Every vector bundle admits a connection

Let $M$ be a smooth manifold and let $E\to M$ be a smooth vector bundle.

Corollary (existence of vector bundle connections)

There exists at least one connection on the vector bundle $E\to M$, i.e. a covariant derivative $\nabla:\Gamma(E)\to \Omega^1(M;E)$ satisfying the Leibniz rule.

One proof strategy is to pass to the frame bundle $\mathrm{Fr}(E)\to M$, use the existence of connections on principal bundles (see existence of principal connections ), and then induce a connection on $E$ from a principal connection on $\mathrm{Fr}(E)$.

Examples

1. Trivial bundle. On $E=M\times \mathbb R^k$, the operator $d$ (componentwise differentiation in a trivialization) is a connection.
2. Tangent bundle. On a Riemannian manifold, the Levi–Civita connection gives a connection on $TM$ and hence on all tensor bundles built from $TM$ and $T^*M$.
3. Complex line bundles. A Hermitian metric on a complex line bundle admits a compatible unitary connection; for holomorphic line bundles this includes the Chern connection.