Gauge transformation of a manifold
A gauge transformation of a manifold is the gauge transformation of the tangent bundle of that manifold. In other words, it is a differentiable automorphism of the tangent bundle that sends the fiber over every point of the manifold to itself, and is a linear automorphism of each such fiber.
The term gauge transformation in general stands for the gauge transformation of a vector bundle, which is a more general concept.
The gauge group of a manifold is its group of gauge transformations.