I'm trying to understand some basic definitions in relation to line bundles (following Woodhouse's 'Geometric Quantization').
Let $V\rightarrow M$ be a vector bundle (in particular, a complex line bundle). Furthermore, let the space of smooth sections over $M$ be denoted by by $C^\infty_V(M)$.
A connection on V is an operator $\nabla$ which assigns to any $s\in C^\infty_V(M)$ a map $\nabla s:TM\rightarrow V$. $\nabla$ distributes over addition and satisfies $\nabla(fs)=(df)s+f\nabla s$ for any function $f:V\rightarrow V$.
I understand that this is something like the version of one-forms relevant in the context of vector bundles. OK. So far, so good. The following definition is where I'm confused:
The potential 1-form $\Theta\in\Omega^1(U)$ ($U\subset M$ is a trivialization) is implicitly defined (for a given connection and trivialization) by $\nabla s=-\textrm{i}\Theta s$. The curvature is subsequently defined as the two-form $d\Theta$.
First of all, I'm unclear on the notation. Is this supposed to be the product of the two functions, subsequently multiplied with iota? If so, the left-hand side is a map $TU\rightarrow V$, and the right-hand side is a map $U\times TU\rightarrow \mathbb{C}\times V$. How can they match up?
Edit: The section $s$ is, in particular, the unit section $\tau(\cdot, 1)$, where $\tau$ is the mapping associated with the trivialization.
The equation $\nabla s=-i\Theta s$ means for each vector field $X$ on $U$, and each $p\in U$, \begin{align} (\nabla_Xs)(p)&=-i\cdot [\Theta(X)](p)\cdot s(p). \end{align} Notice what each object is. By plugging in $X$ into $\nabla s$, we now get a section $\nabla_Xs$ of the vector bundle $V$ over the open set $U$. Evaluating this on $p$ gives us an element $(\nabla_Xs)(p)\in V_p$. On the RHS, $-i$ is just a complex number. You can evaluate $\Theta$ on $X$ to get a function $\Theta(X):U\to\Bbb{R}$ (or maybe $\Bbb{C}$, idk how you want to proceed), so evaluating on $p$ gives a number $\Theta(X)(p)$. Lastly, $s(p)\in V_p$, belongs to this particular fiber, which is a vector space so it can be multiplied by the other two numbers.
Or what amounts to the same thing, we can phrase everything pointwise: for each $p\in U$ and tangent vector $X_p\in T_pM$, \begin{align} \nabla_{X_p}s&=-i\Theta_p(X_p)\cdot s(p), \end{align} where again this is an equality in the vector-bundle fiber $V_p$.
Btw, it should be emphasized that the $s$ in this equation is NOT arbitrary; this tripped me up initially. The $s$ here is the “unit section” determined by the trivialization (i.e what we’d usually write as $e_1$ if there were more dimensions).