Interpretation for the curvature and monodromy of a connection - Reality check

Question

Let $P \to M$ be a principal $G$-bundle with connection form $\omega \in \Omega^1(P,\mathfrak{g})$. Here are the statements I'm basing my viewpoint on:

1. A connection is flat (vanishing curvature) iff it is locally the pullback of the maurer cartan form on $G$ i.e. for all $p \in P$ there's a neighborhood $p \in U$ and a map $f:U \to G$ satisfying $\omega|_U = f^*\omega_G$, where $\omega_G$ is the maurer cartan form of $G$ (this can be proved via an integrable distribution argument).
2. The monodromy of a flat connection is zero iff it is globally given by the pullback of the maurer cartan form on $G$. i.e. iff there's a function $f:P \to G$ satisfying $\omega=f^*\omega_G$.

Here's what I want to be able to say:

1. A connection $P$ is flat iff the $TP \to P$ admits covariantly constant local sections everywhere. Meaning, for every point $p \in P$ there's a neighborhood $p \in U$ and a section $X: U \to TP$ satisfying $\omega(X)=0$.

2. A flat connection on $P$ has zero monodromy iff $TP \to P$ admits a covariantly constant global section. Meaning there's a global section $\sigma : P \to TP$ satisfying $\omega(\sigma)=0$.

I get a bit confused though whenever I try to formalize a proof of the above. Sometimes I think the covariantly constant sections should be of the bundle $P \to M$ and that $TP \to P$ always has a covariantly constant section in the sense I defined, here I also get confused. My questions has two parts:

• Is the above interpretation a valid one? If so how can I formalize this with minimal effort and confusion? (a hint might suffice). If not how could it be fixed?
• Does this picture still hold when moving to the category of associated bundles? In particular, do covariantly constant local (or global) vector fields all arise in this manner?

Answer 1

As far as I can see, the interpretation you give is not correct. To explain, I'll use the usual terminology of the horizontal distribution, i.e. for a point $p\in P$, the horizontal subspace $H_pP\subset T_pP$ is the kernel of $\omega(p)$. By definition of a connection, this subspace is complementary to the (canonical) vertical subspace $V_pP$, the kernel of the tangent map of the bundle projection $\pi$ in $p$. Correspondingly, one calls a vector field horizontal, if its values all lie in the horizontal distribution.

So the condition you propose in 1. for flatness of the connection is that locally there are horizontal vector fields on $P$, whereas the condition in 2. which you intend to use for vanishing monodromy just is existence of a global horizontal vector field on $P$. But for any principal bundle endowed with a principal connection, there are many global horizontal vector fields, for example the horizontal lifts of vector fields on the base of the bundle.

The standard interpretation of flatness of a connection is that the horizontal distribution $H$ is involutive. This is equvialent to the criterion on flatness that you use in point 1. of the first block: If $H$ is involutive, then there are local integral submanifolds for the distribution $H$. By definition, the bundle projection restricts to a local diffeomorphism on each such integral submanifold. Local inverses to this are smooth sections $\sigma:V\to P$ for $V\subset M$ open such that $\sigma^*\omega=0$. Conversely, the image of such a section is an integral submanifold for $H$. Hence the existence of such sections is equivalent to flatness of the connection (and this should be the correct version of what you propose as 1.). A local section $\sigma$ defines a local trivialization $V\times G\to \pi^{-1}(V)$ via $(x,g)\mapsto\sigma(x)\cdot g$. Calling $f$ the second component of the inverse of this isomorphism, the pullback of the Maurer Cartan form on $G$ along $f$ has the same horizontal subspaces as $\omega$, which easily implies that the two connections coincide. Conversely, if you have a $f:U\to G$ such that which pulls back the Maurer Cartan connection to $\omega$, it is easy to see that $f$ is a submersion, so locally around each point $p\in U$, $f^{-1}(f(p))$ is a smooth submanifold of $P$, which has the same dimension as $M$, and it is easy to see that this is an integral submanifold for the horizontal distribution.

You can also bring the monodromy nicely into the picture of the horizontal distribution and conclude that vanishing monodromy is equivalent to a global section $\sigma:M\to P$ such that $\sigma^*\omega=0$ (which in particular implies that $P$ is a trivial principal bundle).

This carries over to associated vector bundles to a certain extent, in the form of local or global frames made up of parallel sections. For the existence of single local or global parallel sections, one does not need flatness of the connection. The right concept here is holonomy of a connection.