# Parallel Ricci condition - Status report and bibliography

First I'd like to point out that I'm not a mathematician but a physicist. Dealing with a (hopefully) new affine theory of gravity we have find that the equation of motion are not the usual Einstein's equation (In Besse's notation) $$r - \frac{1}{2} s\, g = 0,$$ but something related with a well known generalization known as parallel Ricci condition, i.e. $Dr = 0$. Our condition is (in components) $$\nabla_{[a} R_{b]c} = 0.$$

As you noticed, I found some information about parallel Ricci manifolds in Arthur Besse book. However I'd like to know if you know recent works on the subject, specially since our condition is different.

I found this post where @vladimir-s-matveev and @robert-bryant discuss a related topic.

Comments, explanations, references (books, lecture notes, articles or other) are all welcome. Thank you.

PD: I forgot to tell that we're interested in general integrability (solutions) of parallel Ricci condition, not necessarily on Riemannian (or Lorentzian) manifolds... they could be affine spaces, or metric spaces without metricity condition ($\nabla g \neq 0$).

• Is your affine connection torsionfree and is the dimension you are working in 4? Actually this condition is a natural condition in the framework of the projective geometry, what stays on the right hand side is sometimes is called the cotton-york tensor of a connection. It is not projectively invariant thouhg – Vladimir S Matveev Dec 12 '14 at 16:04
• Oh! yes, it is supposed to be a four-dimensional space and a torsion-free connection. – Dox Dec 12 '14 at 16:51
• According to 16.4 in Besse, your condition when $n=4$ and the connection is a metric connection is equivalent to the metric having harmonic Weyl tensor. Have you done a literature search on "harmonic Weyl tensor", which is a well-known condition? I know that you are interested in the non-metric case, but my question is whether you have at least considered what is known in the metric case. $$\$$ Also, for an affine torsion-free connection, your equation is very underdetermined, so it is likely that not much is known of a global nature and, locally, there will be many solutions. – Robert Bryant Dec 15 '14 at 12:31