexion to a torsion-free connexion on (the positive part of) the total space of the bundle of all $a$-densities for nonzero $a$, by lifting the symbols in a projectively invariant way (they turn out to be in bijection to the space of all $\real^+$-equivariant divergence-free symmetric tensor fields on the total space), and by using the standard ordering procedure (`all the covariant derivatives to the right') on the total space. For Ricci-flat manifolds we show that this ordering prescription coincides --with the appropiate replacements-- with an explicit formula in $\real^m$ obtained by Duval, Lecomte and Ovsienko.