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I find the following viewpoint helpful to translate between the different incarnation of a connection. To every vector bundle $\pi: E \to M$ (in your case $E = TM$) we have an associated exact sequen …

The easiest way to see that the norm of the curvature corresponds to the energy is to consider the special case of an abelian U(1)-Yang-Mills theory (i.e. electrodynamics). If you write out the norm s …

First note that the adjoint bundle $ad(E_G)$ can be canonically identified with the vertical tangent bundle $V E_G / G$: send the pair $(p, \xi)$ consisting of a point $p \in E_G$ and a Lie algebra el …

Ad 1.: Since every vector can be decomposed in its horizontal and vertical part. Thus it is enough to consider the case a) where all vectors are horizontal (this is trivial) and b) where at least one …

Finally, equivalence classes of flat connections are parametrized by $Hom(\pi_1(M), U(1)$. …

Let us assume that the growth condition at infinity is implemented by requiring that all fields (connections and gauge transformations) extend to a given compactification $M$ of $\mathbb R^4$. … The center is the minimal orbit type in the sense that the space of connections whose stabilizer subgroup is the center is dense in the space of all connections (this is a consequence of the approximation …

The space of principal connections on $P$ can be identified with sections of an affine bundle $QP \to M$. … In order to realize $QP$ as an associated bundle, we need a first-order jet bundle of $P$ (since connections are first-order operators). …