Second Bianchi identity

If we take the exterior covariant derivative of the curvature, we get

\(\displaystyle \mathrm{D}\check{R}=0. \)

This is called the second Bianchi identity, and can be verified algebraically from the definition \({\check{R}\equiv\mathrm{d}\check{\Gamma}+\check{\Gamma}\wedge\check{\Gamma}}\). We can write this identity more explicitly as

\begin{aligned}0 & =\mathrm{D}\check{R}(u,v,w)\vec{a}\\ & =\nabla_{u}\check{R}(v,w)\vec{a}+\nabla_{v}\check{R}(w,u)\vec{a}+\nabla_{w}\check{R}(u,v)\vec{a}\\ & \phantom{{}=}-\check{R}([u,v],w)\vec{a}-\check{R}([v,w],u)\vec{a}-\check{R}([w,u],v)\vec{a}, \end{aligned}

where we have used the antisymmetry of \({\check{R}}\) and the covariant derivative acts on the value of \({\check{R}}\) as a tensor of type \({\left(1,1\right)}\). Working this expression into tensor notation and using the tensor expression for the torsion in terms of the commutator, we find that

\begin{aligned}0 & =\nabla_{e}R^{c}{}_{dab}+\nabla_{a}R^{c}{}_{dbe}+\nabla_{b}R^{c}{}_{dea}\\
& \phantom{{}=}-R^{c}{}_{dfe}T^{f}{}_{ab}-R^{c}{}_{dfa}T^{f}{}_{be}-R^{c}{}_{dfb}T^{f}{}_{ea},
\end{aligned}

or

\(\displaystyle R^{c}{}_{d[ab;e]}=R^{c}{}_{df[e}T^{f}{}_{ab]}, \)

and in the case of zero torsion, \({R^{c}{}_{d[ab;e]}=0}\).

Geometrically, the second Bianchi identity can be seen as reflecting the same “boundary of a boundary” idea as that of \({\mathrm{d}^{2}=0}\) when considering the exterior derivative of a 2-form, except that here we are parallel transporting a vector \({\vec{a}}\) around each face that makes up the boundary of the cube. As in the previous section, we can take advantage of the fact that \({\check{R}(v,w)\vec{a}}\) only depends upon the local value of \({\vec{a}}\), constructing its vector field values such that e.g. \({\vec{a}\left|_{p+\varepsilon u}\right.=\parallel_{\varepsilon u}(\vec{a}\left|_{p}\right.)}\), giving us

\begin{aligned}\varepsilon\nabla_{u}\check{R}(v,w)\vec{a} & =\check{R}(v\left|_{p+\varepsilon u}\right.,w\left|_{p+\varepsilon u}\right.)\vec{a}\left|_{p+\varepsilon u}\right.-\parallel_{\varepsilon u}\check{R}(v,w)\parallel_{\varepsilon u}^{-1}\vec{a}\left|_{p+\varepsilon u}\right.\\
& =\check{R}(v\left|_{p+\varepsilon u}\right.,w\left|_{p+\varepsilon u}\right.)\parallel_{\varepsilon u}\vec{a}-\parallel_{\varepsilon u}\check{R}(v,w)\vec{a}.
\end{aligned}

The first term parallel translates \({\vec{a}}\) along \({\varepsilon u}\) and then around the parallelogram defined by \({v}\) and \({w}\) at \({p+\varepsilon u}\), while the second parallel translates \({\vec{a}}\) around the parallelogram defined by \({v}\) and \({w}\) at \({p}\), then along \({\varepsilon u}\). Thus in the case of vanishing Lie commutators (e.g. a holonomic frame), we construct a cube from the vector fields \({u}\), \({v}\), and \({w}\), and find that the second Bianchi identity reflects the fact that \({\mathrm{D}\check{R}(u,v,w)\vec{a}}\) parallel translates \({\vec{a}}\) along each edge of the cube an equal number of times in opposite directions, thus canceling out any changes.

75.second-bianchi-identity

The above depicts how the second Bianchi identity reflects the fact that for vanishing Lie commutators, \({\mathrm{D}\check{R}(u,v,w)\vec{a}}\) parallel translates \({\vec{a}}\) along each edge of the cube made of the three vector field arguments an equal number of times in opposite directions, thus canceling out any changes. Above, \({\varepsilon\nabla_{u}\check{R}(v,w)\vec{a}=\check{R}(v\left|_{p+\varepsilon u}\right.,w\left|_{p+\varepsilon u}\right.)\parallel_{\varepsilon u}\vec{a}-\parallel_{\varepsilon u}\check{R}(v,w)\vec{a}}\) is highlighted by the bold arrows representing the path along which \({\vec{a}}\) is parallel translated in the first term, and by the remaining dark arrows representing the path along which \({\vec{a}}\) is parallel translated in the second term.

In the case of a non-vanishing commutator, e.g. \({[u,v]\neq0}\), we find that the cube gains a “shaved edge,” and that the extra non-vanishing term \({-\check{R}([u,v],w)\vec{a}}\) in \({\mathrm{D}\check{R}}\) maintains the “boundary of a boundary” logic by adding a loop of parallel translation of \({\vec{a}}\) in the proper direction around the new “face” created.

76.second-bianchi-commutator

The above depicts how in the case of a non-vanishing commutator, the extra term \({-\check{R}([u,v],w)\vec{a}}\) in \({\mathrm{D}\check{R}}\) maintains the cancellation of face boundaries by adding a loop \({L}\) around the new “shaved edge” created.

An Illustrated Handbook