The missing term vanishes for metric-compatible connections. (Thanks to Bert Janssen for pointing this missprint to us.)
(Thanks to Gregory Giecold for this one).
We can immediately define the dual basis of 1-forms {ea = ea μdxμ} defined by...
(Thanks to Gregory Giecold for this one).
(Thanks to Masoud Soroush).
The last equation can be rewritten in the form
and, using the Ricci identities for the l.h.s.
Now, using the Bianchi identities for the metric-compatible torsinless curvature we get
Combining this equation with the Ricci identity
we get Eq. (1.108).
Another important case is when k = p + 2 and F(k) is the field-strength of
the potential A(p+1), so F(p+2)μ1μ(p+2) = (p + 2)∂[μ1A(p+1)μ2
μ(p+2)].
(Thanks to Diego Blas)
We can also take the divergence of the tensor and contract it with the volume
element dd-n+1Σ
μ1μn-1.