I recently asked about the intuition behind connection 1-forms on pricipal bundles (Intuition behind connection 1-forms and Ehresmann connections). Thanks to the phenomenal answer I received, I now have a good idea of the geometry connection 1-forms and Ehresmann connections are trying to capture.
I would like to build a similar intuition for curvature 2-forms. I am familiar with the technical definition that if $\pi: P \rightarrow M$ is a principal $G$-bundle, $\pi_H: TP \rightarrow H$ the projection map from the tangent bundle on $P$ onto the horizontal vector bundle $H$, and $A$ a connection 1-form, then the curvature 2-form $F$ is defined as
$$F(X,Y) = dA(\pi_H(X), \pi_H(Y)) \tag{1}$$
for all $X, Y \in T_pP$ and $p \in P$.
Equivalently, we may define $F$ through the structure equation:
$$F = dA + \frac{1}{2} [A \wedge A] \tag{2}$$
where
$$[A \wedge A](X,Y) = [A(X), A(Y)] – [A(Y), A(X)]$$
with $[\cdot]$ being the Lie bracket on the Lie algebra $\mathfrak{g}$.
An answer given by user Tim van Beek to this question (Geometric interpretation of connection forms, torsion forms, curvature forms, etc) says
The curvature measures how much the parallel transport on the surface deviates from the parallel transport in the ambient $\mathbb{R}^3$.
and another answer given by user ಠ_ಠ says
Now, to interpret the curvature, lets just think about any bundle $\pi: E \to M$ with connection $\nabla$. Let's use the notation $x \sim y$ to indicate that two points are infinitesimal neighbours. If we have three points $x,y,z \in M$ such that $x \sim y$, $y\sim z$ and $z\sim x$. These three define an infinitesimal 2-simplex in $M$. Lets consider the transport around (the boundary of) this simplex:
$$R(x,y,z) = \nabla(z,x) \circ \nabla(y,z) \circ \nabla(x,y): E_x \to E_x$$
If we transport a point $w \in E_x$ around the simplex, we have no guarantee that we end up back where we started. This is precisely the notion of curvature. The curvature measures the extent to which parallel transport around infinitesimal 2-simplices deviates from the identity.
Other posts answer in terms of Riemannian metrics, which I'm not too familiar with, or in terms of other concepts such as parallel transport or covariant derivatives, which the book I am using has not introduced yet.
Based on the above answers, my understanding is that the connection 2-form measures how "flat" the transport of vectors are in a given connection. The picture I have in mind is that flat curvature means the vectors are transported around in a perfect circle, whereas some curvature means the vectors are transported around in a sort of spiral.
Is this the right picture to have in mind? If so, how does this naturally lead to the definition (1) or (2)? For example, why does the exterior derivative appear here? It seems natural to measure such a thing by mapping the curvature into a scalar field. Instead, why does curvature take values in a Lie algebra?
Best Answer
I think it's easier to understand this first on a vector bundle, then on the frame bundle associated with the vector bundle, and finally a principal bundle. Using a connection, you can always extend a vector in a vector bundle over a point to a parallel (i.e., "constant") section along a curve through that point. The natural next question is whether you can extend it to a parallel section on a surface containing the point. It's not necessarily possible, and curvature arises naturally when you formulate the obstruction infinitesimally.
On the frame bundle, it's the same ides, except you are trying to extend a frame over a point to a parallel frame on a surface containing that point.
This view of curvature is known as holonomy. Here is an explanation of it for a vector bundle.