Bundle of connections

Let $\pi:P\to M$ be a principal $G$ -bundle. The bundle of connections of $P$ is the affine bundle

\mathcal{C}(P) \to M

defined as the quotient

\mathcal{C}(P) := J^1P / G,

where $J^1P$ is the 1-jet bundle of $P$ and $G$ acts by prolongation of the principal right action. Its smooth sections are naturally in bijection with principal connections on $P$ .

Affine structure

The bundle $\mathcal{C}(P)\to M$ is modeled on the vector bundle

T^*M \otimes \mathrm{ad}(P),

so the difference of two connections is an $\mathrm{ad}(P)$ -valued $1$ -form. Here $T^*M$ is the cotangent bundle and $\mathrm{ad}(P)$ is the adjoint Lie-algebra 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

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 $T^*M\otimes (M\times \mathfrak{g})$ .
Line bundles. For a principal $U(1)$ -bundle, the difference of two connections is an ordinary real 1-form on $M$ , reflecting that $\mathrm{ad}(P)\cong M\times i\mathbb R$ is a trivial real line bundle.
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.