  This is an introductory chapter in a series in which we take a systematic
study of the Yang-Mills equations on curved space-times. In this first, we
provide standard material that consists in writing the proof of the global
existence of Yang-Mills fields on arbitrary curved space-times using the
Klainerman-Rodnianski parametrix combined with suitable Gr\"onwall type
inequalities. While the Chru\'sciel-Shatah argument requires a simultaneous
control of the $L^{\infty}_{loc}$ and the $H^{2}_{loc}$ norms of the Yang-Mills
curvature, we can get away by controlling only the $H^{1}_{loc}$ norm instead,
and write a new gauge independent proof on arbitrary, fixed, sufficiently
smooth, globally hyperbolic, curved 4-dimensional Lorentzian manifolds. This
manuscript is written in an expository way in order to provide notes to
Master's level students willing to learn mathematical General Relativity.
