# Intuition of local connection forms

In the past I had the feeling that I understood the mathematics and ideas behind principal connections and connection one-forms rather well. However, while trying to explain these ideas in simple terms to a nonmathematical audience, I noticed that my understanding might not extend flawlessly beyond the pure formulas.

When you have a principal $$G$$-bundle $$\pi:P\rightarrow M$$, you can canonically define the vertical subbundle as the kernel of the projection $$\pi_*$$. A connection is then equivalent to a choice of complement inside the tangent bundle $$TP$$. Locally, where $$P|_U\cong U\times G$$, one can identify the vertical spaces with the Lie algebra $$\mathfrak{g}$$. A complement can then be defined by assigning to every basis $$\partial_i$$ of a tangent space on $$M$$ a ''horizontal lift'' $$\widetilde{\partial}_i:=\partial_i+\chi_i,$$ where $$\chi_i\in\mathfrak{g}$$ (these two terms should be mapped in the right way to a tangent space of $$P|_U$$ and extended to a local frame).

The associated connection one-form $$\omega$$ on $$P$$ is then, as far as I understand, the form that assigns to any vector field on $$P$$, at any point, the contribution in $$\mathfrak{g}$$ that does not arise from these $$\chi_i$$'s: $$X = \sum_i\lambda_i(\partial_i+\chi_i) + \omega(X),$$ for some scalars numbers $$\lambda_i$$. It measures the change in the fibres that does not arise from a mere change on the base manifold.

When we then, locally, pullback the connection one-form $$\omega$$ to a one-form on the patch $$U$$, we get $$A = s^*\omega.$$ By definition of the pullback I would then think that this evaluates to the following formula on any vector field: $$A(Y) = \omega(s_*Y)$$. By then combining the above statements, I would expect that $$A$$ assigns to any vector field on $$M$$ the difference between its pushforward (along a section) to the bundle and its horizontal lift, resulting in a linear combination of the $$\chi_i$$'s. Is this correct?

• I haven't read the details of your question, but I encourage anyone who is trying to understand how a connection on a principal bundle works to first learn about a connection on a vector bundle. This in turn defines a natural connection on the bundle of frames of the vector bundle, by applying the connection to each section in the frame. In that setting it is, at least for me, a lot easier to derive the formulas for a connection on a principal $GL(k)$-bundle since the action of $G$ on the frame bundle is quite explicit. After that, it's not hard to translate everything into the abstract case. Jan 15, 2022 at 20:05
• I know about the situation on vector bundles and how one obtains the formulas. These are not the problem. Its rather how to obtain a simple interpretation of what the local one-forms do. Jan 15, 2022 at 20:35
• I would support Deane's suggestion specifically with what you're saying. Think of the case where $P$ is the frame bundle (say orthonormal frames). Then when you pull back by a section you're looking specifically at how the frame is twisting. If $s(x)=(e_i(x))$, then $s^*\omega = (\omega_{ij}(x))$ and $\nabla e_i = \sum\omega_{ij}e_j$. You can interpret this, of course, in terms of local parallel frame fields. Jan 15, 2022 at 21:52
• Right. Twisting by the negative of $(s^*\omega)_p(Y)$ makes the frame $s(p)$ parallel as you move instantaneously in direction $Y$. Jan 16, 2022 at 0:16
• I don't think there's any contradiction. I'm saying that if you are driving along turning to the left, then you must compensate by turning to the right in order to go straight. :) Jan 16, 2022 at 16:43