Gauge transformation of a vector bundle

Definition

Given data

• A differential manifold $M$
• A vector bundle $E$ over $M$

Definition part

A gauge transformation is a differentiable automorphism of $E$ as a vector bundle over $M$. In other words, it is a map from $E$ to $E$ that:

• Is a diffeomorphism of $E$ as a manifold
• Preserves the vector space over any point in $M$ and restricts to a linear automorphism in each such subspace.

The collection of all gauge transformations of a vector bundle is termed the gauge group of the bundle.