Newtonian mechanics in a Riemannian manifold

The work done by Isaac Newton more than three hundred years ago, continues being a path to increase our knowledge of Nature. To better understand all the ideas behind it, one of the finest ways is to generalize them to wider situations. In this report we make a review of one of these enlargements, the one that bears the mechanical systems from the elementary homogeneous three dimensional Euclidean space to the more abstract geometry of a Riemannian manifold.