Connections on vector bundles via frame bundles

Let $E\to M$ be a smooth real rank-$n$ vector bundle over a smooth manifold $M$. Denote by $\mathrm{Fr}(E)\to M$ the principal $\mathrm{GL}(n,\mathbb R)$-bundle of frames of $E$ (ordered bases of each fiber).

A connection on a vector bundle can be described either by a covariant derivative on sections of $E$ or by a principal connection on $\mathrm{Fr}(E)$.

Theorem (TFAE: vector bundle connections and frame bundle connections)

The following data are equivalent, naturally and bijectively:

1. A vector bundle connection $\nabla$ on $E$ (a covariant derivative satisfying the Leibniz rule).

2. A principal connection on the principal bundle $\mathrm{Fr}(E)\to M$.

3. A $\mathrm{GL}(n,\mathbb R)$-equivariant horizontal distribution $H\subset T\,\mathrm{Fr}(E)$, i.e. a smooth subbundle such that

   $$T_p\mathrm{Fr}(E)=H_p\oplus \ker(d\pi_p)\quad\text{and}\quad (dR_A)(H_p)=H_{p\cdot A}$$

   for all $A\in \mathrm{GL}(n,\mathbb R)$, where $R_A$ is the right action on frames.

4. A connection 1-form $\omega\in\Omega^1(\mathrm{Fr}(E);\mathfrak{gl}(n,\mathbb R))$ satisfying the standard axioms (reproduces fundamental vertical fields and is equivariant under the right action).

Moreover, under this correspondence:

• Given a principal connection on $\mathrm{Fr}(E)$, one obtains $\nabla$ by declaring that a section of $E$ is parallel if and only if its equivariant function on $\mathrm{Fr}(E)$ is constant along horizontal lifts (equivalently, parallel transport in $\mathrm{Fr}(E)$ induces parallel transport in $E$).

• Given $\nabla$, one obtains a principal connection on $\mathrm{Fr}(E)$ by transporting frames along curves using the induced parallel transport and taking the resulting horizontal lifts of tangent vectors.

A useful special case is $E=TM$: connections on the tangent bundle correspond to principal connections on the frame bundle of $M$.

Examples

1. Trivial bundle and matrix-valued 1-forms.
   For the trivial vector bundle $E=M\times\mathbb R^n$, a connection is determined by a matrix of 1-forms $A\in\Omega^1(M;\mathfrak{gl}(n,\mathbb R))$ via

   $$\nabla = d + A$$

   in the standard frame. The corresponding principal connection on $M\times \mathrm{GL}(n,\mathbb R)$ has connection form obtained from $A$ in that trivialization.

2. Riemannian geometry.
   On a Riemannian manifold, the Levi–Civita connection on $TM$ corresponds to a principal connection on the orthonormal frame bundle (a reduction from $\mathrm{GL}(n)$ to $\mathrm{O}(n)$).

3. Line bundles.
   For a real line bundle $L\to M$, the frame bundle is a principal $\mathrm{GL}(1,\mathbb R)\cong \mathbb R^\times$-bundle. A connection on $L$ is equivalent to a principal connection on that frame bundle; in a local trivialization, it is described by an ordinary 1-form.