Flores Castillo, Nicolas VirgilioWe review Synge's trick and present some of its applications to Riemannian Geometry. We use it to prove Frankel, Weinstein-Synge and Wilking's theorems, which concern manifolds of positive sectional curvature. First, we give the classical proofs of these theorems and then we present a reformulation of Synge's trick in terms of a lower bound for the index of a special kind of geodesics.en-USThesisMathematics