We consider minimal maps f: M→ N between Riemannian manifolds (M, g _M) and (N, g _N) , where M is compact and where the sectional curvatures satisfy sec _N≤ σ≤ sec _M for some σ> 0. Under certain assumptions on the differential of the map and the second fundamental form of the graph Γ(f) of f, we show that f is either the constant map or a totally geodesic isometric immersion.