This article contains proof of formulas in Riemannian geometry which involve the
Start with the Bianchi identity:.
Contract both sides of the above equation with a pair of
: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:, "
Q.E.D." ("return to article")
The last equation in Proof 1 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 derivativeof 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: " Q.E.D." ("return to article")
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