The purpose of this article is to describe briefly some investigations of the properties of certain integrals associated with the Riemannian manifold of an algebraic variety, which, besides having an interest of their own, appear to be a fruitful source of new properties of the more classical Abelian integrals usually considered in connection with algebraic varieties. The investigations are still at a very early stage, and up to the present only deal with the Abelian integrals of the first kind (those which are without singularities), and in what follows we shall limit our attention to these.