In this paper we first introduce metallic maps between metallic Riemannian manifolds where with some additions conditions these maps will be harmonic.

Then we study the constancy of certain maps from metallic Riemannian manifolds to various manifolds by addition condition and geometric structure. Then we consider the reverse case and show that all such maps are constant.

Also, we will investigate the problem of integrability for metallic Riemannian structures by using $phi$-operator. In the end, we find some properties of curvature for metallic Riemannian metric.