Abstract. This paper establishes and extends a conjecture posed by M. Gromov which states that every riemannian metric on that strictly dominates the standard metric must have somewhere scalar curvature strictly less than that of . More generally, if is any compact spin manifold of dimension which admits a distance decreasing map of non-zero degree, then either there is a point with normalized scalar curvature , or is isometric to . The distance decreasing hypothesis can be replaced by the weaker assumption is contracting on -forms. In both cases, the results are sharp. An explicit counterexample is given to show that the result is no longer valid if one replaces 2-forms by -forms with .