I'm taking a GR course at the moment, completely stumped on this step here: starting from the Bianchi identity:
$$R^n{}_{ikl;m}+R^n{}_{imk;l}+R^n{}_{ilm;k} = 0 $$
Then it says "Contracting the Bianchi identity..."
$$R^i{}_{klm;n}+R^i{}_{knl;m}+R^i{}_{kmn;l} = 0 $$
How does this work and what does it actually mean to "contract" it?
To contract a tensor is to set two of the indices equal and sum over them, so given a tensor $A^i_j$ the contraction is $A=A^i_i=A^1_1+A^2_2+A^3_3+A^4_4$ The Bianchi identities you list have five indices. To contract them, you would set some pair equal and sum over them. Your second version is the same as the first, it just has the indices renamed. Mathworld shows the usual contraction is $n$ with $k$ in the first equation
I guess you have some typos in your question (or your lecture notes), since the two lines you give just differ by the index names (as already commented) . Starting from $$ R^n_{ikl;m} +R^n_{imk;l} +R^n_{ilm;k} =0 $$ you can rewrite that with the symmetry relations $R^n_{ikl}=- R^n_{ilk}=-R^i_{nkl}$ to $$ R^n_{ikl;m} - R^n_{ikm;l} +R^n_{ilm;k}=0. $$ Now let $n=k$, use Einstein's sum convention, and the definition of the Ricci tensor $R_{ik} :=R^j_{ijk} $: $$ R^n_{inl;m} - R^n_{inm;l} +R^n_{ilm;n}=0\\ R_{il;m} - R_{im;l} +R^n_{ilm;n}=0 $$
