In this paper, we first introduce two new classes of sets and functions called geodesic $E$-invex sets and geodesic $E$-preinvex functions on a Riemannian manifold, respectively. Moreover, we present the definition and properties of geodesic $E$-quasi-preinvex functions on Riemannian manifolds. Finally, we investigate the properties and characterizations of these two classes of sets and functions.