Curvature of a connection (Ehresmann)

General fiber bundles

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

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

such that

where $P_H=id-v$ is the horizontal projection in $TE$.
I guess that, again can be seen like a vector bundle map

or. for $p\in E$:

Adapted from Wikipedia.

Principal $G$-bundles

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 $P$:

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

and the curvature of the connection is defined by

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.

Vector bundles

(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 $\mathrm{\Theta }$, or better said, a $\mathfrak{gl}(n)$-valued 1-form. Then the curvature is

or, in components,

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 related to the Riemann curvature tensor of the metric by the Cartan's second structural equation.

See the note Gaussian curvature#Relation to the curvature of a connection.