Flat connection on a trivial bundle

Let $P=M\times G\to M$ be the trivial principal bundle over a smooth manifold $M$.

Definition (product connection / zero gauge potential)

Define a horizontal subspace at $(x,g)\in M\times G$ by

This $G$-invariant splitting defines a principal connection on $P$, often called the product connection.

In the global trivialization given by the canonical section $s(x)=(x,e)$, the associated gauge potential is the $\mathfrak g$-valued 1-form $A\in\Omega^1(M;\mathfrak g)$ given by $A=0$.

Properties

• The curvature of this connection vanishes identically.
• Parallel transport along a curve $\gamma$ keeps the $G$-coordinate constant in the product chart.
• The holonomy group is trivial (contained in the identity element), for each basepoint.

The induced connection on any associated vector bundle $M\times V$ is the “ordinary derivative” connection: in the standard frame, $\nabla=d$.

Examples

1. Flat connections on trivial line bundles.
   On $M\times U(1)$, this is the standard flat circle-bundle connection; parallel transport is literally constant phase in the chosen trivialization.

2. Associated vector bundle: constant sections are parallel.
   For $E=M\times\mathbb R^n$, the induced connection satisfies $\nabla s=ds$; in particular, constant maps $s:M\to\mathbb R^n$ are parallel sections.

3. Restriction to open subsets.
   If $U\subset M$ is open, restricting the trivial bundle and this connection to $U$ gives the same product connection on $U\times G\to U$.