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Let $(M,g_{ab})$ be a Riemannian manifold. I know of the following scalars that one can construct them out of the metric and its derivatives:

  1. Ricci scalar $R$
  2. $R_{ab}R^{ab}$
  3. $R_{abcd}R^{abcd}$

Are there more independent scalars that can be constructed or are all diffeomorphism invariant scalars functions of these?

What about $R_{ab}R_{cd}R^{acbd}$?

I am looking for a list of independent scalars such that any other scalar can be expressed as a function of scalars in the list.

Any relevent references are most welcome.

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On searching about it on the internet, I came across this Wikipedia page and this article.Though this is perhaps limited to 4 dimensions and Lorentzian metrics and does not settle the general question asked above. – user90041 Nov 24 '13 at 4:44

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