Riemannian curvature in a vector bundle and the curvature 2-form in principal bundle On a vector bundle, I can define the Riemannian curvature tensor
$$\nabla_X \nabla_Y - \nabla_Y \nabla_X - \nabla_{[X, Y]}$$
In the frame bundle of this vector bundle, I can define the curvature form
$$\textrm{d}\omega(X^H, Y^H) = \textrm{d}\omega(X, Y) + \frac{1}{2}[\omega\wedge\omega](X, Y)$$
How to directly see (a geometrical interpretation) that the two is equivalent?
 A: I write this as an answer since it is too long for the comment section.
The curvature form of a connection form $\omega\in\Omega^1(P,\mathfrak{g})$, that is
$$F(X,Y):=\textrm{d}\omega(X^H, Y^H) = \textrm{d}\omega(X, Y) + \frac{1}{2}[\omega\wedge\omega](X, Y)$$
is also a differential form $F\in\Omega^2(P,\mathfrak{g})$ with values in the Lie algebra $\mathfrak{g}$ of the group $G$, but contrary to the connection form it is horizontal. Moreover, along the fibers of $P$ it transforms under the the adjoint representation of $G$, hence by the isomorphism in the first two comments it is associated with a differential form on the base manifold $M$ which takes values in the adjoint bundle
$$\textrm{Ad}(P)=P\times_\text{ad}\mathfrak{g}.$$
In the link in the comments you can now find how some other representation $\rho$ of $G$ on some space $W$ gives rise to a bundle homomorphism from $\textrm{Ad}(P)$ to the endomorphism bundle of $E=P\times_\rho W$. This bundle homomorphism extends to the section spaces and tothe spaces of differential forms with values in these bundles.
The curvature form of $\omega$, as the $\textrm{Ad}(P)$-valued form on $M$, and the curvature form $\nabla^\omega_X \nabla^\omega_Y - \nabla^\omega_Y \nabla^\omega_X - \nabla^\omega_{[X, Y]}$ of the covariant derivative $\nabla^\omega$ on $E$ induced by $\omega$, as the $\mathrm{End}(E)$-valued form on $M$, are related by the the above homomorphism. If I remember correctly I'm missing some details, but it's essentially that.
And no, I don't say that they don't have a geometrical interpretation, but the single terms cannot have a rigorous correspondence term-wise, since the natural identifications and bundle homomorphisms from above only identify forms with the respective behaviors (e.g. being horizontal and transforming under $\rho$ for forms on $P$).
However, I guess to work out whether there is some correspondence you'd have to dive into the dirty work and prove the above correspondence of the different notions of curvature (and work out the details that I have skipped). I wouldn't expect so, but maybe there is somebody with a deeper view on that.
