----- > [!proposition] Proposition. ([[second Bianchi identity]]) >The [[curvature form]] of any [[connection on a vector bundle|connection]] $A$ on a [[vector bundle]] $E$ is [[covariant constant section|covariant constant]]: $d_{A}F(A)=0$. Here, $d_{A}$ denotes the [[higher covariant derivative|(higher)]] [[covariant derivative]]. ^proposition > [!note] Note. > The term 'covariant constant' is applicable here because $F(A)$ is indeed a [[section]] — not of $E$ (obviously), but of $\Lambda^{2}T^{*}B \otimes \text{End }E$, [[differential form with values in a vector bundle|i.e.]], $F(A) \in \Omega_{B}^{2}(\text{End }E)$. ^note > [!proof]+ Proof. ([[second Bianchi identity]]) > Let $s \in \Gamma(E)$ and thrice-covariant-differentiate, exploiting associativity of function composition. On the one hand >$\begin{align} d_{A}(d_{A}d_{A})s &= d_{A} \big( F(A) \wedge s \big) \\ &= (d_{A}F(A)) \wedge s + \textcolor{Thistle}{F(A) \wedge d_{A}s} \end{align}$ where we have used the Leibniz rule characterizing the [[higher covariant derivative]]. On the other hand, $\begin{align} (d_{A}d_{A})d_{A}s &= \textcolor{Thistle}{F(A) \wedge d_{A}s}. \end{align}$ So $(d_{A}F(A)) \wedge s$ is zero. ----- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```