I would like to know as to what is the definition and significance of what are called "Euler density" and "Weyl invariants" (of weight $-d$ on a $d-$manifold)

Do many (which?) of them vanish when integrated on a compact $d$-manifold? (at least $S^d$?)

And for the case of $d=2$ do all of these just collapse into one quantity the scalar curvature?

One might want to look at this (partially answered) previous question of mine to see my motivations.