Projective and Conformal Schwarzian Derivatives and Cohomology of Lie Algebras Vector Fields Related to Differential Operators

Abstract

Let $M$ be either a projective manifold $(M,Pi)$ or a pseudo-Riemannian manifold $(M,g).$ We extend, intrinsically, the projective/conformal Schwarzian derivatives that we have introduced recently, to the space of differential operators acting on symmetric contravariant tensor fields of any degree on $M.$ As operators, we show that the projective/conformal Schwarzian derivatives depend only on the projective connection $Pi$ and the conformal class $[g]$ of the metric, respectively. Furthermore, we compute the first cohomology group of $Vect(M)$ with coefficients into the space of symmetric contravariant tensor fields valued into $delta$-densities as well as the corresponding relative cohomology group with respect to $sl(n+1,R).$

33 pages, no figures, Latex2e. A completely rewritten version, new results have been added; we extend the projective/conformal Schwarzian derivatives to tensor fields of any degree; we compute the first-cohomology group of the Lie algebra of smooth vector fields with values into the space of differential operators acting on contravariant tensor fields valued into d-densities, generalizing a result of Lecomte-Ovsienko