Proofs involving covariant derivatives

This article contains proof of formulas in Riemannian geometry that involve the Christoffel symbols.
Contracted Bianchi identities.
Proof.
Start with the Bianchi identity
Contract both sides of the above equation with a pair of metric tensors:
The first term on the left contracts to yield a Ricci scalar, while the third term contracts to yield a mixed Ricci tensor,
The last two terms are the same (changing dummy index "n" to "m") and can be combined into a single term which shall be moved to the right,
which is the same as
Swapping the index labels "l" and "m" yields
The covariant divergence of the Einstein tensor vanishes.
Proof.
The last equation in the proof above can be expressed as
where δ is the Kronecker delta. Since the mixed Kronecker delta is equivalent to the mixed metric tensor,
and since the covariant derivative of the metric tensor is zero (so it can be moved in or out of the scope of any such derivative), then
Factor out the covariant derivative
then raise the index "m" throughout
The expression in parentheses is the Einstein tensor, so 
this means that the covariant divergence of the Einstein tensor vanishes.
The Lie derivative of the metric.
Proof.
Starting with the local coordinate formula for a covariant symmetric tensor field formula_16, the Lie derivative along a vector field formula_17 is
here, the notation formula_19 means taking the partial derivative with respect to the coordinate formula_20.      "Q.E.D."     ("return to article")