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 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, 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, the Lie derivative along a vector field is
here, the notation means taking the partial derivative with respect to the coordinate. Q.E.D.

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