---- Everything is smooth, and Einstein summation convention is in effect. > [!definition] Definition. ([[curvature]]) > Let $A$ be a [[connection on a vector bundle|connection]] on a [[vector bundle]] $E \xrightarrow{\pi}B$. The **curvature form** of $A$ is defined in a local trivialization to be the [[differential form with values in a vector bundle|(matrix of differential) 2-form(s)]][^1] $F(A):=dA + A \wedge A \in \Omega_{B}^{2}(\text{End }E).$ > > > Further assuming a [[coordinate chart|local coordinates]] $(x^{k})_{k=1}^{n}$ on $B$ and assuming $\text{rank }E=m$, we use the various notations $\begin{align} > F(A) &= \boldsymbol F_{\ k \ell} \, dx^{k} \wedge dx^{\ell} \\ > &= [F ^{i}_{j, k \ell}] _{i \in [m]}^{j \in [m]} \, dx^{k} \wedge dx^{\ell} \\ > &= [F^{i}_{j, k \ell}\, dx ^{k} \wedge dx^{\ell}]_{i \in [m]}^{j \in [m]} \, \\ > &= [F ^{i}_{j}]_{i \in [m]}^{j \in [m]}, > \end{align}$ > where: > - $\boldsymbol F_{\ k \ell} \in C^{\infty}(B, \mathbb{R}^{m \times m})$; > - $F^{i}_{j, k \ell}:= (F_{\, k \ell})^{i}_{j} \in C^{\infty}(B)$; > - $F^{i}_{j} \in \Omega^{2}(B)$. > [!justification] More Information. > In light of the material in [[higher covariant derivative]], we get a chain of maps $\Gamma(E)=\Omega_{B}^{0}(E) \xrightarrow{d_{A}}\Omega_{B}^{1}(E) \xrightarrow{d_{A}} \Omega^{2}_{B}(E) \xrightarrow{d_{A}} \cdots $ > but, unlike in the case of the [[exterior derivative]] $d$, this is not a [[chain complex of modules|cochain complex]]: we need not have $d_{A}d_{A}=0$. Instead (moving to work locally), let $s \in \Omega_{B}^{*}(E)$ be of any degree, then $\begin{align} > d_{A}d_{A}s &= d_{A}(ds + A \wedge s) \\ > &= d(ds + A \wedge s) + A \wedge (ds + A \wedge s) \\ > &= \cancel{d d}^{0} s + d(A \wedge s) + A \wedge ds + A \wedge (A \wedge s) \\ > &= dA \wedge s + \cancel{(-1)^{1} A \wedge ds + A \wedge ds} + A \wedge A \wedge s \\ > &= (dA + A \wedge A) \wedge s, > \end{align}$ > where the penultimate step used the Leibniz rule for the [[exterior derivative]] (recall that $A$ 'is a $1$-form'). > > So in general $d_{A}d_{A} \neq 0$. But something special *does* still happen: it is 'an operator of algebraic type' — there is no differentiation of $s$ happening! $d_{A}d_{A}$ is characterized by some fixed (matrix of) two-form(s) depending on the [[connection on a vector bundle|connection]] $A$ — namely, the form $dA + A \wedge A$ so-defined above. Some consequences: > > 1. Not only is $d_{A}d_{A}$ $\mathbb{R}$-linear, but it is also $C^{\infty}$-linear: $d_{A}d_{A}(fs)=f \ d_{A}d_{A} s$ for all $f \in C^{\infty}(B)$. > 2. $d_{A}d_{A}$ is not only a local operator (as $d_{A}$ [[covariant derivative on a vector bundle#^properties|is]]), but is in fact a *pointwise* operator: $s_{1}(b)=s_{2}(b) \implies (d_{A}d_{A}s_{1})(b)=(d_{A}d_{A}s_{2})(b)$. To be convinced this is a big deal, consider how absurd such a claim would be for the ordinary Euclidean derivative $\frac{d}{dx}$. > > One might ask about composing more covariant derivatives. This turns out to not be interesting: [[second Bianchi identity]]. > > If $F(A)=0$ then $A$ is called a [[flat connection]]. [[flat connection#^basicexample|It turns out that]], in a sense, $F(A)$ is an obstruction to the covariant derivative and exterior derivative agreeing. genreal thing: $(\omega \wedge \eta)(X,Y)=\omega (X) \eta(Y)-\omega(Y) \eta(X)$ is (by definition of atlernative / wedge prouct ) ---- #### [^1]: Recall that notation $A \wedge A$ is understood to mean 'matrix multiplication, but with scalar products replaced with [[algebra of alternating multilinear forms|wedge products]]'. Explicitly: $(A \wedge A)_{j}^{i}=\Gamma_{pk}^{i} dx^{k} \wedge \Gamma_{j \ell}^{p} dx^{\ell}.$ Unlike for the wedge of ordinary ('scalar-valued') $1$-forms, this is not 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 > ```