Bianchi identities

Remarks

For part 1, apply $d$ to the structure equation: $d\Omega = \frac12 d[\omega \wedge \omega] = \frac12\big([d\omega \wedge \omega] - [\omega \wedge d\omega]\big)$, substitute $d\omega = \Omega - \frac12[\omega \wedge \omega]$, and use the Jacobi identity to see that $[[\omega \wedge \omega] \wedge \omega] = 0$. Part 1 is often written $D\Omega = 0$, with $D$ the exterior covariant derivative. For the first identity in part 2, expand $R$ and use $\nabla_X Y - \nabla_Y X = [X, Y]$; the cyclic sum reduces to the Jacobi identity for the Lie bracket. Without the torsion-free hypothesis the identities get extra terms involving the torsion and its covariant derivative. Example: tracing the second identity for the Levi-Civita connection gives $\mathrm{div}\, \mathrm{Ric} = \frac12 \, d\,\mathrm{scal}$, so the Einstein tensor $\mathrm{Ric} - \frac12 \mathrm{scal}\, g$ is divergence-free.

Going deeper

For a student

The first identity, together with the symmetries of the Riemann tensor, cuts down its number of independent components to $n^2(n^2 - 1)/12$ in dimension $n$. The contracted second identity is the reason the left side of Einstein's field equations is divergence-free, which matches conservation of energy and momentum. In gauge theory, part 1 says that the field strength $F$ of a gauge field satisfies $D F = 0$; for electromagnetism this gives the two homogeneous Maxwell equations. Part 1 is also the key step in showing that the forms $f(\Omega)$ of Chern–Weil theory are closed.