  Let $(M,g^{TM})$ be an odd dimensional ($\dim M\geq 3$) connected oriented
noncompact complete spin Riemannian manifold. Let $k^{TM}$ be the associated
scalar curvature. Let $f:M\to S^{\dim M}(1)$ be a smooth area decreasing map
which is locally constant near infinity and of nonzero degree. Suppose
$k^{TM}\geq ({\dim M})({\dim M}-1)$ on the support of ${\rm d}f$, we show that
$\inf(k^{TM})<0$. This answers a question of Gromov.
