Bundle of connections

Let $\pi\colon P\to M$ be a principal G-bundle . The set of principal connections on $P$ is an affine space; the “bundle of connections” packages this affineness pointwise over $M$.

Definition (Connection bundle; principal case)

The bundle of connections of $P$ is the affine bundle

defined as the quotient

where $J^1P$ is the 1-jet bundle of $P$ and $G$ acts by prolongation of the principal right action.

It has the property that:

• Sections of $\mathcal{C}(P)\to M$ are in natural bijection with principal connections on $P$.

Moreover, $\mathcal{C}(P)\to M$ is an affine bundle modeled on the vector bundle

so the difference of two connections is an $\mathrm{Ad}(P)$-valued 1-form (a section of $T^*M\otimes \mathrm{Ad}(P)$). Here $T^*M$ is the cotangent bundle and $\mathrm{Ad}(P)$ is the adjoint bundle associated to $P$.

Vector bundle variant

If $E\to M$ is a vector bundle, the set of connections on E is an affine space modeled on $\Omega^1(M;\mathrm{End}(E))$, and there is an analogous affine bundle over $M$ whose sections correspond to connections on $E$.

Examples

1. Trivial principal bundle. If $P=M\times G$, then choosing the product trivialization identifies connections with $\mathfrak{g}$-valued 1-forms on $M$, so $\mathcal{C}(P)$ is (noncanonically) isomorphic to an affine bundle modeled on $T^*M\otimes (M\times \mathfrak{g})$.
2. Line bundles. For a principal $U(1)$-bundle, the difference of two connections is an ordinary real 1-form on $M$, reflecting that $\mathrm{Ad}(P)$ is a trivial real line bundle.
3. Levi–Civita as a section. The Levi–Civita connection determines a distinguished section of the connection bundle of the orthonormal frame bundle of a Riemannian manifold.