Construction: Connection on Fr(E) induced by a vector bundle connection (and conversely)

Let $\pi:E\to M$ be a rank-$n$ smooth vector bundle over a smooth manifold $M$, and let $P=\mathrm{Fr}(E)$ be its frame bundle (a principal $\mathrm{GL}(n,\mathbb R)$-bundle).

From a vector bundle connection to a principal connection

Let $\nabla$ be a connection on the vector bundle $E$. Define horizontality in $P$ by parallel transport of frames:

For a point $u\in P_x$ (a frame $u:\mathbb R^n\to E_x$) and a tangent vector $v_x\in T_xM$, choose a smooth curve $\gamma:(-\varepsilon,\varepsilon)\to M$ with $\gamma(0)=x$ and $\dot\gamma(0)=v_x$. Let $\tau_t:E_x\to E_{\gamma(t)}$ denote parallel transport in $E$ induced by $\nabla$ along $\gamma$. Then define a curve of frames

The horizontal lift of $v_x$ at $u$ is $\dot u(0)\in T_uP$, and the span of all such vectors defines a horizontal subspace $H_u\subset T_uP$.

The assignment $u\mapsto H_u$ is $\mathrm{GL}(n,\mathbb R)$-equivariant and complementary to the vertical subspace, hence defines a principal connection on $P$.

From a principal connection to a vector bundle connection

Conversely, suppose $P=\mathrm{Fr}(E)$ carries a principal connection. The associated bundle

is canonically isomorphic to $E$. A principal connection on $P$ induces a covariant derivative on every associated vector bundle, hence in particular a vector bundle connection $\nabla$ on $E$.

These two constructions are inverse to each other: connections on $E$ are in bijection with principal connections on $\mathrm{Fr}(E)$.

Examples

1. Levi-Civita connection. For $E=TM$, a Riemannian metric gives a unique torsion-free metric connection on $TM$, and the induced principal connection on $\mathrm{Fr}(TM)$ can be described by horizontals consisting of parallel transported frames.

2. Trivial bundle and flat connection. If $E\cong M\times\mathbb R^n$ with the standard flat covariant derivative, then the induced principal connection on $\mathrm{Fr}(E)\cong M\times \mathrm{GL}(n,\mathbb R)$ has horizontal subspaces equal to the tangent directions along $M$.

3. Parallel transport viewpoint. In both directions, the induced notion of parallel transport agrees: transporting a frame in $P$ horizontally along a curve is the same as transporting each vector in the frame by the covariant derivative on $E$.