  We consider minimal maps $f:M\to N$ between Riemannian manifolds
$(M,\mathrm{g}_M)$ and $(N,\mathrm{g}_N)$, where $M$ is compact and where the
sectional curvatures satisfy $\sec_N\le \sigma\le \sec_M$ for some $\sigma>0$.
Under certain assumptions on the differential of the map and the second
fundamental form of the graph $\Gamma(f)$ of $f$, we show that $f$ is either
the constant map or a totally geodesic isometric immersion.
