I have to deal the reduction of Riemann^2. I cannot understand how I should make it use the symmetries of Riemann tensor when I split indices:
{\alpha,\beta,\gamma,\delta,\kappa,\lambda,\eta,\mu,\nu,\rho,\sigma,\chi#}::Indices(full, position=independent);
{a,b,c,d,m,n,p,q,r,s,t,u,v,w,x,y,z,m#,a#,b#,c#,d#}::Indices(subspace, position=independent, parent=full);
\partial{#}::PartialDerivative.
g_{\mu\nu}::Metric.
g^{\mu\nu}::InverseMetric.
g_{\mu? \nu?}::Symmetric.
g^{\mu? \nu?}::Symmetric.
h_{m n}::Metric.
h^{m n}::InverseMetric.
\delta^{\mu?}_{\nu?}::KroneckerDelta.
\delta_{\mu?}^{\nu?}::KroneckerDelta.
F_{m n}::AntiSymmetric.
todo:= R_{\alpha \beta \gamma \delta} R_{\mu \nu \rho \sigma} g^{\alpha \mu} g^{\beta \nu} g^{\gamma \rho} g^{\delta \sigma} ;
split_index(_, $\mu, m, 3$, repeat=True);
It does not regognise that many expression (taking into account simmetries of Riemann and also the metric contractions) are basically the same. How to do it? For now I set by hand the symmetries of Riemann. but still many terms should collapse together because due to metric conractions they are the same, but it does not recognise it. I tried rename dummies, canonicalise and many other commands...