  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 $sl(n+1,R).$
