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  • 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.

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The Euler density is defined as $\mathcal{R}^n$ in these equations

enter image description here

enter image description here

Its integral over a manifold is the Euler characteristic up to a constant. That is its main significance. It also means that the integral is a topological invariant and vanishes when the Euler characteristic is zero.

In 2-d the Euler characteristic is the Ricci scalar.

This answer and a section of this blog post blog post have more information on the Euler density.

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