I am an analyst struggling through some geometry used in physics.

**Some background:** For some Lie group $G$, let $P$ be a principal $G$-bundle over the smooth manifold $M$. Let $\omega$ be a connection 1-form on $P$ (a "principal connection"). This is a Lie algebra-valued 1-form.

As for the curvature two-form, either you see definitions with no explanation at all, e.g. "The curvature is given by $\Omega = d\omega + \frac{1}{2}[\omega,\omega]$". This is obviously less than ideal for improving one's intuitive appreciation. Or one defines something called the exterior covariant derivative $D$ (see wiki) and then the curvature is simply the exterior covariant derivative of the connection one-form.

**The issue:** I can't get round the following observation though: From the point of view of the manifold $P$, $\omega$ is just a one-form with values in some vector space which happens to be $\mathfrak{g}$. Usually when you need to covariantly differentiate such an object, you would need a connection in a bundle $E \to P$ with fibre $\mathfrak{g}$, no? Then $\omega$ would be an $E$-valued one-form on $P$, *i.e.* in $\Gamma(E) \otimes\Omega^1(P)$, and you can differentiate covariantly in the normal way using the connection. Why is this scenario different?

The exterior covariant derivative $D$ satisfies $D^2\phi = \Omega\wedge\phi$... So you have some sort of covariant differentiation $D$ which differentiates forms $\eta$ taking values in $\mathfrak{g}$ and for which $D^2$ is some sort of curvature... but of $P \to M$ and not of the bundle in which $\eta$ is taking values. Isn't this strange? Or is this indeed just how things are? This prompts my more precise question:

Is the Lie algebra-valued curvature two-form on a principal bundle P the curvature of some vector bundle over P with fibre $\mathfrak{g}$?

havea connection $1$-form $\omega$. This is equivalent to an exterior covariant derivative. In order for one to be able to sensibly differentiate a $\mathfrak{g}$-valued form, one needs a connection. Try looking at Volume 1 of Kobayashi-Nomizu. Your questions should all be answered there. $\endgroup$isa $\mathfrak g$-valued 1-formon the principle bundle, satisfying some properties. Depending on your interests, a good place to read about the geometry might be Section 1 of Dan Freed's article "Classical Chern Simons Theory, Part 1" arxiv.org/abs/hep-th/9206021 . Freed defines the the curvature of a connection by working with the connection as a $\mathfrak g$-valued 1-form on the total space of the bundle; he defines it as $\Omega=d\omega+\frac12[\omega\wedge\omega]$, and then proves (or just quotes) various properties. $\endgroup$alwaysa $\mathfrak{g}$-valued $1$-form, whether the connection is on a principal $G$-bundle or on a vector bundle $E$ (with a right $G$-action). But even in the latter case $\mathfrak{g}$ isnotthe fiber of $E$. $\endgroup$5more comments