2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/65879A coordinate-free proof of the Maximum Principle is provided in the specific case of an optimal control problem with fixed time. Our treatment heavily relies on a special notion of variation of curves that consist of a concatenation of integral curves of time-dependent vector fields with unit time component, and on the use of a concept of lift over a bundle map. We further derive necessary and sufficient conditions for the existence of so-called abnormal and strictly abnormal extremals.38p, accepted for publication in Acta Appl. MathDifferential GeometryOptimization and Control49Kxx; 53Cxxtext