A connection on a fiber bundle is a particular kind of distribution, so we can think of the curvature of it. In this case, we have an isomorphism
$$ TE/H \equiv V $$being $V$ the vertical bundle of here, so the structure 1-form specified here takes value in the tangent spaces to the fibres.
If $v$ is the connection 1-form, then the curvature is the map
$$ R:\mathfrak{X}(E)\times \mathfrak{X}(E) \rightarrow \Gamma(V) $$such that
$$ R(X,Y)=v([P_H(X),P_H(Y)]), $$where $P_H=id-v$ is the horizontal projection in $TE$.
I guess that, again can be seen like a vector bundle map
$$ R:\Lambda^2E\rightarrow V $$or. for $p\in E$:
$$ R_p:T_p E\times E_p P\rightarrow V_p $$Adapted from Wikipedia.
If we are in the context of a connection not in a general bundle but in a $G$-principal bundle $P$, being $\mathfrak{g}$ the Lie algebra of the Lie group $G$, then the curvature is a $\mathfrak{g}$-valued 2-form on $P$:
$$ \Omega_p: T_p P\times T_p P \rightarrow \mathfrak{g} $$and can be written in a simpler expression. Since in this case the connection 1-form $\omega$ can be seen as $\mathfrak{g}$-valued 1-form it turns out that
$$ \Omega(X, Y)=d \omega(X, Y)+\frac{1}{2}[\omega(X), \omega(Y)] $$and the curvature of the connection is defined by
$$ \Omega=d\omega+\frac{1}{2}[\omega,\omega] $$for certain bracket defined in Lie algebra-valued differential forms.
The Riemann curvature tensor is a special case of curvature of a connection on a principal bundle.
(see wikipedia )
In this case the connection would be a vector bundle connection and the connection 1-form can be expressed in local frames with the connection 1-forms $\Theta$, or better said, a $\mathfrak{gl}(n)$-valued 1-form. Then the curvature is
$$ \Omega=d \Theta-\Theta \wedge \Theta $$or, in components,
$$ \Omega_{j}^{i}=d \Theta_{j}^{i}-\sum_{k} \Theta_{j}^{k}\wedge \Theta_{k}^{i} $$called the curvature 2-forms.
If the vector bundle is the tangent bundle and the connection is the Levi-Civita connection of a metric then this curvature is the same as the Riemann curvature tensor of the metric.
See the note Gaussian curvature#Relation to the curvature of a connection.
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Author of the notes: Antonio J. Pan-Collantes
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