nLab
connection on a vector bundle

Idea

One way to define a connection on a vector bundle $E\to X$ is as a principal connection of the $\mathrm{GL}(n)$-principal bundle that the vector bundle is an associated bundle to. Another is as a covariant derivative on the space of sections of the vector bundle.

Examples

• Bott connection

(…)

References

An exposition with an eye towards applications in gauge theory is in section 16.3 of

• Theodore Frankel, The Geometry of Physics - An Introduction