The theory of geodesics on a surface is a subject standing at the intersection of various fields : differential geometry, the calculus of variations, the theory of differential equations, and mechanics. At the outset of the 19th century, the local theory of geodesics was well known as an instance of the application of the methods of the infinitesimal calculus to geometry. The equations for geodesics, however, were generally unamenable to easy resolution; and direct methods of treatment yielded but little of interest, as regards the global behaviour of geodesics on a given surface.
With the work of Jacobi, Sturm, and Liouville, the emergence of a qualitative approach in the study of the resolution of differential equations allowed for a global, and qualitative, treatment of geodesics on a surface — most conspicuously through the study of the conjugate locus, and of the cut locus, and by the examination of the incidence of such global topological or geometrical conditions as the sign, positive or negative, of the curvature.