  We extend the deep and important results of Lichnerowicz, Connes, and
Gromov-Lawson which relate geometry and characteristic numbers to the existence
and non-existence of metrics of positive scalar curvature (PSC). In particular,
we show: that a spin foliation with Hausdorff homotopy groupoid of an
enlargeable manifold admits no PSC metric; that any metric of PSC on such a
foliation is bounded by a multiple of the reciprocal of the foliation K-area of
the ambient manifold; and that Connes' vanishing theorem for characteristic
numbers of PSC foliations extends to a vanishing theorem for Haefliger
cohomology classes.
