I am interested in learning tricks that people have developed to speed up / clean up calculations in Riemannian Geometry.
I am hopeful about this question because there is often a lot of symmetry in the formulas, so I am wondering if someone has figured out a way to systematize that observation.
I have seen things like the cycle notation $C ( F(x,y,z)) = F(x,y,z) + F(y,z,x) + F(z,x,y)$, and I am looking for similar tricks. For instance, something I have just started to do is add signs changes to the cycle notation in subscripts, which makes the definition of the Levi-Civita connection look nice: $\langle Z, \nabla_Y X \rangle = 1/2( C_{++-} ( X\langle Y, Z \rangle) + C_{-++}( \langle [X,Y], Z \rangle) )$. The main problem with tricks like this is that they are not very convenient for more elaborate computations (which is still okay in my current opinion).
I also know about the Einstein sum trick.